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+CLASS-INCREMENTAL LEARNING WITH REPETITION
+Hamed Hemati1, Andrea Cossu2, Antonio Carta3, Julio Hurtado3, Lorenzo Pellegrini4
+Davide Bacciu3, Vincenzo Lomonaco3, Damian Borth1
+[email protected], [email protected], [email protected],
+[email protected], [email protected], [email protected],
+[email protected], [email protected]
+1University of St. Gallen, Switzerland
+2 Scuola Normale Superiore, Italy
+3University of Pisa, Italy
+4University of Bologna, Italy
+ABSTRACT
+Real-world data streams naturally include the repetition of previous concepts.
+From a Continual Learning (CL) perspective, repetition is a property of the en-
+vironment and, unlike replay, cannot be controlled by the user. Nowadays, Class-
+Incremental scenarios represent the leading test-bed for assessing and comparing
+CL strategies. This family of scenarios is very easy to use, but it never allows
+revisiting previously seen classes, thus completely disregarding the role of repe-
+tition. We focus on the family of Class-Incremental with Repetition (CIR) sce-
+narios, where repetition is embedded in the definition of the stream. We propose
+two stochastic scenario generators that produce a wide range of CIR scenarios
+starting from a single dataset and a few control parameters. We conduct the first
+comprehensive evaluation of repetition in CL by studying the behavior of existing
+CL strategies under different CIR scenarios. We then present a novel replay strat-
+egy that exploits repetition and counteracts the natural imbalance present in the
+stream. On both CIFAR100 and TinyImageNet, our strategy outperforms other
+replay approaches, which are not designed for environments with repetition.
+1
+INTRODUCTION
+Continual Learning (CL) requires a model to learn new information from a stream of experiences
+presented over time, without forgetting previous knowledge (Parisi et al., 2019; Lesort et al., 2020).
+The nature and characteristics of the data stream can vary a lot depending on the real-world en-
+vironment and target application. Class-Incremental (CI) scenarios (Rebuffi et al., 2017) are the
+most popular ones in CL. CI requires the model to solve a classification problem where new classes
+appear over time. Importantly, when a set of new classes appears, the previous ones are never
+seen again. However, the model still needs to correctly predict them at test time. Conversely, in a
+Domain-Incremental (DI) scenario (van de Ven & Tolias, 2019) the model sees all the classes at the
+beginning and continue to observe new instances of the classes over time.
+The CI and DI scenarios have been very helpful to promote and drive CL research in the last few
+years. However, they strongly constrain the properties of the data stream in a way that it sometimes
+considered unrealistic or very limiting (Cossu et al., 2021). Recently, the idea of Class-Incremental
+with Repetition (CIR) scenarios has started to gather some attention in CL (Cossu et al., 2021).
+CIR scenarios are arguably more flexible in the definition of the stream, since they allow both the
+introduction of new classes and the repetition of previously seen classes. Crucially, repetition is a
+property of the environment and cannot be controlled by the CL agent. This is very different from
+Replay strategies (Hayes et al., 2021), where the repetition of previous concepts is heavily structured
+and can be tuned at will.
+CIR defines a family of CL scenarios which ranges from CI (new classes only, without repetition) to
+DI (full repetition of all seen classes). Although appealing, currently there exists neither a quantita-
+1
+arXiv:2301.11396v1  [cs.LG]  26 Jan 2023
+
+tive analysis nor an empirical evaluation of CL strategies learning in CIR scenarios. Mainly, because
+it is not obvious how to build a stream with repetition, given the large amount of variables involved.
+How to manage repetition over time? How to decide what to repeat? What data should we use?
+In this paper, we provide two generators for CIR that, starting from a single dataset, allow to build
+customized streams by only setting few parameters. The generators are as easy to use as CI or DI
+ones.
+We leveraged our generators to run an extensive empirical evaluation of the behavior of CL strategies
+in CIR scenarios. We found out that knowledge accumulation happens naturally in streams with
+repetition. Even a naive fine-tuning, subjected to complete forgetting in CI scenarios, is able to
+accumulate knowledge for classes that are not always present in an experience. We observed that
+Replay strategies still provide an advantage in terms of final accuracy, even though they are not
+crucial to avoid catastrophic forgetting. On one side, distillation-based strategies like LwF (Li &
+Hoiem, 2018) are competitive in streams with a moderate amount of repetition. On the other side,
+existing Replay strategies are not specifically designed for CIR streams. We propose a novel Replay
+approach, called Frequency-Aware Replay (ER-FA) designed for streams with unbalanced repetition
+(few classes appear rarely, the other very frequently). ER-FA surpasses by a large margin other
+Replay variants when looking at infrequent classes and it does not lose performance in terms of
+frequent classes. This leads to a moderate gain in the final accuracy, with a much better robustness
+and a reduced variance across all classes. Our main contributions are:
+1. The design of two CIR generators, able to create streams with repetition by only setting few
+control parameters. We built both generators with Avalanche (Lomonaco et al., 2021) and
+we will make them publicly available to foster future research. The generators are general
+enough to fit any classification dataset and are fully integrated with Avalanche pipeline to
+run CL experiments.
+2. We perform an extensive evaluation of the properties of CIR streams and the performance
+of CL strategies. We study knowledge accumulation and we showed that Replay, although
+still effective, is not crucial for the mitigation of catastrophic forgetting. Some approaches
+(e.g., LwF) look more promising than others in CIR scenarios. We consolidate our results
+with an analysis of the CL models over time through Centered Kernel Alignment (CKA)
+(Kornblith et al., 2019) and weights analysis.
+3. We propose a novel Replay variant, ER-FA, which is designed based on the properties of
+CIR scenarios. ER-FA surpasses other Replay strategies in unbalanced streams and provide
+a more robust performance on infrequent classes without losing accuracy on the frequent
+ones.
+2
+CLASS-INCREMENTAL LEARNING WITH REPETITION GENERATORS
+𝑒!
+CI
+DI
+Concepts:
+𝑒"
+𝑒#
+𝑒!
+𝑒"
+𝑒#
+𝑒$
+…
+𝑒!
+𝑒"
+𝑒#
+𝑒$
+…
+CIR
+New Inst.
+New/Old 
+Inst.
+Figure 1: Illustration of scenario types
+that can be generated with episodic par-
+tial access to a finite set of concepts.
+The shape colors indicate whether in-
+stances are new in each episode or can
+be a mixture of old and new instances.
+CL requires a model to learn from a stream of N expe-
+riences S = {e1, e2, ..., eN}, where each experience ei
+brings a dataset of examples Dei = {Xi, Yi} . Many
+CL scenarios, like CI or DI, are generated from a fixed
+dataset D = {(x, y); x ∈ X, y ∈ Y }, where x is the in-
+put example, y is the target and Y = {1, · · · , C} is the la-
+bel space (closed-world assumption). Depending on how
+classes from the entire dataset D are shown or revisited
+in the stream, this configuration can lead to CI, CIR or DI
+scenarios (Figure 1). In Table 1, we formally present and
+compare the properties of the three scenario types.
+In CIR, streams with repetition are characterized by mul-
+tiple occurrences of the same class over time. To study
+this scenario, we propose two stream generators designed
+to create a stream from a finite dataset: the Slot-Based
+Generator (Gslot) and the Sampling-Based Generator
+(Gsamp). Gslot generate streams by enforcing constraints
+on the number of occurrences of classes in the stream us-
+ing only two parameters. Gslot does not repeat already observed samples, therefore the stream length
+is limited by the number of classes. However, it guarantees that all samples in the dataset will be
+2
+
+Property
+CI
+DI
+CIR
+Instance Repetition *
+Xei ∩ Xj = Ø
+Xi ∩ Xj = Ø
+|Xi ∩ Xj| ≥ 0
+Domain Coverage
+�i=N
+i=1 Xi = X
+�i=N
+i=1 Xi = X
+�i=N
+i=1 Xi ∈ P(X) \ Ø
+Concept Repetition *
+Yi ∩ Yj = Ø
+Y1 = . . . = YN = Y
+|Yi ∩ Yj| ≥ 0
+Codomain Coverage
+�i=N
+i=1 Yi = Y
+�i=N
+i=1 Yi = Y
+�i=N
+i=1 Yi ∈ P(Y ) \ Ø
+Table 1: Comparison of scenario properties in CI, DI and CIR. P(A) and |A| represent the power
+set and the cardinality of set A. *: ∀1 ≤ i, j ≤ N , i ̸= j.
+observed exactly once during the lifetime of the model. Instead, Gsamp generates streams according
+to several parametric distributions that control the stream properties. It can generate arbitrarily long
+streams in which old instances can also re-appear with some probability.
+2.1
+SLOT-BASED GENERATOR
+The Slot-Based Generator Gslot allows to carefully control the class repetitions in the generated
+stream with a single parameter K. Gslot takes as input a dataset D, the total number of experiences
+N and the number of slots per experience K. It returns a CIR stream composed by N experiences,
+where each of the K slots in each experience is filled with samples coming from a single class.
+1
+10
+20
+...
+C
+Number of Experiences (N)
+1
+10
+20
+C
+Number of Slots (K)
+CIR
+Class-Incremental
+Domain Incremental
+Figure 2: Illustration of how various
+scenarios can be generated by Gslot, by
+changing K and N. The red area under
+the blue curve represents invalid scenar-
+ios.
+Gslot constrains the slot-class association such that all the
+samples present in the dataset are seen exactly once in the
+stream. Therefore, Gslot considers repetition at the level
+of concepts. To implement this logic, Gslot first partitions
+all the samples in the dataset into the target number of
+slots. Then, it randomly assigns without replacement K
+slots per experience. At the end, the N mod K blocks
+remaining are assigned to the first experience, such that
+the rest of the stream is not affected by a variable number
+of slots.
+The Slot-Based Generator is useful to study the transition
+from CI scenarios to DI scenarios, obtained by simply
+changing the parameter K (Figure 2). For example, let
+us consider a dataset with 10 classes such as MNIST. By
+choosing N = 5 and K = 2 we obtain the popular Split-
+MNIST, a CI scenario with no repetition and 2 classes for
+each experience. Conversely, by setting N = 5 and K =
+10 we obtain a DI stream where all the 10 classes appear
+in each experience with new unseen samples. More in
+general, given a dataset with C classes, we obtain a CI scenario by setting K = C
+N (N must divide
+C). We obtain a DI scenario by setting K = C. In Appendix B we illustrate the overall steps of
+stream generation (Figure 12), and provide a step-by-step formal definition of Gslot (Algorithm 2).
+2.2
+SAMPLING-BASED GENERATOR
+The Sampling-Based Generator (Gsamp) generates arbitrarily long streams and controls the repe-
+titions via probability distributions. The stream generator allows to control the first occurrence of
+new classes and the amount of repetitions of old classes. Unlike Gslot, it allows to generate infinite
+and even unbalanced streams.
+Gsamp parameters:
+• N: Stream length, i.e. number of experiences in the stream.
+• S: Experience size which defines the number of samples in each experience.
+• Pf(S): Probability distribution over the stream S used for sampling the experience ID of
+the first occurrence in each class.
+• Pr: List of repetition probabilities for dataset classes.
+3
+
+Scenario 
+Matrix 
+Generator
+𝑁
+𝐾
+CL Scenario
+ℙ!(𝒮)
+𝑃"
+Scenario Matrix
+𝐺!"#$
+Parameters
+Sampler
+…
+𝑒#
+𝑒! 𝑒" 𝑒# 𝑒$ 𝑒%
+𝑐!
+𝑐"
+𝑐#
+𝑐$
+𝑐%
+𝑐&
+𝑐'
+𝑐(
+𝑐)
+…
+…
+…
+…
+…
+…
+…
+…
+…
+…
+𝒟
+𝒮$"%&'
+𝒮$()$
+𝑒*
+…
+𝑒#
+𝑒*
+Instances
+Concepts
+Figure 3: Schematic view of Gsamp generator. Each concept is shown with a different color.
+Note that Pf is a probability mass function over the stream S which means it sums up to 1.0 and
+determines in which parts of the stream it is more likely to observe new classes for the first time.
+However, the list of probabilities {p1, p2, ..., pC} in Pr are independent and each probability value
+0.0 ≤ pi ≤ 1.0 indicates how likely it is for each class to repeat after its first occurrence.
+For each experience, Gsamp samples instances from the original dataset D according to a two step
+process. First, Gsamp defines a C × N binary matrix T called Occurrence Matrix that determines
+which classes can appear in each experience. Then, for each experience ei, 1 ≤ i ≤ N we use the
+i-th column of T to sample data points for that experience. The generator uses the inputs N, Pf(S)
+and Pr to generate T. Therefore, it first initializes T as a zero C × N matrix. Then for each class
+c in the dataset, it uses Pf(S) to sample the index of the experience in which class c appears for
+the first time. Different probability distributions can be used to either spread the first occurrence
+along the stream or concentrate them at the beginning, which allows a good flexibility in the stream
+design. After sampling the first occurrences, the classes are then repeated based on Pr probability
+values to finalize matrix T. In the simplest case, Pr can be fixed to the same value for all classes to
+generate a balanced stream.
+Once the matrix T is constructed, a random sampler is used to sample patterns for each experience.
+Since each experience may contain an arbitrary number of classes, another control parameter that
+could be added here is the fraction of samples per class in experience size S. For simplicity we keep
+the fractions equally fixed and thus the number of datapoints sampled from each class in experience
+ei is ⌊ S
+|Ci|⌋ where |Ci| indicates the number of classes present in ei. Since the sampler is stochastic,
+each time we sample from a class, both new and old patterns can appear for that class. Given a large
+enough stream length N, the final stream will cover the whole dataset with a measurable amount of
+average repetition. In Figure 3 we demonstrate the schematic of the generator Gsamp. We provide
+the pseudo-code for Gsamp in Appendix C (Algorithm 2).
+Although we assume a fixed number of instances per class in D, Gsamp can be easily extended
+to settings where the number of instances can grow over time. Moreover, the sampler can also be
+designed arbitrarily depending on the stochasticity type, e.g., irregular or cyclic.
+3
+FREQUENCY-AWARE REPLAY
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+Figure 4: Ratio of buffer slots for infre-
+quent classes for three random seeds.
+Experience Replay (ER) is the most popular CL strategy
+due to its simplicity of use and high performance in class-
+incremental scenarios. The storage policy, which deter-
+mines which samples to keep in a limited buffer, is the
+major component of ER methods. Class-Balanced (CB)
+and Reservoir Sampling (RS) Vitter (1985) are the most
+popular storage policies in ER methods. CB keeps a fixed
+quota for each class, while RS samples randomly from the
+stream, which leads to the class frequency in the buffer
+being equal to the frequency in the stream. CB and RS
+are great choices for balanced streams such as class incremental scenarios, where the number of
+samples per class is the same over the whole stream. However, as in most real-world scenarios, CIR
+scenarios are naturally unbalanced, and different classes may have completely different repetition
+frequencies. Accordingly, CB and RS storage policies may suffer a big accuracy drop in the infre-
+quent classes of an unbalanced stream. For example, in highly unbalanced streams, RS will store
+an unbalanced buffer replicating the the original distribution of the stream, which is sub-optimal be-
+4
+
+cause the less frequent classes will require more repetition to prevent forgetting, while the frequent
+classes will be repeated naturally via the stream occurrences.
+We propose Frequency-Aware (FA) storage policy that addresses the imbalance issue in CIR streams
+by online adjustment of the buffer slots in accordance with the amount of repetition for each class.
+Given a buffer B with a maximum size of M, a list of previously observed classes P initialized as
+P = {} with a corresponding list O indicating the number of observations per class c in C, and a
+dataset Di from experience ei, the algorithm first checks the present classes Pi in Di and adds them
+to P (P ← P ∪ Pi). Then, for each class c in Pi it increments the number of observations O[c] by
+1 if the class was previously seen, otherwise it initializes O[c] = 1. After updating the number of
+observations, FA computes the buffer quota Q for all observed classes by inverting the number of
+observations (Q = [
+1
+O[c]∀c ∈ C]) and normalizes it . This way, the algorithm offers the less frequent
+classes a larger quota. Finally, a procedure ensures the buffer is used to its maximum capacity by
+filling unused slots with samples from more frequent classes sorted by their observation times. This
+is a crucial step since it is possible that an infrequent class which is not present ei will be assigned
+with a larger quota than its current number of samples in B, and therefore the buffer will remain
+partially empty. In Figure 4 we show how our method assigns higher ratio of samples for infrequent
+classes to overcome the imbalance issue in the stream. For further analysis and pseudo-code of FA
+policy refer to Appendix E. We present examples of unbalanced scenarios in Appendix D.
+4
+EMPIRICAL EVALUATION
+We study CIR scenarios by leveraging our generators Gsamp and Gslot. First, by using Gslot we
+provide quantitative results about forgetting in CL strategies when transitioning from CI to DI sce-
+narios (Sec. 4.1). Then, by using Gsamp we focus on long streams with 500 experiences and
+study the performance of Replay and Naive (Sec. 4.2). The long streams give us the opportunity
+to study knowledge accumulation over time in the presence of repetition. We also provide an intu-
+itive interpretation of the model dynamics over long streams (Sec. 4.3). Finally, we show that our
+Frequency-Aware Replay is able to exploit the repetitions present in the stream and to surpass the
+performance of other replay approaches not specifically designed for CIR scenarios (Sec. 4.4).
+The experiments were conducted using the CIFAR-100 Krizhevsky et al. (2009) and Tiny-ImageNet
+LeCun et al. (1998) datasets with the ResNet-18 model. For Gslot, we run experiments for Naive (in-
+cremental fine tuning), LwF Li & Hoiem (2018), EWC Kirkpatrick et al. (2017), Experience Replay
+with reservoir sampling Kim et al. (2020) (ER-RS) and AGEM Chaudhry et al. (2018) strategies.
+For Gsamp we run experiments for Naive and ER (CB/RS/FA) strategies. We set the default buffer
+size for CIFAR-100 to 2000, and for Tiny-ImageNet to 4000 in the replay strategies. We evaluate
+all strategies on the Average Test Accuracy (TA).
+4.1
+TRANSITION FROM CLASS-INCREMENTAL TO DOMAIN INCREMENTAL
+DI and CI scenarios are heavily studied in the CL literature. However, little is known about what
+happens to the performance of popular CL strategies when gradually transitioning from one scenario
+to the other. By changing the value of K in Gslot, we provide a quantitative analysis of such
+behavior in CIR scenarios. Figure 5 shows the Average Test Accuracy over all classes for different
+CL strategies when transitioning from CI (left-most point of each plot) to DI (right-most point of
+each plot).
+Replay is one of the most effective strategies in CI scenarios. As expected, in CIR scenarios the
+advantage provided by ER-RS with respect to other CL strategies diminishes as the amount of rep-
+etition increases. However, in order for the other strategies to match the performance of ER-RS, the
+environment needs to provide a large amount of repetition.
+LwF guarantees a consistent boost in the performance, both in CIFAR-100 and Tiny-ImageNet.
+In particular, and quite surprisingly, on Tiny-ImageNet LwF is able to quickly close the gap with
+ER-RS and even surpass it as the amount of repetition increases. The positive interplay between
+distillation and repetition provides an effective way to mitigate forgetting in CIR scenarios, without
+the need to explicitly store previous samples in an external memory. EWC showed different sen-
+sitivity to the regularization hyper-parameter λ. We experimented with λ = 0.1, 1, 10, 100. While
+on MNIST we did not see any difference in performance, on CIFAR-100 and Tiny-ImageNet large
+values of λ lead to a dramatic decrease, dropping as low as Naive. We found 0.1 to be the best value
+5
+
+2
+3
+5
+8
+10
+K
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Average Test Accuracy
+MNIST
+ER-RS
+LwF
+EWC
+A-GEM
+Naive
+10
+30
+50
+80
+100
+K
+0.0
+0.2
+0.4
+0.6
+CIFAR-100
+20
+60
+100
+160
+200
+K
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Tiny-ImageNet
+Figure 5: Average Test Accuracy for different values of K in CIR scenarios generated with Gslot.
+Class-Incremental scenarios are represented by the left-most point of each plot, Domain-Incremental
+scenarios by the right-most point. Results averaged over 3 seeds.
+50
+100
+150
+200
+250
+300
+350
+400
+450
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Class Accuracy
+Class Status
+Present
+Missing
+Figure 6: Accuracy of a particular class over the stream. The target class is either present or absent
+in the experiences indicated by the blue and orange points, respectively.
+on both CIFAR-100 and Tiny-ImageNet. This configuration only provides a low amount of regular-
+ization. Overall, the role played by the natural repetition already guarantees a sufficient stability of
+the model, which is additionally boosted only in the case of LwF.
+4.2
+IMPACT OF REPETITION IN LONG STREAMS
+We investigate the impact of repetition in long streams (N = 500) generated with Gsamp. For
+the long-stream experiments we also report the missing-classes accuracy (MCA) and seen-classes
+accuracy (SCA). MCA measure the accuracy over the classes that were seen before but are missing
+in the current experience, and SCA measure the accuracy over all seen classes up to the current
+experience.
+Missing Class Accuracy Increases Over Time
+In CI scenarios, a Naive strategy catastrophically
+forgets each class as soon as it starts learning on new classes. Surprisingly, we found that in CIR
+scenarios there is knowledge accumulation over time for all the classes. Figure 6 shows the accuracy
+of a single class over time, highlighting whether the class is present or not in the current experience.
+At the beginning of the stream missing classes are completely forgotten, which can be noticed by
+the instant drop of the accuracy to zero. However, over time the model accumulate knowledge
+and the training process stabilizes. As a result, the accuracy of missing classes tends to increase
+over time, suggesting that the network becomes more resistant to forgetting. Notice that this is an
+example of continual learning property that is completely ignored when testing on CI scenarios.
+This finding prompts the question, ”What is happening to allow knowledge accumulation even for
+Naive finetuning?”. We investigate this question by analysing the model’s accuracy over time and
+the properties of the learned model in the next experiments.
+Accuracy Gap Between Naive and Replay Decreases Over Time
+To study the impact of long
+streams with repetitions we monitor the accuracy gap between ER and Naive fine-tuning by com-
+paring their accuracy after each experience. For the scenario configuration, we set Pf(S) as a
+Geometric distribution with a coefficient of 0.01 and fix the probability of repetition Pr as 0.2 for
+all classes. For more details and illustrations on distribution types refer to Appendix C. In such
+scenarios, the majority of classes occur for the first time in the first quarter of the stream, and then
+repeat with a constant probability of 0.2 which makes them appropriate for our experiments since
+all classes are observed before the middle of the stream and the repetition probability is low enough.
+As can be seen in Figure 7, while the accuracy of ER saturates over time, the accuracy of Naive
+increases, closing the gap between the two strategies from around 25% in experience 100 to 7% in
+6
+
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Average Test Accuracy
+Strategy
+ER-CB
+Naive
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Average Seen Class Accuracy
+Strategy
+ER-CB
+Naive
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Average Missing Class Accuracy
+Strategy
+ER-CB
+Naive
+Figure 7: Average test accuracy and average missing class accuracy plots for long streams streams
+with 500 experiences.
+experience 500. This supports our hypothesis that neural network’s ability to consolidate knowledge
+is significantly influenced by ”natural” repetition in the environment.
+The Role of Repetition
+The amount of repetition is one of the key aspects of a CIR scenario.
+To find out how strategies perform under different repetition probabilities, we consider a setting
+where all components of a scenario are fixed except for Pr. For this experiment, we set Pf(S)
+as geometric distribution with p = 0.2 and let Pr change. In Figure 8 we demonstrate the seen
+class accuracy (SCA) for the Naive and ER-CB strategies in CIFAR-100. It is clear from the plots,
+that the model’s rate of convergence can be significantly affected by the amount of repetition in the
+scenario. Although, it may seem obvious that higher repetition leads to less forgetting, it is not very
+intuitive to what extent different strategies may gain from the environment’s repetition. While the
+Naive strategy gains a lot from increased repetition, the replay strategy saturates after some point
+for higher repetitions and the gaps close between different repetition values.
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+ER-CB
+p
+0.1
+0.3
+0.7
+1.0
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+Naive
+p
+0.1
+0.3
+0.7
+1.0
+Figure 8: Retained accuracy for different values of p in Pr.
+4.3
+MODEL SIMILARITY AND WEIGHT SPACE ANALYSIS
+Weight Interpolation
+Based on the ”gradual loss drop” observation in missing classes, we study
+how the loss surface changes over time if we perturb the weights. We interpolate between the model
+weights from two consecutive checkpoints with an interval of 10 experiences in various segments
+of the stream. Assuming that w∗
+t and w∗
+t+10 are the obtained solutions for experiences t and t + 10
+respectively, we generate eight in-between models wk = α ∗ w∗
+t + (1 − α) ∗ w∗
+t+50 by by increasing
+α from zero to one, and then compute the accuracy of wk for the data of experience t. We show
+the interpolation accuracy for various pairs of experiments in different segments of the stream for
+the Naive strategy in Figure 9 (left). In the beginning of the stream, the accuracy of experience t
+in each pair drops significantly, while we observe a milder loss drop towards the end of the stream.
+The findings suggest that, towards the end of the stream, even a relatively big perturbation does not
+have a large negative effect on the model’s accuracy and the optimal solutions of the consecutive
+experiments are connected with a linear low-loss path.
+Weight changes
+Another approach to analyzing the gradual drop of the accuracy is by dissecting
+how much, when, and where the weight changes occurs. As shown in Figure 9 (right), we can
+observe that within the first experiences, there is a significant difference for blocks 0, 1, and 2. This
+difference then stalls, showing that as we continue training experiences, the weights of these blocks
+stabilize. On the other hand, blocks 3 and 4 show a linear increase in the difference with the number
+of experiences. An explanation for this phenomenon, is that the first layers of the model capture
+knowledge that can be useful for several classes (more general), so it is unnecessary to change them
+after several experiences. On the other hand, the last blocks are the ones that memorize or learn
+more specific patterns, so they adapt to each experience.
+7
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Interpolation Weight
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Accuracy
+Accuracy over Linear Path
+Start
+End
+50
+100
+100
+150
+200
+250
+300
+350
+400
+450
+0
+100
+200
+300
+400
+500
+Experience
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+Difference
+Difference over the Experience
+Blocks
+Block 0
+Block 1
+Block 2
+Block 3
+Block 4
+Figure 9: (left) Interpolation accuracy. (right) Weight changes in each block. The difference used in
+(right) is calculated as Dj =
+1
+|θ0|
+�θb
+i
+��� (θ0,i−θj,i)
+∥θ0,i∥2
+���, where the weights of experience j are compare
+with the initialization θ0 for each block i
+CKA Analysis
+Finally, we show the CKA Kornblith et al. (2019) of the model in the beginning,
+middle and the end of the stream with an interval difference of 50 experiences. As shown in the
+visualizations in Figure 10, the longer the model is trained on more experiences, the less significant
+the changes in the representations become especially for the final layers. We can see that the diag-
+onal of the CKA becomes sharper propagating forward with more experiments. This indicates that
+although the model is trained on different subsets of classes in each experiment, the representations
+change less after some point in the stream.
+Experience-15
+Experience-5
+5
+15
+Experience-35
+Experience-25
+25
+35
+Experience-65
+Experience-55
+55
+65
+Experience-200
+Experience-190
+190
+200
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Figure 10: CKA of the model in different parts of the stream.
+4.4
+FREQUENCY-AWARE REPLAY IN UNBALANCED SCENARIOS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Accuracy
+Strategy
+ER-FA
+ER-CB
+ER-RS
+Naive
+Figure 11:
+Accuracy of Infrequent
+Classes.
+We conduct experiments for bi-modal unbalanced scenar-
+ios where classes can have a high frequency of 1.0 or a
+low frequency of 0.1. We use a fraction factor that de-
+termines the amount of infrequent classes in the scenario,
+e.g., Fraction=0.3 means that 30% of the classes are infre-
+quent. In Table 2 we compare ER-FA with the Naive, ER-
+RS and ER-CB strategies. The numbers show the MCA
+and average Test Accuracy (TA) metrics for each strategy
+in the end of the stream averaged over three runs. Our
+strategy outperforms all other scenarios in almost all set-
+tings in both CIFAR-100 and TinyImageNet datasets in
+terms of TA, and significantly outperforms other methods
+in terms of MCA (in the last experience). Moreover, we
+illustrate the accuracy of infrequent classes in CIFAR-100 experiments for Fraction=0.3 in Figure
+11 where ER-FA achieves considerably higher accuracy in the whole stream by assigning larger
+quota to infrequent classes without losing its performance on frequent classes (refer to Appendix G
+for further illustrations).
+5
+RELATED WORK
+Current CL methods are mainly focused on two types of benchmarks namely, Multi Task (MT) and
+Single Incremental Task (SIT) Maltoni & Lomonaco (2019). MT divides training data into distinct
+tasks and labels them during training and inference. SIT splits a single task into a sequence of
+unlabeled experiences. SIT can be further divided into Domain-Incremental (DI) where all classes
+are seen in each experience, and Class-Incremental (CL) where each experience contains only new
+8
+
+DS
+Strategy
+Fraction= 0.1
+Fraction= 0.3
+Fraction= 0.5
+MCA
+TA
+MCA
+TA
+MCA
+TA
+C-100
+Naive
+5.0 ± 0.7
+58.0 ± 0.1
+7.3 ± 2.2
+49.0 ± 0.8
+8.0 ± 2.0
+40.8 ± 1.4
+ER-RS
+11.4 ± 0.9
+57.7 ± 0.7
+16.7 ± 3.6
+51.1 ± 0.4
+20.5 ± 1.9
+45.6 ± 0.8
+ER-CB
+30.9 ± 2.7
+59.5 ± 0.1
+34.5 ± 1.7
+55.3 ± 0.1
+35.7 ± 0.5
+52.0 ± 1.5
+ER-FA
+52.2 ± 1.1
+60.8 ± 0.3
+44.7 ± 1.5
+57.8 ± 0.4
+40.9 ± 1.2
+54.2 ± 1.2
+TIN
+Naive
+2.0 ± 0.8
+33.5 ± 0.4
+2.0 ± 0.1
+29.1 ± 0.1
+2.0 ± 0.1
+24.0 ± 0.4
+ER-RS
+3.7 ± 0.6
+31.8 ± 1.2
+4.4 ± 0.7
+28.1 ± 0.2
+6.0 ± 0.1
+24.0 ± 0.1
+ER-CB
+10.4 ± 0.2
+32.2 ± 0.7
+10.0 ± 1.0
+28.8 ± 0.2
+11.0 ± 0.2
+26.0 ± 0.3
+ER-FA
+22.0 ± 1.0
+33.0 ± 0.9
+15.3 ± 1.0
+30.4 ± 0.1
+13.6 ± 0.1
+27.0 ± 0.1
+Table 2: Unbalanced scenario results for the CIFAR-100 (C-100) and TinyImageNet (TIN) dataset.
+“Fraction” refers to the fraction of infrequent classes having repetition probability of only 10%.
+(unseen) classes van de Ven & Tolias (2019). Both DI and CI are extreme cases and are unlikely to
+hold in real-world environments Cossu et al. (2021). In a more realistic setting, the role of natural
+repetition in CL scenarios was studied in the context of New Instances and Classes (NIC) scenario
+Lomonaco et al. (2020) and the CRIB benchmark Stojanov et al. (2019). NIC mainly focuses on
+small experiences composed of images of the same object, and repetitions in CRIB are adapted to
+a certain dataset and protocol. The Class-Incremental with Repetition (CIR) scenario was initially
+formalized in Cossu et al. (2021), however the work lacks a systematic study of CIR scenarios as
+the wide range of CIR scenarios makes them difficult to study.
+To counter the lack of repetition in CI, replay has been extensively used as a CL strategy (Rebuffi
+et al., 2017; Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2018; Wu et al., 2019; Castro et al.,
+2018; Belouadah & Popescu, 2019; Kim et al., 2020; Douillard et al., 2020). In such methods,
+natural repetition is artificially simulated by storing past data in an external memory, and replaying
+them alongside the scenario stream data. Repetition reduces catastrophic forgetting through implicit
+regularization of model’s weights Hayes et al. (2021). In CI benchmarks, replay seems to be the
+only working strategy van de Ven et al. (2020). In other words, replay seems to be a necessity when
+no natural repetition happens.
+Although replay can be seen as a method to simulate natural repetitions artificially, the two concepts
+are fundamentally different. Repetition in replay strategies occurs with the same data seen in pre-
+vious experiences, which is neither realistic nor biologically plausible Gupta et al. (2010). On the
+other hand, natural repetitions of already seen objects occur in different real-world environments,
+and better fit the CIR scenario studied in this paper. Recently, Lesort et al. (2022) scaled the number
+of tasks in a finite world setting (Boult et al., 2019; Mundt et al., 2022) where the model has access to
+a random subset of classes in each experience. The authors proposed naive fine-tuning with masking
+techniques to improve retained accuracy. Our work is different in the sense that we compare among
+different strategies and study various types of repetitions with two flexible generators.
+6
+DISCUSSION AND CONCLUSION
+We defined CIR scenarios which represent CL environments where repetition is naturally present in
+the stream. Although the concept of repetition is quite intuitive, it is not obvious how to realize it in
+practice for research purposes. Therefore, we proposed two CIR generators that can be exploited to
+address this issue. Through empirical evaluations, we showed that, unlike CI scenarios, knowledge
+accumulation happens naturally in CIR streams, even without applying any CL strategy. This raised
+the question of whether the systematic repetition provided by Replay is critical in all CIR scenarios.
+With several experiments in long streams, we demonstrated that although Replay provides an ad-
+vantage in general, even random repetition in the environment can be sufficient to induce knowledge
+accumulation given a long enough lifetime.
+Moreover, we found that existing Replay strategies are exclusively designed for classical CI scenar-
+ios. Thus, we proposed a novel strategy, ER-FA, to exploit the properties of CIR scenarios. ER-FA
+accumulates knowledge even in highly unbalanced stream in terms of class frequency. ER-FA out-
+performs by a large margin other Replay approaches when monitoring the accuracy for infrequent
+classes while preserving accuracy for the frequent ones. Overall, ER-FA guarantees a more ro-
+bust performance on a wide range of real-world scenarios where classes are not homogeneously
+distributed over time.
+9
+
+The framework defined in this work opens new research directions which depart from the existing
+ones, mainly focused on the mitigation of forgetting in CI scenarios. We hope that our experiments
+and results will promote the study of CIR scenarios and the development of new CL strategies, able
+to exploit the inner semantics of repetition, a natural trait of real-world data streams, in many other
+clever and imaginative ways.
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+12
+
+A
+RELATED WORK (CONTINUED)
+Considering benchmark formalization frameworks, De Lange & Tuytelaars (2021) recently pro-
+posed a subdivision aimed at framing continual learning setups by categorizing them based on the
+batch and observable horizon that the learning agent is able to access at each time. With this frame-
+work, the authors aim to better formalize the online learning setup. While the concept of observ-
+able horizon may be useful in evaluating the significance and local (in time) usefulness of natural
+repetition in a training stream, this work does not consider the concept of natural repetition in its
+framework.
+Recently, Koh et al. (2022) proposed to introduce blurry task boundaries in class incremental bench-
+marks. Their proposal is based on previous works Bang et al. (2021); Aljundi et al. (2019) that tried
+to produce more realistic benchmarks by blurring the class-incremental scenario, which however
+resulted in a setup in which no classes are added to new tasks. They argue that this idea moves the
+focus too far away from the class-incremental setup and it is still not quite realistic. The resulting
+setup, named i-Blurry, aims at resolving the aforementioned issues and moving toward a more re-
+alistic scenario by partitioning the classes available in the source dataset into two groups: Disjoint
+and Blurry. Classes of the disjoint group are gradually added in successive experiences while sam-
+ples of classes from the blurry group always appear in all experiences with their numerosity being
+controlled through a blur ratio M. The authors show that, based on the degree of disjunction N
+and blurry M, this framework can produce class-incremental (no blurring), domain-incremental (no
+disjunction), and blurred setups. This setup is the one that most moved towards the direction of
+introducing repetition in Continual learning benchmarks in a controlled way so far. However, the
+proposed blurring mechanism is too coarse-grained to simulate a natural repetition of concepts as
+the significance of the repetition introduced by blurring relies too much on i) a random (uniform)
+sampling of the concepts to be repeated, ii) the static subdivision of classes in the Disjoint and Blurry
+groups.
+B
+SLOT-BASED GENERATOR
+Following the properties of CIR scenarios in Section 2, Gslot generates a subset of CIR streams that
+hold the assumptions below in the defined properties:
+• |Xi ∩ Xj| = 0 : new instances appear in each experiences
+• �i=N
+i=1 Xi = X: all samples are used
+• |Yi ∩ Yj| ≥ 0, ∀1 ≤ i, j ≤ N where i ̸= j
+• �i=N
+i=1 Yi = Y : all classes are used
+• X is constant.
+These assumptions allow transitioning through different CIR scenario types between the two ex-
+tremes of CI and DI.
+B.1
+ALGORITHM
+The overall steps of Gslot are illustrated in Figure 12. In Algorithm 2 we present all steps of Gslot
+used to generate arbitrary CIR scenarios given a dataset D, number of experiences N and number
+of slots K. The output of the algorithm is a CL stream.
+B.2
+TRANSITIONING
+Transitioning in Gslot for a scenario with a fixed number of experiences can be done by increasing
+K. When K = 1 the generated scenario will be class-incremental and as K get closer to the total
+number of classes in D, the scenarios moves towards a domain incremental setting. In Figure 13 we
+show an example of how generated scenarios change by increasing K.
+13
+
+…
+Concepts:
+Replicate each concept !×#
+$
+times
+and shuffle the sequence. 
+𝒆𝟏
+𝒆𝟐
+𝒆𝟑
+…
+Example: 𝐾 = 4
+Slot 1
+Slot 2
+Slot 3
+Slot 4
+New Instances
+𝒆𝑵
+𝓢𝒕𝒓𝒂𝒊𝒏
+Figure 12: Illustrations of the overall steps of Gslot. Each shape represents a concept, and the green
+color means that new instances of that concept are used in each experience.
+Class
+Experience
+Figure 13: From left to right: transitioning from CI to DI in Gslot. Each class is represented with a
+unique color.
+C
+SAMPLING-BASED GENERATOR
+Following the properties of CIR scenarios in Section 2, Gsamp generates a subset of CIR scenarios
+that hold all defined properties. Additionally, for any stream S = {e1, e2, ..., eN}, Gsamp defines
+a probability distribution for the first occurrence of concepts over S and per-class probabilities for
+each concept c ∈ Y . Gsamp can generate arbitrarily long stream (N ≥ 1) and even from a growing
+set of samples X where Y remains constant.
+C.1
+ALGORITHM
+The overall steps of the Gslot are shows in Algorithm 2.
+C.2
+DISTRIBUTION TYPES
+In this section we show some examples of different discrete distributions that can be used for Pf(S)
+and Pr. For Pr we use the unnormalized version of the final distribution. Distributions used for
+Gsamp can be any arbitrary discrete distribution and are not limited to the ones we describe here.
+C.2.1
+ZIPFIAN
+Given the number of elements N and scalar e ≥ 0 , the probability mass function of a Zipfian
+distribution over a list of N elements is defined in Equation 1. When used for the probability of first
+occurrence, the distribution can be defined over the experiences of a stream. For example, N can
+be considered as the number of experiences and i can indicate the ith experience in the stream. By
+increasing e, the distribution over the stream will be skewed towards the beginning. In figure 14 we
+demonstrate some examples of first occurrence probabilities over a stream of length 10 generated
+with Zipf distribution with increasing values of e. Many natural distribution follow Zipf distribution
+14
+
+Algorithm 1 Slot-Based Generator (Gslot) Pseudo-Code.
+Require: Dataset D = {(xi, yi)}i=1,...,P with C classes, number of experiences N, experience
+size S present in each experience.
+Ensure: K ≤ N
+Ensure: C mod N = 0
+Ensure: NK mod C = 0
+cls-idxs = {}
+▷ Empty dictionary
+for y ∈ set({yi}i=1,...,P ) do
+cls-idxs[y] = []
+▷ Empty list init
+end for
+for i = 1, . . . , P do
+cls-idxs[yi].append(i)
+end for
+slots={}
+▷ Empty dictionary
+for y ∈ cls-idxs do
+slots[y] = []
+ksample = int(len(cls-idxs[y]) /K)
+for k = 1, . . . , N×K
+C
+do
+subset-idxs = pop(cls-idxs[y], ksample)
+subset-samples = [xidx for idx ∈ subset-idxs]
+slots[y].append(subset-samples)
+end for
+end for
+stream = []
+for n = 1, . . . , N do
+experience = dataset()
+seen-classes = []
+for k=1,...,K do
+repeat
+y = sample(slots)
+until y /∈ seen-classes
+seen-classes.append(y)
+experience.add(pop(slots[y], 1))
+end for
+stream.append(experience)
+end for
+return stream
+0
+5
+10
+0.0
+0.5
+1.0
+Probability
+e = 0.0
+0
+5
+10
+0.0
+0.5
+1.0
+e = 0.5
+0
+5
+10
+Experience
+0.0
+0.5
+1.0
+e = 1.0
+0
+5
+10
+0.0
+0.5
+1.0
+e = 1.5
+0
+5
+10
+0.0
+0.5
+1.0
+e = 2.0
+Figure 14: Zipf distribution with varying values of e.
+and it can be used to generate highly skewed distributions both for first occurrence and repetition
+probabilities.
+f(i; e, N) =
+1
+ie
+�N
+n=1
+1
+ne
+(1)
+15
+
+Algorithm 2 Sampling-Based Generator (Gsamp) Pseudo-Code.
+Require: Dataset D = {(xi, yi)}i=1,...,P with C classes, number of experiences N, number of
+slots K, probability distribution for first occurrence Pf(S), and list of repetition probabilities Pr.
+T = {0}C×N
+▷ Initialize occurrence matrix with zeros
+for c ∈ {0, 1, . . . C} do
+i ∼ Pf(S)
+▷ Samples the first occurrence of class c
+T[c, i] = 1
+for j ∈ {i, i + 1, . . . N} do
+r ∼ U(0, 1)
+▷ U: uniform distribution over [0, 1.0]
+if r < Pr[c] then T[c, j] = 1
+end if
+end for
+end for
+E = {}
+for ei ∈ {1, 2, . . . N} do
+Ci = RetrieveClasses(ei)
+Dei = Sample(D, Ci, S)
+▷ Sample S instances from dataset D for classes Ci
+E ← E ∪ Dei
+end for
+Strain, Stest = GenerateStream(E)
+▷ Generate streams using E
+return Strain, Stest
+C.2.2
+POISSON
+The PMF for Poisson distribution is given in Equation 2 where µ ≥ 0. Poisson with larger values
+of µ can be used for distributions where the probability of occurrence/repetition first rises and then
+gradually decreases over time.
+f(i; µ) = µie−µ
+i!
+(2)
+0
+5
+10
+0.0
+0.5
+1.0
+Probability
+= 0.0
+0
+5
+10
+0.0
+0.5
+1.0
+= 0.5
+0
+5
+10
+Experience
+0.0
+0.5
+1.0
+= 1.0
+0
+5
+10
+0.0
+0.5
+1.0
+= 1.5
+0
+5
+10
+0.0
+0.5
+1.0
+= 2.0
+Figure 15: Poisson distribution with varying values of µ.
+C.2.3
+GEOMETRIC
+Another useful distribution that can be used for the first occurrence probabilities over a stream is
+Geometric distribution with its PMF given in Equation 3. This distribution is in particular inter-
+esting for transitioning from domain incremental to class incremental. By setting p = 1, only the
+probability of experience i = 0 will be equal to 1.0 and the rest will be zero, and by decreasing p,
+the probability will spread over the stream. In figure 17 we show examples for generated scenarios
+with Gsamp with Geometric first occurrence and fixed probability of repetition.
+f(i, p) = (1 − p)i−1p
+(3)
+16
+
+0
+5
+10
+0.0
+0.5
+1.0
+Probability
+p = 0.01
+0
+5
+10
+0.0
+0.5
+1.0
+p = 0.2
+0
+5
+10
+Experience
+0.0
+0.5
+1.0
+p = 0.5
+0
+5
+10
+0.0
+0.5
+1.0
+p = 0.7
+0
+5
+10
+0.0
+0.5
+1.0
+p = 1.0
+Figure 16: Geometric distribution with varying values of p.
+Experience
+Class
+Experience
+Class
+Experience
+Class
+Figure 17: Scenarios generated with Geometric first occurrence and probability of repetition equal
+to 1.0 for all classes. The p values for the Geometric distributions from left to right are 0.01, 0.2 and
+1.0 respectively.
+D
+UNBALANCED SCENARIOS
+In this section we present a particular type of unbalanced scenarios where a subset of classes in
+the stream have a low probability of repetition and the rest repeat very often. We refer to such
+scenarios bi-modal scenarios, where each mode refers to a subset of classes with a distinct repetition
+probability. More specifically, we have a stream of experiences S = {e1, e2, ..., eN} where Y S =
+�N
+1 Yei indicates the set of all available concepts in S. In bi-modal scenarios Y = Y if ∪Y fr where
+Y if and Y fr are the set of frequent and infrequent concepts respectively, and Y if ∪ Y fr = Ø. In
+Figure 18 we show examples of unbalanced bi-modal scenarios.
+Experience
+Class
+Experience
+Class
+Experience
+Class
+Figure 18: Unbalanced scenarios with two modes of repetition. The fractions of infrequent classes
+from left to right are 0.2, 0.4 and 0.6 respectively. Repetition probabilities for frequent and infre-
+quent classes are set to 0.2 and 0.9 accordingly.
+E
+FREQUENCY-AWARE REPLAY
+E.1
+ALGORITHM
+In Algorithm 3 we present the steps for updating the buffer in FA storage policy.
+E.2
+ANALYSIS: VARYING THE FRACTION OF INFREQUENT CLASSES
+In this section, we study the behavior of FA, CB, and RS storage policies by changing the fraction
+of infrequent classes. In our analysis, we consider an unbalanced stream generated with Gsamp
+where N = 100 and the probability of repetition for frequent and infrequent classes are 0.9 and
+0.1, respectively. In such streams, the large probability gap between frequent and infrequent classes
+17
+
+Algorithm 3 Frequency-Aware Buffer.
+Require: Current Buffer Set B, Maximum buffer size M, List of Seen Classes C, Number of
+Observation per Seen Class O
+D = GetExperienceDataset(ei)
+P = DetectPresentClasses(D)
+C ← C ∪ P
+for c ∈ P do
+▷ For each present class, increment the number of observations
+if c ∈ O then O[c]+ = 1
+else O[c] = 1
+end if
+end for
+Q = [
+1
+O[c]∀c ∈ C]
+▷ Calculate quota per class
+ˆQ =
+Q
+|Q|
+▷ Normalize quota values
+S = {⌈Q[c] ∗ M⌉∀c ∈ C}
+▷ Calculate buffer slot size for each class
+UpdateSlots(S)
+▷ Update assigned slots according to the current state of B
+UpdateBuffer(B, D, S)
+return B, M, C, O
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+Figure 19:
+Ratio of samples for infrequent classes in unbalanced scenarios for the FA,
+CB and RS policies.
+Fraction of infrequent classes from top-left to bottom-right are
+20%, 40%, 60%, 80%, 100%.
+helps us observe the difference more clearly. We report the ratio of samples assigned to infrequent
+classes in the buffer in the lifetime of the model in the stream for scenarios where the fraction of
+infrequent classes is equal to {20%, 40%, 60%, 80%, 100%}. For this experiment, we set the buffer
+size to 500 for all methods.
+As demonstrated in Figure 19, when the fraction of infrequent classes is equal to 20%, i.e. only 20%
+of classes are infrequent, the ratio is very low for RS policy as it tries to replicate the true distribution
+of the stream while CB assigns exactly 20% of the buffer space to the infrequent samples. However,
+we can observe that FA starts to assign more samples over time to the infrequent classes over time as
+it adapts the buffer slots based on the frequency of repetition. Moreover, it is evident in the plots that,
+by increasing the fraction of infrequent classes, the ratio gap between FA and CB gets smaller as
+the quota for CB stays the same while the number of infrequent classes increases. Eventually, when
+the fraction of infrequent classes is equal to 100%, i.e. all classes have the same (low) probably of
+repetition, all buffers have exactly the same ratio since all classes are infrequent.
+In conclusion, FA buffer slots can be very helpful in highly unbalanced streams where a smaller
+fraction of classes have a low probability of repetition. When the stream moves towards becoming
+18
+
+balanced, the FA and CB get closer, and all methods become similar in the extreme case of a fully
+balanced stream with similar probability of repetition.
+F
+CHANGING Pf(S)
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+ER-CB
+p
+0.1
+0.3
+0.7
+1.0
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+Naive
+p
+0.1
+0.3
+0.7
+1.0
+Figure 20: Average test accuracy for different values of p in first occurrence.
+We conduct experiments to find the differences between situations in which all classes occur early
+versus those in which new classes also appear late in the stream in order to analyze the role of first
+occurrence type. How early or late in the stream we observe all classes of a dataset, depends on
+the the parameters that control Pf(S). In this experiment, we fix the probability of repetition Pr
+and change Pf(S)’s parameters. In particular, we opt the geometric distribution for Pf(S) and
+choose the values {0.1, 0.3, 0.7, 1.0} for its only parameter 0 < p ≤ 1.0. Increasing p is inversely
+proportional to the spread factor in the first occurrence distribution, i.e. when p is close to 0 all
+classes happen in the first experience and as we move p toward 1.0 the classes start to spread along
+the stream. Figure 20 shows the CIFAR-100 results for the Naive and ER-CB strategies. The results
+suggest that when the spread factor is low, the model initially has difficulty to learn since there
+are more classes in the initial experiments and thus the model has to learn from fewer instances.
+However, with more experiences, all first occurrence types, reach almost the same SCA.
+G
+ER-FA RESULTS
+Results in Figure 21 illustrate the total test accuracy and accuracy of frequent classes over time.
+Although the discrepancy between the accuracies of frequent classes is very small, the total test ac-
+curacy can significantly vary due to the difference in the accuracy of infrequent classes as presented
+in Section 4.4.
+0
+20
+40
+60
+80
+100
+Experience
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Strategy
+ER-FA
+ER-CB
+ER-RS
+Naive
+0
+20
+40
+60
+80
+100
+Experience
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+Strategy
+ER-FA
+ER-CB
+ER-RS
+Naive
+Figure 21: TA over all classes (left) and frequent classes (right) in a bi-modal unbalanced scenario
+with Fraction=0.3.
+19
+

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+1
+Enabling Augmented Segmentation and Registration
+in Ultrasound-Guided Spinal Surgery via Realistic
+Ultrasound Synthesis from Diagnostic CT Volume
+Ang Li†, Jiayi Han†, Yongjian Zhao, Keyu Li, Li Liu‡
+Abstract—This paper aims to tackle the issues on unavailable
+or insufficient clinical ultrasound (US) data and meaningful
+annotation to enable bone segmentation and registration for US-
+guided spinal surgery. While the US is not a standard paradigm
+for spinal surgery, the scarcity of intra-operative clinical US data
+is an insurmountable bottleneck in training a neural network.
+Moreover, due to the characteristics of US imaging, it is difficult
+to clearly annotate bone surfaces which causes the trained neural
+network missing its attention to the details. Hence, we propose an
+In silico bone US simulation framework that synthesizes realistic
+US images from diagnostic CT volume. Afterward, using these
+simulated bone US we train a lightweight vision transformer
+model that can achieve accurate and on-the-fly bone segmen-
+tation for spinal sonography. In the validation experiments,
+the realistic US simulation was conducted by deriving from
+diagnostic spinal CT volume to facilitate a radiation-free US-
+guided pedicle screw placement procedure. When it is employed
+for training bone segmentation task, the Chamfer distance
+achieves 0.599mm; when it is applied for CT-US registration,
+the associated bone segmentation accuracy achieves 0.93 in
+Dice, and the registration accuracy based on the segmented
+point cloud is 0.13∼3.37mm in a complication-free manner.
+While bone US images exhibit strong echoes at the medium
+interface, it may enable the model indistinguishable between thin
+interfaces and bone surfaces by simply relying on small neighbor-
+hood information. To overcome these shortcomings, we propose
+to utilize a Long-range Contrast Learning Module (LCLM) to
+fully explore the Long-range Contrast between the candidates
+and their surrounding pixels. In the ablation experiments, it is
+verified that the proposed Long-range Contrast Learning module
+is effective in the precise positioning of the US region of interest.
+On top of that, the training data is entirely generated by our
+proposed US simulation framework without fine-tuning based
+on real clinical data, which demonstrates its effectiveness of the
+bone realistic US simulation framework.
+Note to Practitioners—The motivation of this paper is to
+address the issues on unavailable or insufficient bone US images
+and annotation labels. We employ such a data augmentation
+technique to generate realistic simulated bone US and annotation
+associated with the corresponding CT volume. The problems
+of current US data augmentation approaches are mainly the
+inability to generate continuous context-accurate data (neural
+network-based algorithm) and the lack of real-time physical
+simulation capability (traditional acoustics-based algorithm).
+In this paper, we enhance the US simulation derived from
+CT images by supplementing techniques such as extrusion
+simulation, moving space simulation, and image augmentation
+to yield higher quality images, and hence apply these images
+directly to the target task, i.e., train a neural network to
+† These authors contributed equally
+‡ represents the corresponding author. Email: [email protected]
+Ang Li, Yongjian Zhao, Keyu Li and Li Liu is with Chinese University of
+Hong Kong.
+Jiayi Han is with Fudan University.
+guide US-guided spinal surgery. Besides, we demonstrate the
+effectiveness of the realistic US simulation framework and each
+new module of the neural network using ablation experiments. It
+can be concluded that the neural network trained with the data
+generated by US simulation framework promises to enable US-
+guided pediclescrew placement procedures. In the future work,
+we will apply the In silico bone US simulation as a reinforcement
+learning environment and deploy the trained agents to directly
+guide US-guided procedures.
+Index Terms—Realistic US Simulation, Vision Transformer,
+Bone Surface Segmentation, CT-US Registration, US-Guided
+Spinal Navigation
+I. INTRODUCTION
+Segmentation of bone surfaces from intra-operative US data
+followed by CT-US registration is two critical steps for US-
+guided spinal surgery. Recent research has focused on the
+use of deep learning-enabled methods for accurate, robust,
+and real-time segmentation and registration of bone surfaces.
+However, scarcity of data size, due to a lack of standardized
+data and patient privacy concerns, is a major challenge in
+applying deep learning-enabled methods in the intra-operative
+imaging field. This is specifically a challenge due to the fact
+that the US is not a standard imaging modality in spine-related
+surgeries and US-guided spinal surgeries are not common;
+even if spinal sonography can be available for pre-clinical
+collection procedures, annotations would be still a severe
+challenge due to the vast amounts of data. Another limiting
+factor is the manual data acquisition process: sub-optimal
+orientation of the US transducer with respect to the imaged
+spinal anatomy will result in the user-dependent acquisition of
+low-quality bone scans.
+Increasing the size of existing datasets through data aug-
+mentation in order to improve models’ performance is exten-
+sively investigated. Among them, a fundamental approach to
+obtaining a large labeled dataset is In silico realistic simulation
+of spinal US images. The US simulation methods could be
+broadly categorized into three types: acoustic model-based US
+simulation, image-based US simulation, and generation of a
+virtual image using a generative adversarial network (GAN).
+US generation algorithms based on acoustic models are usually
+very slow [1]. Generating US by GAN may require training
+different networks for different organs, meanwhile, obtaining
+corresponding US volume and CT volume pairs aligned in
+the same space is difficult, allowing for the generation of
+large-scale synthetic US images using GAN a challenging
+task. Image-based approaches attempt to utilize a simulated
+arXiv:2301.01940v1  [eess.IV]  5 Jan 2023
+
+2
+Fig. 1. The figure shows examples of generated Ultrasound images which are the junctional surfaces of the two vertebrae, the plane where the vertebral plate
+could be entirely verified, the plane swept along the spinous, and an arbitrary scan plane, respectively.
+US probe to re-sample the original image (such as a CT scan)
+and then consider the image scalar as acoustic parameters of
+organs and then simulate propagation with acoustic properties.
+Nonetheless, different from other imaging methods such as
+CT and MRI, the US is significantly influenced by gas. To
+avoid its influence, the operator presses the probe to remove
+the gas between the probe and the skin, which leads to the
+distortion of tissues in the original images. However, this
+distortion is ignored by the former researchers. Moreover,
+because each tissue has a different reflection rate, the former
+works segment the image and design specific transfer functions
+to each tissue, which is time-consuming. In addition, the
+conventional use of US image generation is only to assist
+in 2d-3d image registration [2] or for training examining
+physicians [3] without further application of synthetic US
+images.
+In this paper, we propose a novel CT-derived realistic US
+synthesis framework incorporating automated image gener-
+ation with sampling methods, as shown in Fig. 3. From
+(a)∼(b), each column represents the real US, corresponding
+CT scan, reflection map, and transmission map, respectively.
+We simulate the distortion resulting by pressing the probe via
+warping the original image and propose an adaptive transfer
+function that could be directly adopted to the whole image
+which eliminates the transfer function designing process for
+each tissue and highly speeds up the US simulation task.
+To fully take the advantage of the proposed simulation and
+conduct real-time US image segmentation, we further propose
+a lightweight vision transformer with Long-range Contrast
+Learning Module (LCLM) which utilizes a designed cascaded
+dilated convolution layers to achieve dense super-large recep-
+tive field which enhances the US image segmentation. Exper-
+iments demonstrate the proposed simulation system achieves
+state-of-the-art performance compared with other approaches
+and benefits the proposed vision transformer for real US image
+segmentation.
+Our contributions are listed as follows:
+1) We propose an In silico bone US simulation framework
+that synthesizes realistic US images from diagnostic CT
+volume.
+2) We develop a lightweight vision transformer model that
+achieves precise and real-time bone segmentation for
+spinal sonography images.
+3) Experiments demonstrate that the proposed In silico
+bone US simulation approach dramatically enhances the
+segmentation performance in comparison with initial CT
+scans, indicating that the proposed data augmentation
+method is capable of pre-training models for real clinical
+spinal sonography.
+II. RELATED WORK
+In this section, the related work on realistic US simulation
+and associated bone segmentation are discussed.
+A. Realistic US Simulation
+Traditional acoustics-based US simulation software was
+pioneered by Jenson et al. [4] in 1996. K-space method for
+fast computation of pulsed photo-acoustic fields was proposed
+by B. T. Cox et al. [5] in 2005 and Treeby et al. [6] optimized
+the US image propagation model on the K-space method and
+then the widely used K-wave tools have been developed. The
+advantage of this conventional acoustic-based algorithm is that
+it can simulate various types of US imaging systems with the
+physical effect of the images as realistic as possible.
+As these algorithms based on physical models of acoustics
+have efficiency problems to generate large datasets, researchers
+started to develop ray-tracing-based approaches. Burger et al.
+[7] developed a US simulation system by segmenting the
+CT dataset into different tissues, and then assigning velocity,
+impedance, scattering factor, and other acoustic properties
+to each tissue; hence, it was used to simulate the sound
+propagation, absorption, and scattering processes. Cong et al.
+[8] proposed a multi-scale enhancement method to augment
+tubular structures to simulate blood flow, and allow for US
+images more realistic. Piorkowski et al. [3] in 2013 applied
+the algorithms of Wein [9] and Kutter [10] to make a Trans-
+esophageal Echocardiography (TEE) Simulator which has a
+tremendously positive effect on training doctors to perform
+TEE examinations. To further accelerate the image generation
+speed, Wang et al. [11] used NVIDIA’s Optix 6.0 ray-tracing
+
+3
+engine to do the Monte Carlo simulation of US with a good
+result compared to GAN and Field-II [12].
+In recent years, generative adversarial network (GAN) mod-
+els have been used extensively in realistic US simulation
+research. It generates realistic US images after learning from a
+large dataset in the US domain. Hu et al. [13] applied a GAN
+model to yield US images for Freehand scanning, the model
+takes the spatial location information as a conditional input,
+and then outputs the US image at the current location. This
+work contributes to producing US data for the corresponding
+location, yet they were unable to create synthetic US images
+for each specific patient. To address this shortcoming in 2018,
+Tom et al. [14] employed cascaded GAN with an image
+segmentation label as conditional input to create more realistic
+US images and to be able to produce synthetic US images for
+different segmentation results. Nonetheless, according to their
+report, it can be concluded that even with precise segmenta-
+tion the GAN-based virtual US system still has difficulty in
+ensuring the edge intensity and shape as the real US in same.
+B. US Segmentation
+Some early study utilized handcrafted features for US
+segmentation, such as active contour [15], [16]. In recent years,
+the most popular model in US segmentation is UNet [17].
+Many works adopt different reinforcements in UNet-based
+models for US segmentation. [18] proposed a multi-task UNet
+which combines classification and segmentation tasks. [19]
+proposed a lightweight UNet model which alternately adopt
+3 × 3 and 1 × 1 convolution layers. They also introduced a
+false output suppression mechanism that combines patch-wise
+classification and segmentation results to eliminate false pos-
+itive. [20] adopts spatial attention on US image segmentation
+task.
+III. METHOD
+A. Realistic US Simulation from CT Volume
+The entire US simulation progress is divided into several
+parts: data sampling, data filtering, US transmitting, and image
+blending.
+a) Image Probe and Press simulation: Since the goal
+of the proposed US simulation system is to train doctors or
+artificial intelligence agents for surgical robots, the moving
+space of probe motion cannot be the entire 3D space. The
+probe in the system can move in the provided scanning space,
+but the probe is a curved surface, it is the probe surface, and
+the scanning space will not be fitted perfectly. The scanning
+space will wrap around and fit the probe if the pressure is
+simulated. However, the CT volume will not deform with the
+surface. Hence, an algorithm needs to be proposed to make
+the CT data deform with the probe surface along the scanning
+space. Using the spring-mass model to simulate deformations
+in scanning space by using the moving least squares algorithm
+to fit the CT deformation is certainly good, but also has
+two demerits. The first is solving the spring-mass simulation
+and the moving least squares equation will consume a lot of
+computational resources. Secondly, the algorithm requires a
+lot of extra work to mark the anchor points on the images.
+Under this situation, we changed the local translation image
+warping algorithm proposed by A. Gustafsson for CT data
+and apply it with the shape of the probe to the image. The
+algorithm only needs to know the shape of the probe and the
+HU value of the CT data of that slice to simulate the motion of
+the tissues. If further acceleration is desired, the weight value
+of the HU parameter can be set to 1, then the UV coordinate
+mapping only needs to be calculated once, without affecting
+the imaging efficiency at all.
+The equation is formulated as Equ. 1
+�
+�
+�
+�
+�
+⃗u = ⃗x
+r2
+max−|⃗x−⃗c|2
+r2max−|⃗x−⃗c|2+D(⃗m − ⃗c)
+D = 100
+f α(hu)|⃗m − ⃗c|2
+.
+(1)
+In the above equation, f controls the ratio of deformation.
+In our case, we want to push the intersection point of the
+center-line of the probe and the top line of the sampled image,
+as the blue dot is shown in Fig.4, to the top of the probe
+which is shown in the figure as the red dot. According to
+the mentioned algorithm, all the tissue between the blue and
+red dots needs to be pressed downwards. Assuming that the
+tissue is a rigid body, the thickness of the tissue does not
+change by pressing with a very big force. In this case, the
+original position at the red dot should be pushed to the top of
+the green curve. Then a green arc that represents the limit of
+tissue movement can be drawn. The squeezing of the tissue
+in the area between the probe arc and the green arc will
+cause high reflections or absorption in this area. Hence, in the
+process of simulating transmission, it is significant to make
+the sound waves attenuate less in this area, otherwise, the
+squeezed tissue generated by the algorithm will completely
+block the transmission of US. As you can see from Fig. 3,
+there are a few bright lines below the probe curve, which are
+the signal of the squeezed tissue and it is identical to the vivo
+US image.
+Fig. 2. This figure illustrates the pressure simulation algorithm. As an enlarged
+part of the dashed box in the figure, the yellow arc represents the shape of
+the probe, and the area between the yellow arc and the green arc is the target
+space being squeezed into.
+b) Restricted Movement Space of Probe: As mentioned
+above, the probe cannot move in space arbitrarily. To automate
+the acquired US images, two types of probe moving regulation
+are designed in this paper using spinal US as an instance.
+Define M as a mesh in 3d space, clipping a 3D sub-mesh Mr
+
+4
+Fig. 3.
+Figure (a) shows the CT images after being processed by the pressure simulation, the skin and muscles are compressed into the probe’s curved
+surface. Figure (b) shows the virtual US image produced with simulated extrusion. Figure (c) shows the reflection map after extrusion. Figure (d) shows that
+the processed image will not affect the propagation.
+Fig. 4. The green part of the figure is the object to be scanned. Figure (a) shows the relationship between the entire feasible space (skin) and the scanned
+object(spine). Figure (b) shows the first type of movement, with the yellow dot representing the tip of the probe and the arrow representing the two-degree
+freedom of movement in the manifold. Figure (c) shows the second type of movement, the red line represents the discrete surface, the dashed arrow represents
+the probe can switch on different curves, and the arrow represents the probe can be moved along the curve direction.
+of interest by the bounding box of the scanning target, the
+movement of the probe is only meaningful in this manifold.
+The first way is to move freely on the grid, for any point p
+on Mr, the direction of movement of p will be curved along
+the grid normal vector. In this case, a very small dx or dy
+will not make a great dz, which would not make the image
+change dramatically and could not acquire a continuous image.
+The second method is to slice Mr into a series of curves
+in 3D space along the forward direction Mr0...Mrn. In each
+polyline, the points can only move forward or backward along
+the tangent direction by a certain distance. It has the merit
+that keeping the components of the probe’s normal vector
+remain consistent across that curve, which is often used for
+automatic circular or flat scanning of a target and ensuring that
+the target being swept is on a line or a point. Besides, both of
+these movement methods can be employed by a reinforcement
+learning-based agent, and the first method is more suitable for
+continuous control, while the second method is more suitable
+for discrete control.
+c) Sound Reflection: The first step in the simulation is
+to map the CT-HU image into an acoustic impedance image,
+with the mapping look-up table references to [21]. In the
+mapping table, each HU value will be mapped to an acoustic
+impedance Z. The image of this acoustic impedance will then
+be processed instead of the CT image. Based on the acoustic
+physical transfer properties, we first introduce the Fresnel
+equation to calculate the reflection. Dividing sound waves
+into those parallel and those perpendicular to the plane, the
+equations are:
+�
+�
+�
+R⊥ = ( Z1cosθi−Z2cosθt
+Z1cosθi+Z2cosθt )2
+R∥ = ( Z2cosθi−Z1cosθt
+Z2cosθi+Z1cosθt )2
+R = 1
+2(R⊥ + R∥)
+.
+(2)
+When sound waves transmitted from one medium to an-
+other, it is not only be reflected but also be refracted. This
+refracted has the same physical properties as the refracted of
+light passing through different media which is given by Snell’s
+law:
+sinθ1
+sinθ2
+= Z2
+Z1
+(3)
+In fact, when a sound wave is refracted or bent, the direction
+will be kept until the next transmission. It is possible to
+simulate multi-level refraction and reflection values using a
+
+5
+recursive ray-tracing algorithm However, since the amplitude
+of the refracted sound waves is small and has little effect on
+the generated image quality, the proposed method does not
+consider its direction after refraction and only superimposes
+the amplitude on the incident sound wave of the next medium.
+Irradiance: Lambert’s cosine theorem is introduced to take
+the irradiance of the sound wave should into consideration.
+This equation means that the more perpendicular the direction
+of the tissue gradient and the sound wave, the stronger the
+sound reflection and the brighter the tissue.
+Ir
+Ii
+= cosθ
+(4)
+d) Sound Attenuation: Attenuation in the propagation of
+US passing through tissue could be broadly divided into re-
+flection, scattering, and absorption, where absorption accounts
+for the majority of it. The paper [22]states that US absorption
+of substances such as bovine and porcine livers accounts for
+90% or even 100% of US propagation attenuation. Even in
+the tissues that exhibit anisotropy such as the bovine brain
+and leg muscle absorption still makes a major contribution
+to the attenuation. Therefore, mapping the CT scan to an
+absorption image is important. The mapping table is obtained
+by interpolating the dataI in the paper [23]. The formula for
+absorption is given by the Lambert-Beer law. Since different
+tissues may not have the same US absorption capacity in
+the same Hu value, in order to get a better result from the
+generated image, an adjustable parameter α is added to the
+original equation. The modified equation is as follows:
+Ia = I010−α∗d∗f∗
+1
+10∗β
+(5)
+Tissues
+Density
+Velocity
+Impedance
+Attenuation
+Skin
+1100
+1631
+1.794
+0.22
+Fat
+916
+1435
+1.352
+0.975
+Muscle
+1041
+1595
+1.647
+1.47
+TABLE I
+ACOUSTICS PROPERTIES OF DIFFERENT TISSUES. WITH DENSITY(KG
+m−3), VELOCITY(ms−1), IMPEDANCE 106(kgm−2s−1),
+ATTENUATION(dBcm−1MHz−n) [23]
+Fig. 5. Illustration of propagation.
+e) Propagation in Discrete Field: The propagation of
+the US is the most time-consuming part of the generation
+process. Particularly, the steps of finding receiver tissues of
+reflection and refraction would take a lot of time in detecting
+intersections that occur frequently.
+At the same time, the direction of the adjacent pixels is
+discrete, whereas the direction of acoustic transmission is
+continuous. So it may happen that the line segments intersect
+while the pixels do not have an intersection. Each pixel among
+the eight neighborhoods should be regarded as the obstacle of
+the wave propagation and the normalized result of the cosine
+value of the direction of propagation and the direction of the
+pixel connection is used as the weight of the obstacle. That
+is, if the connection direction is in the opposite direction to
+the direction of propagation, the weights will be zero and the
+obstacle weight is maximized when the directions are exactly
+the same.
+In order to calculate the amplitude enhancement due to
+reflection and refraction easily, each point also calculates
+its wave propagation to the neighborhood. In this case, a
+negative cosine value of the pixel connection direction and
+the propagation direction will consider as reflection. Cosine
+values less than 0.8 (a hyper-parameter) will be considered
+as contrast enhancement by refraction and scattering. The
+schematic diagram of propagation is shown in 1
+f) Reflection Enhancement:
+When the transducer re-
+ceives the reflected sound waves from the medium gap, a
+bright stripe appears in the US image. Because the reflection
+only occurs when the properties of the medium change, there
+should be only one reflection between the bone surface and
+the muscle tissue, but the US image often shows a very thick
+bright stripe between bone and muscles. hacihaliloglu et al.
+[24] propose that this bright stripe is produced by the thickness
+of the US in the elevational direction. Based on this theory, this
+paper sampled 3d gradient textures in front of and behind the
+current plane (As the blue line in figure 6). These textures are
+multiplied by a weight value α which is inversely proportional
+to the distance and then accumulated into the current generated
+US image. Simulating thick stripes is significant for neural
+network training; if the thick stripes are not present in the
+training data, the output of the model will deviate from the
+ground truth when segmenting the bone surface on the real
+US data.
+In summary, the generated US image is obtained by cal-
+culating the propagation image with the reflection image and
+the absorption image and then blending these three images
+together with the radial noise.
+B. US Image Segmentation
+In this work, we propose a lightweight segmentation frame-
+work for US segmentation, as shown in Fig. 7. We follow
+[25] as the backbone. After each MobileViT block, we add
+an LCLM for further long-range contrast learning. We then
+utilize a feature pyramid network (FPN) to recover the initial
+resolution of the input US image.
+a) MobileViT block: To cover the global information,
+MobileViT block first utilizes a 3×3 convolution to aggregate
+the neighboring features. The pixels are then grouped into
+patches, each patch contains h × w pixels in which h and w
+represent the height and the width of the patch. The ith token
+
+6
+Fig. 6. The thick stripe is indicated by the arrow in Figure a). Figure b) shows how the thickness of the US waves induces the thick stripes artifact. Figure
+c) is a schematic of sampling in a 3d gradient texture. The Blue line indicates the imaging plane. The reflection value between the blue line and the yellow
+line is multiplied and accumulated into the imaging plane. Figure d) shows the simulated thick stripes in the system.
+Fig. 7. The framework of the proposed lightweight vision transformer for US segmentation. For the first two stages, we only utilize basic convolution layers to
+learn local representations. In the last three stages, we implement MobileViT block and LCLM alternately to cover both long-range dependency and long-range
+contrast.
+of each patch transverses its feature from each other by SA.
+When h <= 3 ∧ w <= 3, each pixel can access information
+from all pixels. Details of the MobileViT block could be found
+in [25].
+b) Why is LCLM needed: We find that the region of
+interest (RoI) in US images has similar textures, but is weak in
+amplitude to other tissues. Basic 3×3 convolutions are capable
+of modeling the neighboring texture but fail to learn long-range
+textures. As the local texture of the to-be-segmented region
+is similar to the other tissues, we have to design a module
+that covers the texture within and outside the to-be-segmented
+region.
+In vision transformers, long-range texture modeling is
+achieved by self-attention (SA) of all tokens as in Equ. 6:
+�x = softmax( xWq(xWk)T
+d
+)(xWv),
+(6)
+in which x ∈ RN×d represents the N input tokens with
+dimension d, Wq ∈ Rd×dq, Wk ∈ Rd×dq, and Wv ∈ Rd×dv
+are learnable parameters. We rewrite the self-attention to the
+formula as in Equ. 7:
+�
+�
+�
+�
+�
+�xi = �
+j∈A⟩
+wj(xjWv)
+wj =
+exp(xiWq(xjWk)T /d)
+�
+t
+exp(xiWq(xtWk)T /d)
+.
+(7)
+where Ai represents the set of accessible tokens of token xi.
+As Wq and Wk only attribute in modeling the relationship
+of queries, once Wv is fixed, the space generated by x is
+therefore limited. Meanwhile, as shown in Fig. 8(a), SA
+tends to aggregate “similar” features from the tokens, which
+helps the model to learn long-range dependency. However,
+because of the nature of US images, long-range contrast is
+also important to be learned, but SA fails to do so.
+Convolution operation could be formulated in the same
+formulation as SA, as shown in Equ. 8:
+�xi = �
+j∈Ai
+λj(xjWv) .
+(8)
+In this case, if rank(xjWv) = d, the space generated by x is
+Rd. We randomize 1000 different pairs of Wq and Wv, 1000
+different λ and fixed Wv. The outputs are shown in Fig. 8. As
+demonstrated in the figure, the outputs of convolution fulfill
+the space, while the outputs of SA are gathered.
+
+C+w
+<
+Transducer
+R=
+krk
+k=c-wS
+MViT
+1/2
+1/2
+MViT
+LCLM
+Up
+Up7
+Fig. 8. The possible outputs (dots in red) of self-attention (a) and convolution
+(b) with fixed Wv.
+A recent work [26] also demonstrates that CNNs with
+super large kernels learn feature representations from the
+super large receptive field (sub-global), and achieve compara-
+ble performance with state-of-the-art transformers with fewer
+parameters.
+As aforementioned, we need a module to gather long-range
+textures. As SA fails to learn long-range contrast, and a super-
+large convolution kernel leads to large computational cost,
+we need to design a light yet effective convolution module
+for long-range contrast learning, namely Long-range Contrast
+Learning Module (LCLM).
+c) Long-range contrast learning with dilation convo-
+lution: To cover long-range contrast with fewer parameters,
+a simple way is to utilize dilation convolution. In this work,
+we adopt 3 cascaded dilation convolution layers with dila-
+tion {3,5,11}. Along with one basic convolution layer which
+appears at the end of the MobileViT block, LCLM covers
+up to 41 × 41 pixels densely. We show the receptive field in
+Fig. 9. Each convolution layer is followed by a normalization
+layer and an activation layer. A depth-wise LCLM only needs
+3×3×3×d parameters, which is even smaller than a standard
+convolution layer (3 × 3 × d × d, d >> 3).
+Fig. 9. The receptive field of the pixel in red. The pixels painted not in grey
+are covered by the receptive field.
+IV. EXPERIMENTS
+A. Realistic US Simulation from CT Volume
+It is difficult to provide quantitative evaluation criteria for
+image generation-related works. Qualitatively, the advantages
+of this paper over deep learning-based US image generation
+are patient-specific and more accurate lesion boundaries, and
+when compared with previous papers based on image gener-
+ation, we make the system produce more realistic images by
+simulating squeezing and scattering in soft tissues and bones.
+Nonetheless, the quantitative evaluation is also of significance,
+and since the quality of image generation is hard to be
+evaluated quantitatively, we evaluate the quality of image
+generation from the purpose of the work. Except for training
+doctors to do US examinations, the most important uses of
+the US image generation algorithms are intra-operative reg-
+istration and providing datasets for deep learning. Therefore,
+we put the generated images into various deep learning-based
+segmentation algorithms to verify the usefulness of this image
+generation system for the pre-training of neural networks.
+B. Deep Learning-Based Segmentation
+a) Implementation Details: To validate the proposed
+US simulation method and the lightweight segmentation
+model, we train the proposed model on synthetic US images
+and inference on real US images. We set the batch size to 32,
+utilize the Adam optimizer, and set the learning rate to 1e-4.
+Fig. 10.
+The validation image examples. Figure (a) is the US image of a
+spinal phantom in gel mimic, Figure (b) is the US image of a 3D-printed
+spine in water, and Figure (c) is the US image of a clinically human body.
+b) Comparison with other models: We first compare
+the proposed model with other US segmentation models
+in Tab. II. All performances reported in the table are re-
+implemented by ourselves. Compared with UNet-based mod-
+els, the proposed model is much smaller and more effective.
+Compared with MViT-FPN, the proposed approach achieves
+much better performance.
+To utilize the model, in this case, the segmentation is to
+achieve better registration, therefore we remove the cases with
+obvious segmentation errors to obtain our (selected) result,
+which is used in the subsequent registration. In this case,
+the effect of the network trained with generated data can be
+equivalent to or better than the effect of the SOTA model
+trained by a large number of real US images with an accurately
+labeled. Our label is acquired by ct segmentation, which is
+more challenging for the convergence of lightweight neural
+networks since he can label bone surfaces which is invisible
+or ambiguous in some situation in US.
+c) Ablation study: To demonstrate the effectiveness of
+synthetic US over initial CT scans, long-range dependency,
+and long-range contrast, we validate the aforementioned set-
+tings and show the results in Tab. III. Training the model with
+CT scans leads to a dramatic IoU decrease, which demon-
+strates that the proposed method can effectively synthesize US
+images. Without LCLM, the model fails to learn the long-range
+contrast of US images and results in a performance decrease.
+Without MViT, the IoU slightly drops, which represents that
+
+query
+key
+output
+(b)8
+Method
+Dice
+CD(TP)
+CD(FN)
+#Parm
+FPS
+UNet [17]
+0.574
+1.085mm
+0.590mm
+131.8M
+4.13
+MViT-FPN [25]
+0.294
+4.664mm
+0.824mm
+20.0M
+28.1
+CNLUnet [19]
+0.329
+2.868mm
+1.287mm
+48.6M
+10.9
+Ours
+0.783
+0.599mm
+1.079mm
+20.0M
+26.3
+Ours(Selected)
+0.926
+0.227mm
+0.184mm
+20.0M
+26.3
+TABLE II
+COMPARISON WITH OTHER SEGMENTATION MODELS. NOTE THAT ALL
+PERFORMANCES REPORTED IN THE TABLE ARE RE-IMPLEMENTED BY
+OURSELVES.
+the model benefits from long-range dependency, but is less
+significant compared with long-range contrast.
+Method
+Baseline
+SUS→CT
+w/o MViT
+w/o LCLM
+Dice
+0.783
+0.040
+0.438
+0.294
+CD(TP)
+0.599mm
+6.062mm
+0.910mm
+4.664mm
+CD(FN)
+1.079mm
+3.938mm
+0.674mm
+0.824mm
+TABLE III
+THE ABLATION STUDY OF THE PROPOSED MODEL. “SUS” REPRESENTS
+THE SYNTHETIC US. “SUS→CT”, “W/O MVIT” AND “W/O LCLM”
+REPRESENT TRAINING THE MODEL BY CT SCANS, REMOVING MVIT
+FROM THE BASELINE AND REMOVING LCLM FROM THE BASELINE,
+RESPECTIVELY.
+Fig. 11.
+The result of segmentation. Figure(a), (b), and (c) presents the
+segmented label, the output of the network, and the ground truth respectively.
+C. Validation of US-CT Registration
+a) Implementation Details: The US simulation system
+is proposed to better obtain labeled data from multiple poses,
+patients, and environments, enabling US-based surgical navi-
+gation to be accomplished. In this paper, we validated whether
+the output of the model trained on synthetic data can align
+the preoperative CT and the planned trajectory into the intra-
+operative phase, and the spinal bone surface registration for
+pedicle screw placement is the instance.
+The validation is divided into two parts, the first is to
+validate the accuracy of the rigid body registration, i.e. trans-
+lation and rotation. The second part is to validate whether
+the pre-operatively planned trajectory with the aforementioned
+transforms will cause complications for the surgery or not, i.e.
+whether the intra-operative trajectory will touch vital organs.
+The experiments were performed on three phantoms, a spine
+phantom in water, a 3D-printed human spine in water, and a
+bovine spine in agar gel. The registration steps were divided
+into a coarse alignment and an individual registration for
+each segment. The coarse registration used all point clouds
+to calculate the approximate position and orientation, and the
+final registration used a de-noised Iterative Closest Point(ICP)
+method. The results of the registration are shown in the table
+below.
+Fig. 12. The phantom model for registration validation.
+Fig. 13. The MSE loss of registration in three dimensions.
+Fig. 14.
+The Translation and Rotation of pedicle screw with registration
+matrix. The rotation is presented as an angle in degrees.
+b) Validation of intra-operative model registration: In
+the registration phase, the model inputs the US image outputs
+the segmentation label, and extracts the upper contour of its
+maximum connectivity, hence the point cloud is created with
+the calibration matrix of the US probe. The calibration of the
+US is done by two cross-wire phantom [27], with 0.97mm
+MSE(min square loss) loss. Since the coordinate system of
+
+3
+Translation in three dimension
+X
+1
+y
+0
+7
+-1
+Li
+L2
+L3
+L4
+L5
+TheLumbarsegment3
+V
+1
+Z
+0
+-1
+Angle
+-2
+Li
+L2
+L3
+L4
+L5
+The Lumbar segment9
+each segment does not coincide, the rotation evaluation in
+angular will be represented by the rotation of pedicle screws in
+the next part. To evaluate the error of registration, the point-
+to-point MSE is used, which is also the objective function
+of the ICP algorithm. The result of registration is shown in
+Fig. 15, which shows the ground truth in the intra-operative
+space(white model) and the results of registration with US
+(green model). In summary, the segmentation algorithm trained
+by the generated US image works in the registration with the
+error 0.133 ∼ 3.366mm. The maximum appears in the z-axis
+of L5, which is around 3 mm, which is probably caused by
+the shape of L5 being more different from the shape of other
+lumbar.
+c) Validation of Pedicle Screw Placement Feasibility: To
+verify the registration effectiveness of the system in the target
+tasks, we compared the translation and rotation of the tip of the
+screw with the US point cloud registration and ground truth
+using an expert-planned intra-operative plan for pedicle screw
+placement. The rotation is determined by the angle between
+two pedicle axis vectors in 3D space which is expressed in
+degrees. As shown in Fig. 14, the error in translation for
+the tip of the screw ranges from 0.027 mm to 4.031 mm,
+with the maximum value still occurring on the z-axis of L5,
+which is consistent with the above assessment. The error in the
+rotation is between 3.05 degrees and 4.47 degrees, therefore,
+it can be concluded that the pre-trained model with generated
+US image can perform the alignment with small errors. The
+results of the pedicle screw placement are shown in figure 16,
+which shows this registration does not put the patient at risk
+of complications.
+Fig. 15. The visualization of registration. The green part of the figure is the
+registration result by US segmentation, and the white part is the ground truth.
+Cases derived from 3d printed patient spine, which is (b) in the figure 12.
+Fig. 16. The visualization of registration-based screw placement. The figure
+shows the posture of the screws with standard length and diameter which
+is planned on preoperative CT by an experienced surgeon, compared to the
+intra-operative spine ground truth by the registration matrix acquired by image
+segmentation. Cases derived from the standard spine phantom, which is (c)
+in the figure 12.
+V. CONCLUSION
+In this paper, we propose a US image simulation method
+based on CT images and acoustic properties that automatically
+generates a US simulation environment based on a specific
+patient CT volume, which allows US-based deep learning and
+reinforcement learning algorithms to be trained and validated.
+In addition, we propose a lightweight vision transformer for
+segmenting US images which is a network structure with better
+segmentation accuracy and modal generalization capabilities.
+Experiments show that the model trained using US im-
+ages generated by the realistic US simulation from the CT
+system achieves higher segmentation accuracy and has the
+capability to perform the intraoperative registration process
+without complications compared to the model trained with
+CT images. Compared to other segmentation algorithms, the
+proposed transformer has real-time computational efficiency,
+better segmentation accuracy, and generalization capability,
+which is particularly important in US surgical applications.
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+Traffic Steering for 5G Multi-RAT Deployments
+using Deep Reinforcement Learning
+Md Arafat Habib1, Hao Zhou1, Pedro Enrique Iturria-Rivera1, Medhat Elsayed2, Majid Bavand2,
+Raimundas Gaigalas2, Steve Furr2 and Melike Erol-Kantarci1, Senior Member, IEEE
+1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada
+2Ericsson Inc., Ottawa, Canada
+Emails:{mhabi050, hzhou098, pitur008, melike.erolkantarci}@uottawa.ca,
+{medhat.elsayed, majid.bavand, raimundas.gaigalas, steve.furr}@ericsson.com
+Abstract—In 5G non-standalone mode, traffic steering is a
+critical technique to take full advantage of 5G new radio while
+optimizing dual connectivity of 5G and LTE networks in multiple
+radio access technology (RAT). An intelligent traffic steering
+mechanism can play an important role to maintain seamless
+user experience by choosing appropriate RAT (5G or LTE)
+dynamically for a specific user traffic flow with certain QoS
+requirements. In this paper, we propose a novel traffic steering
+mechanism based on Deep Q-learning that can automate traffic
+steering decisions in a dynamic environment having multiple
+RATs, and maintain diverse QoS requirements for different
+traffic classes. The proposed method is compared with two
+baseline algorithms: a heuristic-based algorithm and Q-learning-
+based traffic steering. Compared to the Q-learning and heuristic
+baselines, our results show that the proposed algorithm achieves
+better performance in terms of 6% and 10% higher average
+system throughput, and 23% and 33% lower network delay,
+respectively.
+Index Terms—Multi-RAT, traffic steering, reinforcement learn-
+ing
+I. INTRODUCTION
+The dual connectivity between long term evolution (LTE)
+and fifth generation new radio (5G NR) results in multiple
+radio access technologies (multi-RAT) [1], [2]. On the other
+hand, each type of RAT is supposed to have distinctive
+capabilities to serve user equipment (UE) with diverse quality-
+of-service (QoS) requirements. This raises the need of steering
+a specific class of traffic to a certain RAT to fulfill the QoS
+demands. For instance, high throughput video traffic can be
+better served by 5G NR. On the contrary, steering voice
+traffic to LTE base station (BS) with wider coverage can be a
+better decision since such traffic is not throughput hungry but
+requires more coverage to avoid frequent handovers. However,
+steering a specific class of traffic continuously to a certain
+RAT may cause several problems. The system may suffer from
+higher delay due to excessive load and reduced throughput
+because of the packet drops. These issues are quite challenging
+to address, especially when 5G NR facilitates dense network
+deployments and an increased number of users.
+To address the above-mentioned challenges, an AI-enabled
+traffic steering scheme emerges as a promising approach to
+manage densely deployed networks with dynamic require-
+ments. In recent years, AI and machine learning have been
+applied to various other problems in 5G [3]. Even though
+the emergence of the 5G non-stand-alone (NSA) mode has
+drawn the attention of researchers recently, most existing
+works linked with traffic steering lack a comprehensive tool
+to overcome the complexity.
+For instance, in [4], the authors propose a traffic steering
+scheme based on some threshold calculated using parameters
+like load at each type of RAT, channel condition, and service
+type but the method lacks the intelligence to handle dynamic
+wireless environments. Compared with conventional model-
+based optimization methods, machine learning, especially re-
+inforcement learning (RL) algorithms, can significantly reduce
+the complexity of defining a dedicated optimization model
+[5]. Advanced machine learning techniques like deep rein-
+forcement learning (DRL) [6] can not only automate traffic
+steering in a dynamic 5G wireless environment, but also it
+can handle larger state-action space compared to traditional
+reinforcement learning. Therefore, unlike previous works, we
+propose a DRL-based traffic steering scheme that tends to per-
+form RAT specific traffic steering in a multi-RAT environment
+to maintain QoS requirements of different traffic classes in a
+dynamic 5G NSA mode to maintain seamless network activity
+and smooth user experience.
+In this paper, we seek to balance the QoS demands of all the
+traffic classes simultaneously by proposing a Deep-Q-network
+(DQN)-based traffic steering scheme. The reward and state
+functions of the proposed DQN-based traffic steering scheme
+is carefully designed to have satisfactory performance based
+on two crucial key performance indicators (KPIs); i.e. network
+delay and average system throughput. Performance of the
+proposed method is compared with two baseline algorithms:
+Q-learning-based method [7] and a heuristic-based algorithm
+adopted from [4]. It gains 6% and 10% increase in average
+system throughput compared to the Q-learning and heuristic-
+based baseline respectively. Furthermore, it achieves 23% and
+33% decrease in network delay compared to the mentioned
+baselines.
+The rest of the paper is organized as follows: Section II
+presents the related works. We discuss the system model and
+the problem formulation in Section III. Section IV covers the
+proposed DQN-based traffic steering scheme along with the
+baselines. The performance evaluation of the proposed DQN-
+based traffic steering method is presented in Section V. Finally,
+the paper is concluded in Section VI.
+arXiv:2301.05316v1  [cs.NI]  12 Jan 2023
+
+II. RELATED WORKS
+In this section, we summarize the state-of-the-art literature
+on traffic steering. Prasad et al. propose a dynamic traffic
+steering scheme for energy efficient radio access network
+moderation in ultra-dense 5G networks [8]. A unified traffic
+steering scheme by Dryjanski et al. is proposed for LTE-
+advanced pro, aiming at optimal radio resource allocation in
+multi-RAT networks [9]. Most recently, Khaled et al. have
+proposed a cell zooming technique to steer traffic in a software
+defined radio-enabled LTE network that uses renewable energy
+sources to lessen on-grid power consumption [10]. Gijon et
+al. propose a data driven approach to perform traffic steering
+in multi-carrier LTE networks in which traffic steering is
+conducted based on reference signal received quality-based
+handover margins [11].
+Nevertheless, 5G deployments have made it more challeng-
+ing to develop an elegant traffic steering scheme because of
+the increased number of users and dual connectivity. Passas
+et al. propose a pricing oriented network selection process
+for distributed heterogeneous networks based on imposed load
+pressure at a particular RAT [12]. A heuristic-based approach
+proposed in [4] performs traffic steering based on a threshold
+level calculated using parameters like channel condition, load
+level at each RAT, and service type. Priscoli et al. address the
+problem of traffic steering using a Q-learning-based solution
+that aims at maintaining QoS, and performs load balancing in
+a 5G heterogeneous network [13]. Different from the previous
+works, this paper provides automation in the system via DRL-
+based traffic steering scheme that can perform RAT specific
+traffic steering in a multi-RAT environment. Furthermore, the
+proposed method can maintain QoS requirements of different
+traffic classes in a dynamic 5G NSA mode to maintain
+seamless network activity and smooth user experience.
+III. SYSTEM MODEL AND PROBLEM FORMULATION
+A. System Model
+In this work, a multi-RAT network is considered having
+Q classes of RATs where each class of RAT, q represents a
+particular access technology (LTE, 5G, etc.). Multiple users are
+associated with different types of RATs via dual connectivity.
+A UE can maintain K types of traffic classes. Fig. 1 presents
+the network model considered in this study. We represent
+three different classes of traffics: voice, gaming, and video
+as TC1, TC2, and TC3 respectively in the figure. We have
+designed our network environment in a way where small
+cells are within the range of a macro-cell. UEs have dual
+connectivity with LTE or 5G RAT and traffic can be steered
+to either one of these RATs based on our proposed method.
+The total downlink bandwidth, B in MHz is divided into
+NRB resource blocks. A resource block contains a set of
+12 contiguous subcarriers. Consecutive resource blocks are
+grouped to constitute resource block group (RBG) as defined
+in [3]. Each RBG, h is allocated a certain transmission power
+ph,b, by a BS, b. Based on our system model, each BS holds a
+number of transmission buffers corresponding to the number of
+Fig. 1.
+Illustration of network environment with one LTE macro cell and
+several 5G small cells.
+users connected to it. Every transmission time interval (TTI),
+the downlink scheduler assigns resources to the users having
+pending data transmissions.
+The link capacity between the UE, u and BS, b can be
+formulated as follows:
+Cu,b =
+H
+�
+h=1
+ωh log2
+�
+1 +
+ph,bxh,u,bgh,u,b
+ωhN0 + �
+m∈B ph,mxh,u,mgh,u,m
+�
+,
+(1)
+where ωh is the bandwidth of the h, ph,b is the transmit power
+of the BS, b on h, gh,u,b is the channel co-efficient and xh,u,b
+is the RBG’s allocation indicator of the link (h, u, b). N0 is
+the additive white Gaussian noise single-sided power spectral
+density. ph,m is the transmit power of the interfering BS, m,
+gh,u,m is the channel co-efficient, and xh,u,m is the allocation
+indicator of link (h, u, m).
+Each link has a capacity limit. Traffic flows passing through
+a link should not exceed the capacity of the link in the system.
+�
+f∈F
+dfxf
+u,b ⩽ Cu,b
+∀(u, b) ∈ L,
+(2)
+where F is the set of all the flows in the network, df is the
+capacity demand of the flow f ∈ F from UE, u to BS b. xf
+u,b
+represents a binary (0, 1) component that is ‘1’ if the link
+(u, b) has been used from UE,u to BS b. It is ‘0’ otherwise.
+L is the set of links and Cu,b is the capacity of link (u, b). as
+presented in eq. (1)
+In our system model, the delay is considered as the summa-
+tion of transmission and queuing delay which is as follows:
+Dk,b = DT rx
+k,b + Dq
+k,b,
+(3)
+where DT rx
+k,b
+is the transmission delay experienced for a
+particular traffic type k and BS b, and Dq
+k,b is the queuing
+delay experienced for a particular traffic type k at BS b for a
+user u. The transmission delay can be calculated as follows:
+DT rx
+k,b = Lu,b
+Cu,b
+,
+(4)
+where Lu,b is the packet length and Cu,b is the link capacity
+as stated in eq. (1).
+
+MBS
+ SBS“
+UE -.- TC1
+TC2
+TC3B. QoS Requirements and Problem Formulation
+To be able to perform traffic steering for different traffic
+classes with QoS requirements for delay and throughput, first
+two parameters are defined based on delay and throughput. The
+delay parameter associated with our traffic steering problem
+is considered as the ratio of the defined QoS requirement for
+delay and the actual delay experienced in the system for a
+particular traffic class being carried by a certain BS. It can be
+stated as follows:
+rD
+k,b = DQoS
+Dk,b
+,
+(5)
+where DQoS is delay requirement defined in the simulation for
+a particular traffic type and Dk,b is the actual delay achieved.
+Similarly, the throughput parameter is defined as the ratio
+of actual throughput achieved and the required throughput as
+stated in eq. (6):
+rT
+k,b = Tk,b
+TQoS
+,
+(6)
+where TQoS is the throughput requirement defined in the
+simulation for a particular traffic class and Tk,b is the actual
+throughput achieved.
+Since our aim is to improve the system performance in
+terms of the delay and throughput, a new variable is formed
+to represent and meet such targets. It combines the delay and
+throughput parameters in eq. (5) and (6) along with some
+weight factors. The declared variable combined with delay,
+throughput, and weight factors (w1 and w2) is as follows:
+M = w1(rD
+k,b) + w2(rT
+k,b).
+(7)
+The traffic steering problem proposed in this paper is formu-
+lated as the maximization of the variable M (presented in eq.
+(7)) which is as follows:
+max
+�
+u∈U
+�
+k∈K
+�
+b∈B
+Mu,f,b,
+s.t.
+�
+(u,b)∈L
+βfk ⩾ βf
+∀f ∈ F,
+�
+(u,b)∈L
+D(u, b)xf
+u,b ⩽ Df
+∀f ∈ F,
+(8)
+where βfk is the required bitrate for a particular type of traffic
+k, and βf is the available bitrate. Also, Df represents the
+latency demand of flow f ∈ F and D(u, b) is the latency of
+link (u, b).
+IV. PROPOSED DQN-BASED TRAFFIC STEERING SCHEME
+A. DQN-based Traffic Steering Scheme
+For a relatively simplistic RL environment, Q-learning is
+a good solution for optimization. However, as the state-
+space increases, the time needed to traverse all these states
+and iteratively update all the Q-values will increase which
+is computationally inefficient and resource consuming. To
+address this issue, DQN can be used to estimate the Q-values
+for each state-action pair in a given environment using a deep
+neural network (DNN) [6].
+During the training stage of DQN, agent’s experiences at
+each time step is stored in a data set called the replay memory.
+At time τ, the agent’s experience eτ is defined as the following
+tuple:
+eτ = (Sτ, Aτ, Rτ+1, Sτ+1).
+(9)
+The tuple contains the state of the environment, the action
+taken from the state, the reward given to the agent as a
+result of previous state-action pair and the next state of the
+environment. In short, the tuple gives us the summary of the
+agent’s experience at time τ. All the agent’s experiences at
+each time step over all the episodes played by the agent are
+stored in the replay memory. In practice, the replay memory
+is set to some finite size unit (N). Therefore, it will only
+store the last N experiences. The replay memory data set is
+the place from where random samples are chosen to train the
+network.
+The DNN in DQN takes states as inputs from the envi-
+ronment and outputs the Q-values for each action that can
+be taken from that state. Before the training starts, first, the
+replay memory data set, D is initialized to capacity, N. Next,
+DNN is initialized with random weights. For each episode,
+the starting state is initialized. For each time step within the
+episode, the agent either explores the environment and selects
+a random action or the agent exploits the environment and
+selects the greedy action for the given state that provides the
+highest Q-value. This epsilon greedy policy is used to balance
+the exploration and exploitation.
+Aτ =
+�
+random
+action,
+if rand ⩽ ϵ
+argmax(qτ(Sτ, Aτ)),
+otherwise
+(10)
+where ϵ is the exploration probability within 0 ⩽ ϵ ⩽ 1 and
+rand represents a random number between 0 to 1.
+After an action is taken, we observe the reward for the action
+along with the next state of the environment. Therefore, the
+state an agent initialized from, action taken, reward observed
+are all put together in a tuple as described in eq. (9).
+For a single sample, the first pass to the network occurs
+for the state from the experience tuple that was sampled.
+The network then outputs the Q-values associated with each
+possible action that can be taken from that state and then the
+loss is calculated between the Q-values for the action from
+the experience tuple and the target Q-value for this action. To
+calculate the target Q-value, it is required to have a second pass
+to the target network with the next state. The target network
+is the clone of the policy network (which is also the main
+network). Its weights are frozen with the weights same as
+the policy network and the weights are updated in the target
+network after every certain amount of time steps. The loss for
+DQN is calculated using the following equation:
+L(w) = Er(Rτ + γ max
+A q(Sτ+1, A, w′) − q(Sτ, Aτ, w)),
+(11)
+where w and w′ are the weights of the main and the target
+network, and Er represents the error function. Having two
+NNs (main and target) ensures stability.
+
+Fig. 2 describes the schematic of the proposed DQN-based
+traffic steering where we have a main network and a target
+network and minibatch from the replay memory is getting
+fetched.
+Fig. 2. Overall system architecture with DQN.
+The mathematical formulation of DQN depends on Markov
+Decision Process (MDP) that is defined by agents, states,
+actions, and a reward function. Tuples associated with DQN
+is defined as follows:
+• Agent: We implement a centralized agent to control the
+macro base station (MBS) and the small cell base stations.
+It is deployed in the MBS and controls all the incoming
+traffic to each BS.
+• State:
+The
+state
+consists
+of
+three
+elements,
+{Tf, LQ(SINR), qL}. Here, Tf
+represents the traffic
+type. It is assumed that each traffic type has fixed
+QoS
+requirements
+and
+we
+can
+perform
+traffic
+steering to a particular RAT based on that. Users
+periodically
+report
+signal-to-interference
+and
+noise
+ratio (SINR) measurements to the 5G base station
+(gNB) and LTE base station (eNB). It indicates the
+quality of the link associated with a UE and a BS.
+Therefore,
+the
+second
+element
+of
+state
+space
+is:
+LQ(SINR)={SINReNB, SINRgNB}. To represent load
+level, queue length of both types of RATs is used. So,
+the last element of the state space is queue length,
+qL={qL(gNB), qL(eNB)}.
+• Action: The action space contains the action of flow
+admission to the RATs. It is defined as: {ALT E, A5G}.
+Here, (ALT E) stands for flow admission to the LTE RAT
+, and (A5G) stands for flow admission to the 5G RAT.
+• Reward: The reward function is based on eq. (7). To
+keep it normalized, sigmoid function is used. Therefore,
+the reward function is as follows:
+R = sigm(M),
+(12)
+where sigm(M) represents the sigmoid function.
+The proposed DQN-based traffic steering algorithm is sum-
+marized as Algorithm 1.
+Algorithm 1 DQN-based traffic steering
+Initialize: Network and DQN parameters
+1: for TTI = 1
+to
+T do
+2:
+for every u, b, k do
+3:
+if (rand ≤ ϵ) then
+4:
+choose action randomly
+5:
+else
+6:
+select Aτ using greedy policy
+7:
+end if
+8:
+BSs are selected for all the UEs for all k ∈ K
+9:
+Traffic admission is performed
+10:
+Reward calculation based on eq. (12)
+11:
+Agent updates its own state Sτ
+12:
+Save (Sτ, Aτ, Rτ+1, Sτ+1)
+13:
+end for
+14:
+Random sample a minibatch from the experience pool
+15:
+Generate target Q-values, qτ(Sτ, Aτ)
+16:
+Update w using gradient descent to minimize the loss,
+L(w) = Er(qτ(Sτ, Aτ) − q(Sτ, Aτ, w))
+17:
+Copy w to w′ after several training
+18: end for
+19: Output: Optimal traffic steering decisions from TTI =
+1
+to
+T
+B. Baseline Algorithms
+In this section, two baseline algorithms are introduced that
+have been used for the performance comparison. The first
+baseline algorithm for RAT selection is based on a predefined
+threshold [4]. This is called the heuristic baseline. Here, the
+threshold is calculated for each UE based on the metrics like
+load at eNB (le) and gNB (lg), channel condition of a user
+under LTE (che,u) and 5G BS (chg,u), service type of a user
+(Su). The channel condition is determined to be good or bad
+considering a threshold of received SINR values. Similarly, the
+load at each RAT is determined based on a threshold value.
+Based on the mentioned metrics, a value Tu is calculated that
+is used for selecting the RAT for a UE after comparing it with
+a predetermined threshold (Tth). Following equation is used
+to calculate the value for Tu:
+Tu(le, lg, che,u, Su) = αle + βlg + γchg,u + δSu,
+(13)
+where α, β, γ, and δ are the weights associated with consid-
+ered parameters that can be modulated based on the impact of
+any certain metric on system performance. Tth is set to be the
+mean of all the possible values of Tu. The decision of steering
+traffic to a particular RAT is taken the following way:
+Ru =
+�
+1, Tu > Tth
+(1 represents gNB)
+0, Tu ⩽ Tth
+(0 represents eNB).
+(14)
+The Q-learning algorithm has been used as another baseline
+in this work [7]. The goal is to investigate how DQN performs
+against the Q-learning algorithm.
+
+Main network
+Hidden layers
+Action
+selector
+LTE
+()
+Hidden layers
+lnput
+5G NR
+Observation
+Target network
+Reward: Throughput,
+delay
+Minibatch from
+New state: Load level
+experience poolV. PERFORMANCE EVALUATION
+A. Simulation setup
+We have conducted MATLAB based simulations consider-
+ing 1 eNB and 4 gNBs with 30 users in total. There are a total
+of 1 macro-cell and 4 small cells facilitated by the gNBs and an
+eNB. A macro-cell and a small-cell have carrier frequencies of
+3.5 GHz and 0.8 GHz respectively. Specifications of the traffic
+classes used in this study have been summarized in TABLE I.
+For the experimental results, the load has been varied between
+5-10 Mbps. Proportion of the voice, video, and gaming traffic
+is 20%, 50%, and 30% respectively. Higher proportion of
+the video traffic is deliberately considered to observe how
+the system performs with the higher throughput requirements.
+Also, gaming traffic has the most stringent delay requirement
+and we wanted to see if the system performs well enough
+to meet such precise requirement. Therefore it has a higher
+percentage compared to the voice traffic. QoS requirements
+associated with delay and throughput for the three types of
+traffic classes are specified based on the existing literature [14]
+and 3GPP specifications (see TABLE I.). We are using multi-
+RAT dual connectivity architecture, an NSA mode where LTE
+and 5G NR BSs serve together. An architecture specified in
+[15] has been used where the dual connectivity is ensured
+via evolved packet core [16]. Transmission power of the LTE
+BS and 5G NR BSs are set to 40W and 20W. Furthermore,
+bandwidth for the LTE and 5G RAT are fixed to 10MHz and
+20MHz.
+TABLE I
+TRAFFIC CLASS DESCRIPTION AND SIMULATION SETTINGS
+Traffic class specification
+Values
+Traffic model
+Poisson distribution, video
+and gaming traffic [14]
+Voice traffic
+Packet size
+30 bytes
+TQoS, DQoS
+0.1 Mbps , 100ms
+Proportion of the traffic
+20%
+Video traffic
+Packet size
+250 bytes
+TQoS, DQoS
+10 Mbps, 80ms
+Proportion of the traffic
+50%
+Gaming traffic
+Packet size (gaming traffic)
+120 bytes
+TQoS, DQoS
+5 Mbps, 40ms
+Proportion of the traffic
+30%
+B. Simulation results
+The performance of the proposed algorithm is evaluated in
+terms of two KPIs: Average system throughput and network
+delay. In Fig 3, we present a comparison in terms of system
+throughput under different user loads. The proposed DQN
+outperforms heuristic and Q-Learning baselines by gaining 6%
+and 10% increased throughput, respectively.
+Fig. 4 presents the performance comparison of the proposed
+DQN-based traffic steering method with the other baselines in
+terms of delay. The DQN-based method achieves 23% and
+Fig. 3. System throughput against traffic load.
+Fig. 4. System delay against traffic load.
+33% decrease in network delay compared to the baselines.
+Note that, the proposed method and the Q-learning, both have
+a reward function formulated based on throughput and delay.
+Whenever high delay is experienced for steering traffic to a
+particular RAT, the system learns. That is why, both of them
+have better performance compared to the heuristic baseline.
+In Fig. 4, delay is calculated considering all the traffic classes
+together at each load.
+It should be mentioned that the main reason of the improved
+performance of the proposed method is the use of DQN,
+that outperforms Q-learning in terms of exploration efficiency
+and achieves higher average reward. Q-learning suffers due to
+longer exploration period and gets lower average reward since
+it does not have a DNN as an approximator which compels
+the agent to cover larger state and action space.
+In this work, we also want to steer a particular type of
+traffic to a specific RAT. For example, steering the voice
+traffic constantly to a gNB is a waste of resources since the
+throughput requirement is not that high for such traffic. Fig. 5
+is presented which shows what percentage of a traffic class is
+processed by a particular RAT and when the traffic gets steered
+due to higher load. In Fig. 5(a), it is observed that most of
+the voice traffic is processed by the eNB, however, a small
+portion of the traffic is processed by the gNB too whenever
+the system experiences higher load. For the video and gaming
+traffic, it is observed that most of the traffic is processed by
+the gNB.
+
+300
+-DQN
+--Q-learning
+-Heuristic baseline
+250
+200
+150
+5
+6
+7
+8
+9
+10
+Load per user (Mbps)20
+-Heuristic baseline
+-Q-learning
++DQN
+15
+(ms)
+Delay (
+10
+5
+0
+5
+6
+7
+8
+9
+10
+Load per user (Mbps)Fig. 5. Data processing percentage for different traffic types.
+Fig. 6. Traffic steered to other RAT as load changed.
+Lastly, Fig. 6 demonstrates how traffic steering occurs
+whenever a high load is experienced in a BS with a particular
+RAT. We start with one UE at the 300th time slot and increase
+the number of UEs in a small cell up to six for different traffic
+classes. The variable L, in the respective figure represents load
+in terms of queue length. At the 1800th time slot, it can be
+seen that four among six UEs are steering different types of
+traffic to the 5G NR BS. This results in higher load and we can
+see that the third and fourth UEs are experiencing high load
+(value of L changed from 0 to 1). So, in the next observed
+time slot, these two UEs steer the traffic to the eNB. In the
+2100th time slot, we can see four UEs steering voice, video,
+and gaming traffic to the only eNB in our system. This incurs
+high load at eNB and in the next observed slot we can see
+that the sixth UE has switched its traffic to the gNB.
+VI. CONCLUSIONS
+In this study, we have proposed a novel method that can
+perform RAT specific and QoS aware traffic steering using
+DQN. It gains 6% and 10% increase in average system
+throughput compared to the Q-learning and heuristic-based
+baseline respectively. Moreover, it achieves 23% and 33%
+times decrease in network delay compared to the baselines.
+Apart from the better performance in terms of the KPIs, the
+proposed method can perform RAT specific traffic steering
+ensuring efficient use of network resources. Lastly, the pro-
+posed DQN-based traffic steering can successfully perform
+load balancing in an optimal way as whenever high load is
+induced to a particular RAT, traffic is steered to another RAT
+dynamically.
+ACKNOWLEDGEMENT
+This work has been supported by MITACS and Ericsson
+Canada, and NSERC Collaborative Research and Training
+Experience Program (CREATE) under Grant 497981.
+REFERENCES
+[1] V. Ramaswamy, J. T. Correia, and D. Swain-Walsh, “Modeling and
+Analysis of Multi-RAT Dual Connectivity Operations in 5G Networks,”
+in 2019 IEEE 2nd 5G World Forum (5GWF), pp. 484–489.
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+S. Andreev, and Y. Koucheryavy, “Characterizing Throughput and
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+[6] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G.
+Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski
+et al., “Human-Level Control Through Deep Reinforcement Learning,”
+nature, vol. 518, no. 7540, pp. 529–533, 2015.
+[7] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction.
+MIT press, 2018.
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+RAN Moderation and Dynamic Traffic Steering in 5G,” in 2016 IEEE
+84th Vehicular Technology Conference (VTC-Fall), 2016, pp. 1–6.
+[9] M. Dryjanski and M. Szydelko, “A Unified Traffic Steering Framework
+for LTE Radio Access Network Coordination,” IEEE Communications
+Magazine, vol. 54, no. 7, pp. 84–92, 2016.
+[10] H. Khaled, I. Ahmad, D. Habibi, and Q. V. Phung, “A Green Traffic
+Steering Solution for Next Generation Communication Networks,” IEEE
+Transactions on Cognitive Communications and Networking, vol. 7,
+no. 1, pp. 222–238, 2020.
+[11] C. Gij´on, M. Toril, S. Luna-Ram´ırez, and M. L. Mar´ı-Altozano, “A Data-
+Driven Traffic Steering Algorithm for Optimizing User Experience in
+Multi-Tier LTE Networks,” IEEE Transactions on Vehicular Technology,
+vol. 68, no. 10, pp. 9414–9424, 2019.
+[12] V. Passas, V. Miliotis, N. Makris, and T. Korakis, “Pricing Based
+Distributed Traffic Allocation for 5G Heterogeneous Networks,” IEEE
+Transactions on Vehicular Technology, vol. 69, no. 10, pp. 12 111–
+12 123, 2020.
+[13] F. D. Priscoli, A. Giuseppi, F. Liberati, and A. Pietrabissa, “Traffic Steer-
+ing and Network Selection in 5G Networks Based on Reinforcement
+Learning,” in 2020 European Control Conference (ECC), 2020, pp. 595–
+601.
+[14] J. Navarro-Ortiz, P. Romero-Diaz, S. Sendra, P. Ameigeiras, J. J. Ramos-
+Munoz, and J. M. Lopez-Soler, “A Survey on 5G Usage Scenarios and
+Traffic Models,” IEEE Communications Surveys & Tutorials, vol. 22,
+no. 2, pp. 905–929, 2020.
+[15] P.
+Frenger
+and
+R.
+Tano.
+(2019)
+A
+Technical
+Look
+at
+5G
+Energy Consumption and Performance. [Online]. Available: https:
+//www.ericsson.com/en/blog/2019/9/energy-consumption-5{G}-nr
+[16] M. Agiwal, H. Kwon, S. Park, and H. Jin, “A Survey on 4G-5G Dual
+Connectivity: Road to 5G Implementation,” IEEE Access, vol. 9, pp.
+16 193–16 210, 2021.
+
+TCi(Voice traffic)
+TC2(Gamingtraffic)
+100
+100
+-eNB
+Higher load atgNB
+80
++gNB
+Mostofthetraffic
+80
++Packet drop rate
+processedbyeNB
+-gNB
+60
+60
++eNB
++Packet drop rate
+40
+40
+SteeredtraffictoeNB
+Smallpartofthetraffic
+Data
+20
+processed bygNB
+20
+0
+0
+6
+7
+8
+9
+10
+6
+7
+8
+9
+10
+Load per user (Mbps)
+Load peruser (Mbps)
+(b)
+(a)
+TC3(Video traffic)
+100
+80
++gNB
++eNB
+Mostofthetraffic
+-Packet drop rate
+60
+processed by gNB
+40
+Smallpartofthetraffic
+Data
+20
+processedbyeNB
+0
+6
+8
+9
+10
+Load per user (Mbps)
+(c)Traffic steered
+to gNB due to
+eNB
+eNB
+L=Load ("o"
+gNB
+6
+high load
+indicates "low"
+L=0
+L=1
+L=0
+1 indicates “high")
+gNB
+gNB
+gNB
+gNB
+5
+Traffic steered
+△ eNB
+to eNB due to
+=O
+L=0
+L=0
+L=0
+Number of UEs
+O gNB
+high load
+gNB
+gNB
+gNB
+eNB
+eNB
+Tcl (Voice)
+4
+Tc2 (Video)
+TC3 (Gaming)
+L=0
+L=0
+L=1
+L=O
+L=0
+gNB
+gNB
+gNb
+gNB
+eNB
+eNB
+3
+L=0
+L=0
+L=0
+L=1
+L=0
+L=0
+gNB
+gNB
+gNB
+gNB
+gNB
+gNB
+gNB
+2
+L=0
+L=0
+L=0
+L=0
+0=1
+L=0
+L=0
+eNB
+eNB
+eNB
+eNB
+eNB
+eNB
+eNB
+gNB
+1
+L=0
+L=0
+L=0
+L=0
+L=0
+L=0
+L=1
+L=0
+0
+0
+300
+600
+900
+1200 1500 1800 2100 2400 ....
+Time slots

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@@ -0,0 +1,1146 @@
+Discovering Sound Free-choice Workflow Nets
+With Non-block Structures
+Tsung-Hao Huang[0000−0002−3011−9999] and Wil M. P. van der
+Aalst[0000−0002−0955−6940]
+Process and Data Science (PADS), RWTH Aachen University, Aachen, Germany
+{tsunghao.huang, wvdaalst}@pads.rwth-aachen.de
+Abstract. Process discovery aims to discover models that can explain
+the behaviors of event logs extracted from information systems. While
+various approaches have been proposed, only a few guarantee desirable
+properties such as soundness and free-choice. State-of-the-art approaches
+that exploit the representational bias of process trees to provide the guar-
+antees are constrained to be block-structured. Such constructs limit the
+expressive power of the discovered models, i.e., only a subset of sound
+free-choice workflow nets can be discovered. To support a more flexible
+structural representation, we aim to discover process models that provide
+the same guarantees but also allow for non-block structures. Inspired by
+existing works that utilize synthesis rules from the free-choice nets the-
+ory, we propose an automatic approach that incrementally adds activities
+to an existing process model with predefined patterns. Playing by the
+rules ensures that the resulting models are always sound and free-choice.
+Furthermore, the discovered models are not restricted to block struc-
+tures and are thus more flexible. The approach has been implemented
+in Python and tested using various real-life event logs. The experiments
+show that our approach can indeed discover models with competitive
+quality and more flexible structures compared to the existing approach.
+Keywords: Process Discovery · Free-choice Net · Synthesis Rules.
+1
+Introduction
+Process discovery aims to construct process models that reflect the behaviors of
+a given event log extracted from information systems [2]. As it is a non-trivial
+problem, many challenges remain. In most cases, the one and only "best model"
+does not exist as there are trade-offs among the four model quality metrics,
+namely fitness, precision, generalization, and simplicity [2]. In addition to the
+quality metrics, there exist properties that one would like to have for the discov-
+ered models. One of the important properties is being a sound workflow net as
+soundness ensures the absence of deadlocks, proper completion, etc. [1] and it is
+a prerequisite for many crucial automated analyses such as conformance check-
+ing. The other desirable structural property is being free-choice [3]. In free-choice
+nets, choices and synchronizations are separated. This provides an easy conver-
+sion between the discovered models and many process modeling languages such
+arXiv:2301.02185v1  [cs.DB]  3 Jan 2023
+
+2
+T. Huang and W. M. P. van der Aalst
+as Business Process Modeling Notation (BPMN) since the equivalent constructs
+(dedicated split and join connectors) are naturally embedded. Furthermore, free-
+choice nets have been studied extensively and thus supported by an abundance
+of theories [10], which provide efficient analysis techniques.
+While various discovery algorithms have been proposed, only a handful of
+them provides such guarantees. State-of-the-art discovery algorithms like the In-
+ductive Miner (IM) [15] are able to discover sound free-choice workflow nets by
+exploiting its representational bias. However, due to the same reason, the discov-
+ered models are constrained to be block-structured. This limits the expressive
+power of such models, i.e., only a subset of the sound free-choice workflow nets
+can be discovered. As an example, Fig. 1a shows a sound free-choice workflow
+net (with non-block structures) discovered by our approach1. The same language
+can never be expressed by the model discovered by IM, as shown in Fig. 1b.
+b
+c
+d
+f
+g
+a
+e
+h
+(a) A model discovered by our approach. The
+same language cannot be expressed by the
+models discovered using the Inductive Miner,
+which uses process trees internally.
+b
+c
+d
+f
+g
+a
+e
+h
+(b) A model discovered by the IM using the
+log generated by the model in (a). The two
+branches before c need to be synchronized first
+before d can be executed.
+Fig. 1: Examples showing the need for the non-block process models discovery. Note
+that the trace ⟨a, b, c, d, e, f, g, h⟩ that is possible in (a) cannot be replayed by (b).
+In this paper, we aim to discover sound free-choice workflow nets with non-
+block structures. Inspired by the interactive process discovery approach in [11,12],
+we develop an automatic process discovery algorithm that incrementally adds
+activities to an existing net using synthesis rules [10]. Since checking the feasibil-
+ity for the application of the synthesis rules is computationally expensive, we use
+log heuristics to locate the most possible position for the to-be-added activity
+on the existing process model instead of evaluating all possible applications of
+synthesis rules as in [11]. Additionally, we identify the need for an additional
+rule and extend the set of patterns introduced in [12].
+Playing by the rules ensures that the discovered process models by our ap-
+proach are guaranteed to be sound free-choice workflow nets [10,11]. Moreover,
+the discovered models are not constrained to block structures. Last but not least,
+the level of replay fitness is guaranteed via a threshold set by the users. The
+approach has been implemented in Python and evaluated using various public-
+available real-life event logs. The evaluation shows that our approach is able to
+discover non-block structured models with competitive qualities compared to the
+state-of-the-art discovery algorithm.
+1 The proposed approach has dedicated silent transitions for start and end as defined
+later in Def. 5. We dropped them here for ease of comparison.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+3
+The remainder of the paper is organized as follows. We review the related
+work in Sec. 2 and introduce necessary concepts in Sec. 3. Sec. 4 introduces the
+approach. Sec. 5 presents the experiment and Sec. 6 concludes the paper.
+2
+Related Work
+An overview of process discovery is out of the scope of this paper, we refer to
+[7,14] for more details. In this section, we focus on process discovery techniques
+that guarantee soundness (and free-choice) properties. Approaches like [6,8] can
+discover non-block structured models but cannot guarantee both properties.
+While Split Miner discovers models that are deadlock-free, they are not nec-
+essarily sound [8].
+The family of Inductive Miner (IM) algorithms [15] guarantees sound and
+free-choice of the discovered models by exploiting the representational bias of
+the process tree. By design, a process tree represents a sound workflow net. It is
+a rooted tree where the leaf nodes are activities and the non-leaf nodes are the
+operators. The hierarchical representation has a straightforward translation to
+Petri net. However, the resulting models are limited to being block-structured as
+a process tree can only represent process models that can be separated into parts
+that have a single entry and exit [15]. Consequently, process trees can only rep-
+resent a subset of sound workflow nets. The same arguments hold for approaches
+that are based on process trees such as the Evolutionary Tree Miner (ETM) [9]
+and the recently developed incremental process discovery approach [16].
+Applying the synthesis rules [10], the interactive process discovery approaches
+developed in [12,13,11] ensure soundness and free-choice properties. A semi-
+automatic interactive tool, ProDiGy, is proposed in [12] to recommend the best
+possible ways to add an activity to an existing model.
+Our approach differs from [12,13,11] in several ways. First, we adopt an
+automatic setting as the order of adding activities is predetermined and the best
+modification to the existing net is selected based on the model quality. Second,
+we use log heuristics to locate the most suitable position for adding the new
+activity instead of evaluating all the possibilities of synthesis rules applications.
+Moreover, we identify the need for a new rule as the desired models often cannot
+be discovered without going back and forth by a combination of reduction and
+synthesis rules [13]. Lastly, the set of patterns is extended and formally defined.
+3
+Preliminaries
+We denote the set of all sequences over some set A as A∗, the power set of
+A as P(A), and the set of all multisets over A as B(A). For some multiset
+b ∈ B(A), b(a) denotes the number of times a ∈ A appears in b. For a given
+sequence σ = ⟨a1, a2, ..., an⟩ ∈ A∗, |σ| = n is the length of σ and dom(σ) =
+{1, 2, ..., |σ|} is the domain of σ. ⟨⟩ is the empty sequence. σ(i) = ai denotes the
+i-th element of σ. Given sequences σ1 and σ2, σ1 · σ2 denotes the concatenation
+of the two. Let A be a set and X ⊆ A be a subset of A. For σ ∈ A∗ and a ∈ A,
+
+4
+T. Huang and W. M. P. van der Aalst
+we define ↾X∈ A∗→X∗ as a projection function recursively with ⟨⟩↾X = ⟨⟩,
+(⟨a⟩ · σ)↾X = ⟨a⟩ · σ↾X if a ∈ X and (⟨a⟩ · σ)↾X = σ↾X if a /∈ X. For example,
+⟨x, y, x⟩↾{x,z} = ⟨x, x⟩. Projection can also be applied to multisets of sequences,
+e.g., [⟨a, b, a⟩6, ⟨a, b, c⟩6, ⟨b, a, c⟩2]↾{b,c} = [⟨b⟩6, ⟨b, c⟩8].
+Definition 1 (Trace, Log). A trace σ ∈ U∗
+A is a sequence of activities, where
+UA is the universe of activities. A log L ∈ B(U∗
+A) is a multiset of traces.
+Definition 2 (Log Properties). Let L ∈ B(U∗
+A) and a, b ∈ UA.
+– #(a, L) = Σσ∈L|{i ∈ dom(σ)|σ(i) = a}| is the times a occurred in L.
+– #(a, b, L) = Σσ∈L|{i ∈ dom(σ)\{|σ|}|σ(i) = a∧σ(i+1) = b}| is the number
+of direct successions from a to b in L.
+– caus(a, b, L) =
+�
+#(a,b,L)−#(b,a,L)
+#(a,b,L)+#(b,a,L)+1
+if a ̸= b
+#(a,b,L)
+#(a,b,L)+1
+if a = b is the strength of causal rela-
+tion (a, b).
+– Apre
+c
+(a, L) = {apre ∈ UA|caus(apre, a, L) ≥ c} is the set of a’s preceding
+activities, determined by threshold c.
+– Afol
+c
+(a, L) = {afol ∈ UA|caus(a, afol, L) ≥ c} is the set of a’s following
+activities, determined by threshold c.
+– As(L) = {σ(1) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of start activities in L.
+– Ae(L) = {σ(|σ|) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of end activities in L.
+Definition 3 (Petri Net, Labeled Petri Net). A Petri net N = (P, T, F) is
+a tuple, where P is the set of places, T is the set of transitions, P ∩ T = ∅, and
+F ⊆ (P × T) ∪ (T × P) is the set of arcs. A labeled Petri net N = (P, T, F, l) is
+a Petri net (P, T, F) with a labeling function l ∈ T ↛ UA that maps a subset of
+transitions to activities. A t ∈ T is called invisible if t is not in the domain of l.
+For any x ∈ P ∪ T,
+N•x = {y|(y, x) ∈ F} denotes the set of input nodes and
+x
+N• = {y|(x, y) ∈ F} denotes the set of output nodes. The superscript N is
+dropped if it is clear from the context. The notation can be generalized to set.
+For any X ⊆ P ∪ T, •X = {y|∃x∈X(y, x) ∈ F} and X• = {y|∃x∈X(x, y) ∈ F}.
+Definition 4 (Free-choice Net). Let N = (P, T, F) be a Petri net. N is a
+free-choice net if for any t1, t2 ∈ T : •t1 = •t2 or •t1 ∩ •t2 = ∅.
+Definition 5 (Workflow Net (WF-net) [1,11]). Let N = (P, T, F, l) be a
+labeled Petri net. W = (P, T, F, l, ps, pe, ⊤, ⊥) is a WF-net iff (1) it has a dedi-
+cated source place ps ∈ P: •ps = ∅ and a dedicated sink place pe ∈ P: pe• = ∅
+(2) ⊤ ∈ T: •⊤ = {ps}∧ps• = {⊤} and ⊥ ∈ T: ⊥• = {pe}∧•pe = {⊥} (3) every
+node x is on some path from ps to pe, i.e., ∀x∈P ∪T (ps, x) ∈ F ∗ ∧ (x, pe) ∈ F ∗,
+where F ∗ is the reflexive transitive closure of F.
+Definition 6 (Short-circuited WF-net [1]). Let W = (P, T, F, l, ps, pe, ⊤, ⊥)
+be a WF-net. The short-circuited WF-net of W, denoted by SC(W), is con-
+structed by SC(W)=(P, T ∪{t′}, F ∪{(⊥, t′), (t′, ⊤)}, l, ps, pe, ⊤, ⊥), where t′ /∈ T.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+5
+Definition 7 (Paths, Elementary Paths). A path of a Petri net N = (P, T, F)
+is a non-empty sequence of nodes ρ = ⟨x1, x2, ..., xn⟩ such that (xi, xi+1) ∈ F for
+1 ≤ i < n. ρ is an elementary path if xi ̸= xj for 1 ≤ i < j ≤ n.
+Definition 8 (Incidence Matrix [10]). Let N = (P, T, F) be a Petri net. The
+incidence matrix N : (P × T) → {−1, 0, 1} of N is defined as
+N(p, t) =
+�
+�
+�
+�
+�
+0
+if ((p, t) /∈ F ∧ (t, p) /∈ F) ∨ ((p, t) ∈ F ∧ (t, p) ∈ F)
+−1
+if (p, t) ∈ F ∧ (t, p) /∈ F
+1
+if (p, t) /∈ F ∧ (t, p) ∈ F
+For a Petri net N = (P, T, F) and its corresponding incidence matrix N, we use
+N(p) to denote the row vector of the corresponding p ∈ P and N(t) to denote
+the column vector of the corresponding t ∈ T.
+Definition 9 (Linearly Dependent Nodes [10]). Let N = (P, T, F) be a
+Petri net. Q is the set of rational numbers. A place p is linearly dependent if
+there exists a row vector ⃗v : P → Q such that ⃗v(p) = 0 and ⃗v · N = N(p). A
+transition t is linearly dependent if there exists a column vector ⃗v : T → Q such
+that ⃗v(t) = 0 and ⃗v · N = N(t).
+Definition 10 (Synthesis Rules [10,11]). Let W and W ′ be two free-choice
+workflow nets, and let SC(W) = (P, T, F, l, ps, pe, ⊤, ⊥) and SC(W ′) = (P ′, T ′,
+F ′, l′, ps, pe, ⊤, ⊥) be the corresponding short-circuited WF-nets:
+– Linear Dependent Place Rule ψP : W ′ is derived from W using ψP , i.e.,
+(W, W ′) ∈ ψP if (1) T ′ = T, P ′ = P ∪ {p} and p /∈ P is linear dependent in
+SC(W ′), F ′ = F ∪ �F where �F ⊆ (({p} × T) ∪ (T × {p})) (2) Every siphon
+in SC(W ′) contains ps.
+– Linear Dependent Transition Rule ψT : W ′ is derived from W using ψT , i.e.,
+(W, W ′) ∈ ψT if P ′ = P, T ′ = T ∪ {t} and t /∈ T is linear dependent in
+SC(W ′) and F ′ = F ∪ �F where �F ⊆ ((P ×{t})∪({t}×P)), and ∀t∈T ∩T ′l(t) =
+l′(t).
+– Abstraction Rule ψA: (W, W ′) ∈ ψA if (1) there exists a set of transitions
+R ⊆ T and a set of places S ⊆ P such that (R × S ⊆ F) ∧ (R × S ̸= ∅). (2)
+SC(W ′) is constructed by adding an additional place p /∈ P and a transition
+t /∈ T such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ = (F\(R × S)) ∪ ((R × {p}) ∪
+({p} × {t}) ∪ ({t} × S)), and ∀t∈T ∩T ′l(t) = l′(t).
+Applying the three synthesis rules (ψP , ψT , ψA) to derive W ′ from a sound
+free-choice workflow net W ensures that W ′ is also sound [13,11]. Three proper-
+ties need to be hold for a WF-net to be sound (1) safeness: places cannot hold
+multiple tokens at the same time (2) option to complete: it is always possible to
+reach the marking in which only the sink place is marked. (3) no dead transitions.
+Next, we introduce the initial net [11] and show some examples of synthesis rules
+applications.
+
+6
+T. Huang and W. M. P. van der Aalst
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+(a)
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑡1
+𝑝2
+(b)
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+(c)
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+(d)
+Fig. 2: Examples of synthesis rules applications starting from (a) The initial net. (b)
+Using ψA, p2 and t1 are added to the initial net with R = {⊤} and S = {p1}. (c) Using
+ψA, p3 and t2 are added to previous net with R = {⊤} and S = {p2}. (d) p4 is added
+using ψp as p4 is a linear combination of p3 and p2.
+Definition 11 (Initial Net [13]). Let W = (P, T, F, l, ps, pe, ⊤, ⊥) be a free-
+choice WF-net. W is an initial net if P = {ps, p1, pe}, T = {⊤, ⊥}, F =
+{(ps, ⊤), (⊤, p1), (p1, ⊥), (⊥, pe)}.
+The initial net is shown in Fig. 2a. Clearly, it is a sound free-choice workflow
+net. Starting from the initial net, one can incrementally add additional nodes
+according to the synthesis rules. Fig. 2 shows example applications of synthesis
+rules starting from the initial net.
+4
+Approach
+With the necessary concepts introduced, we are now ready to introduce the ap-
+proach. We start by showing the basic idea of the approach with the help of
+Fig. 3 before diving into each step in detail. Internally, the approach incremen-
+tally adds a new activity to an existing net. The figure shows a single iteration.
+In each iteration, we have an existing model from the previous iteration and a
+log projected on the already added activities so far and the to-be-added one.
+We start by locating the most likely position to add the new activity deter-
+mined by log heuristics. The result of this step is a subset of nodes of the existing
+model. The set of nodes will then be used to prune the search space. Then, the
+predefined patterns are applied to the existing net to get a set of candidate nets.
+Lastly, we select the best net (next existing net) out of the candidates in terms of
+fitness and precision. Note that the existing net in the first iteration is initiated
+by the initial net (Def. 11). As a running example, consider the correspond-
+ing log that is used to discover the Petri net in Fig.1a by our approach: Ls =
+[⟨a, b, c, d, f, g, h⟩22, ⟨a, b, c, f, d, g, h⟩14, ⟨a, e, b, c, d, f, g, h⟩13, ⟨a, e, b, c, f, d, g, h⟩13,
+⟨a, e, b, c, f, g, d, h⟩10, ⟨a, b, c, f, g, d, h⟩10, ⟨a, b, e, c, d, f, g, h⟩6, ⟨a, b, e, c, f, g, d, h⟩3,
+⟨a, b, e, c, f, d, g, h⟩3, ⟨a, b, c, d, e, f, g, h⟩2, ⟨a, b, c, e, d, f, g, h⟩2, ⟨a, b, c, e, f, g, d, h⟩1,
+⟨a, b, c, e, f, d, g, h⟩1]. The instances provided in Fig. 3 shows the 3rd iteration for
+the running example Ls. In the following subsections, we introduce the details
+of each step.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+7
+(1) Pruning search 
+space using log 
+heuristics
+(2) Add new activity to 
+the existing net with 
+pre-defined patterns
+[ 𝑑, 𝑔, ℎ 76, 𝑔, 𝑑, ℎ 24]
+Projected Log 𝐿𝑖
+Existing Net
+𝑊𝑖 = (𝑃𝑖, 𝑇𝑖, 𝐹𝑖, 𝑙𝑖, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥)
+(3) Select the best net
+for the next iteration
+Next Existing Net
+𝑊𝑖+1 = (𝑃𝑖+1, 𝑇𝑖+1, 𝐹𝑖+1, 𝑙𝑖+1, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥)
+To-be-added Activity 𝛾(𝑖)
+𝑉𝑖 ⊆ 𝑃𝑖 ∪ 𝑇𝑖
+Set of Candidate Nets 𝐶𝑖
+skip
+loop
+…
+…
+Fig. 3: An example of a single iteration of our approach.
+4.1
+Ordering Strategies for Adding Activities
+Before starting any iteration, we need to come up with an order for adding ac-
+tivities based on a given log L. It is important as the quality of the discovered
+models often depends on the order of adding activities [11]. Moreover, in combi-
+nation with the search space pruning, it can influence the computation time for
+each iteration significantly. In this paper, we introduce two ordering strategies.
+The first one is relatively straightforward. The activities in L are simply ordered
+by their frequency.
+Definition 12 (Activities-Adding Order, Frequency-Based Ordering).
+Let L ∈ B(U∗
+A) and A = �
+σ∈L{a ∈ σ}. γ ∈ A∗ is an activities-adding order for
+L if {a ∈ γ} = A and |γ| = |A|. The frequency-based ordering is orderfreq(L) = γ
+such that γ is an activities-adding order and ∀1≤i<j≤|γ|#(γ(i), L) ≥ #(γ(j), L).
+The second ordering strategy is similar to the Breadth-first Search (BFS) al-
+gorithm. The advantage of this is that it also considers the closeness between
+activities in the log, rather than just frequency. To explain this ordering strategy,
+we first define a sub-function.
+Definition 13 (Directly-Precedes Activities Sorting). Let L ∈ B(U∗
+A) and
+a ∈ UA. A={b ∈ UA|#(b, a, L)>0} is the set of activities directly-precede a in L at
+least once and σ ∈ A∗. Directly-precedes activities sorting is sortPreceded(a, L)=σ
+such that {b ∈ σ} = A and |σ| = |A| and ∀1≤i<j≤|σ| #(σ(i), a, L) ≥ #(σ(j), a, L).
+The function sortPreceded takes an activity a and a log L to return a sequence of
+a’s directly-preceded activities b that are sorted by the frequency of #(b, a, L).
+Finally, we can define the BFS-based ordering strategy.
+Definition 14 (Breadth-First-Search-Based Ordering). Let L ∈ B(U∗
+A)
+and A= �
+σ∈L{a ∈ σ}. BFS-based ordering is defined as orderBFS(L)=γ, where
+γ is an activities adding order for L and γ=γ1 · γ2 · ... · γ|γ|, for each 1 ≤ j ≤ |γ|,
+γj =
+�
+orderfreq(L↾Ae(L))
+if j = 1
+sortPreceded(γ(j − 1), L)↾A\{γ(1),γ(2),...,γ(j−1)}
+otherwise
+
+d
+h
+gdps
+4
+0
+d
+h
+gd
+hh
+g8
+T. Huang and W. M. P. van der Aalst
+The function starts by sorting the end activities Ae(L) according to their fre-
+quency in the log. Then, it enumerates through the sequence γ and sorts the
+preceded activities of γ(j − 1) by the frequency of direct successions. The pro-
+jection function in the second case of Def.14 filters out the activities that are
+already in γ.
+Compared to the frequency-based ordering, the BFS-based ordering considers
+the closeness of the activities. This allows us to add activities that are close
+together. Together with the effect of the search space pruning, it is expected that
+BFS-based ordering would have less computation time. Applying the function
+orderBFS to our running example, Ls, we get the activities adding order as
+γ = orderBFS(Ls) = ⟨h⟩ · ⟨g, d⟩ · ⟨f⟩ · ⟨c, e⟩ · ⟨⟩ · ⟨b⟩ · ⟨a⟩ · ⟨⟩ = ⟨h, g, d, f, c, e, b, a⟩.
+γ is then used to determine the order of adding activities. Given the activities
+adding order γ, we define the artifacts for each iteration i as followed.
+Definition 15 (Projected Log). Let L ∈ B(U∗
+A) and γ be a activities adding
+order for L. The projected log for L in the i-th iteration is Li = L↾{γ(1),γ(2),...γ(i)}.
+For instance, the projected log for the running example Ls for the 3rd iteration
+is then L3
+s = Ls↾{h,g,d} = [⟨d, g, h⟩76, ⟨g, d, h⟩24], as shown in Fig. 3. The to-be-
+added activity is denoted as γ(i), which is γ(3) = d for the 3rd iteration. Also,
+we denote the existing sound free-choice workflow net for iteration i as W i. Note
+that for the running example, W 1, W 2, and W 3 are visualized in Fig. 2a, 2b,
+and 2c, respectively.
+4.2
+Search Space Pruning
+As checking the feasibility of applying linear dependent rules ψT , ψP is compu-
+tationally expensive [11], it is impractical to compute all possible applications
+of the synthesis rules. Also, some of them are not of interest. For example, as
+shown in Fig. 3, it is clear that the to-be-added activity d never happens after
+h in the projected log. Using such information, we can already eliminate the
+constructs (applications of synthesis rules) that allow activity d to be executed
+after h. Therefore, in each iteration i we start by locating the most likely posi-
+tion to add γ(i). This helps us to restrict the application of synthesis rules on
+only a subset of nodes, denoted as V i ⊆ P i ∪ T i, in the existing net W i. To do
+that, we first identify the set of preceding and following activities of γ(i) in the
+projected log Li, which would be Apre
+c
+(γ(i), Li) and Afol
+c
+(γ(i), Li) respectively.
+Recall that c is a threshold for the causal relation and can be given by users
+as an input. We use c = 0.9 as the default value. Then, the corresponding la-
+belled transitions are identified in W i. Finally, V i is the set of all the nodes on
+the elementary paths from the preceding transitions to the following transitions.
+If Apre
+c
+(γ(i), L)/Afol
+c
+(γ(i), L) is an empty set, we use the ⊤/⊥ transitions. For
+instance, in Fig. 3, we identify that Apre
+c
+(d, L3
+s)=∅ and Afol
+c
+(d, L3
+s)={h}. There-
+fore, we find all the nodes on the elementary paths between every node in {⊤}
+and every node in {t1}. As a result, the set V 3 = {⊤, p3, t2, p2, t1} is used to
+prune the search space, i.e., to constrain the application of synthesis rules.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+9
+Constraining synthesis rules For the abstraction rule ψA, this means that the
+set of transitions R and the set of places S used as the preconditions for applying
+ψA need to be a subset of V , i.e., S ⊆ V ∧R ⊆ V . For the linear dependent rules
+ψP /ψT , the new place/transition (p′/t′) cannot have arcs connected to any node
+outside V . This shortens the computation time as certain rules applications can
+be removed and there is no need to check their feasibility.
+4.3
+Patterns
+In this section, we introduce the patterns that are used to add activity γ(i) to
+the existing free-choice workflow net W i. First, we motivate the need for an
+additional rule.
+The need for an additional rule It is proven that any sound free-choice
+workflow net can be constructed by the three synthesis rules ψA, ψP , ψT [10,11].
+However, when applying to discover process models, the desirable model is often
+not possible to derive due to the existing construct. An example is shown in
+Fig. 4. While it is possible to add a transition labeled by a in Fig. 4a, it is not
+possible to derive the same net in Fig. 4b as there is no rule allowing such trans-
+formation. One possible workaround is to go back and forth by a combination
+of reduction and synthesis rules as suggested in [13]. However, once the existing
+net becomes more complex, such a solution becomes infeasible to track.
+b
+c
+L↾〈c,b〉 = [〈c〉3, 〈b〉3]
+b
+c
+a
+L↾〈a,b,c〉 = [〈c,a〉3, 〈b〉3]
+(a)
+L↾〈a,b〉 = [〈a〉3, 〈b〉3]
+L↾〈a,b,c〉 = [〈c,a〉3, 〈b〉3]
+b
+c
+a
+b
+a
+?
+(b)
+Fig. 4: Examples showing the motivation for the dual abstraction rule ψD. Although
+the desirable nets on the right-hand side of (a) and (b) are semantically the same, the
+existing synthesis rules only allow the transformation in (a). There is no rule defined
+for the transformation in (b).
+We observe that in many situations (including the example in Fig. 4b), the
+desired models cannot be constructed because there is no rule allowing one to
+introduce a new transition t and a new place p in between a set of places S and
+a set of transitions R that are fully connected, i.e., S × R ⊆ F. Therefore, we
+define the dual abstraction rule to allow such construct.
+Definition 16 (Dual Abstraction Rule ψD). Let W = (P, T, F, l, ps, pe, ⊤, ⊥)
+and W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be two free-choice workflow nets. (W, W ′) ∈
+ψD if (1) there exists a set of places S ⊆ P and a set of transitions R ⊆ T such
+that S × R ⊆ F ∧ S × R ̸= ∅. (2) W ′ is constructed by adding an additional
+
+10
+T. Huang and W. M. P. van der Aalst
+transition t /∈ T and a place p /∈ P such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ =
+(F \ (S × R)) ∪ ((S × {t}) ∪ ({t} × {p}) ∪ ({p} × R)), and ∀t∈T ∩T ′l(t) = l′(t).
+As we are only interested in sound free-choice workflow nets, we need to make
+sure that the dual abstraction rule ψD preserves soundness.
+Proposition 1 (ψD preserves soundness). Let W = (P, T, F, l, ps, pe, ⊤, ⊥),
+W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be free-choice workflow nets, and W ′ is derived
+from W using ψD, i.e., (W, W ′) ∈ ψD. Then W ′ is sound if W is sound.
+Proof. Let t′ ∈ T ′\T and p′ ∈ P ′\P be the new transition and place in W ′. Let
+R = p′• and S = •t′. The new net W ′ is free-choice in only two cases. Either
+S =
+W• R or R = S
+W• . In either case, any reachable marking in (W, [ps]) that
+does not need to fire tR ∈ R is still reachable in (W ′, [ps]). Also, the reachable
+markings in (W, [ps]) that need to fire tR ∈ R can be reached in (W ′, [ps]) as one
+can just add t′ somewhere before tR in the corresponding firing sequence. Then,
+it is trivial to see that W ′ fulfils the three conditions of soundness if W is also
+sound.
+□
+Next, we extent the linear dependent place rule ψP . As we aim to add a transition
+labeled by γ(i) to the existing labeled free-choice workflow net W i, only adding
+a place p′ by ψP does not suffice. Hence, in our approach, an application of ψP
+is always coupled with a directly followed application of abstraction rule ψA to
+include a transition. ψA is applied between the added place p′ and its preset •p′.
+This is possible because every transition in •p′ is connected to every place in
+{p′} by definition, which satisfies the precondition of ψA. An example is shown
+in Fig. 5, p5 and t3 are added by ψA directly after the addition of p4 by ψP . To
+be more precise, we define the extended rule, ψ′
+P , that describes the pattern.
+Definition 17 (Extended Linear Dependent Place Rule ψ′
+P ). Let W=(P,
+T, F, l, ps, pe, ⊤, ⊥) and W ′′=(P ′′, T ′′, F ′′, l′′, ps, pe, ⊤, ⊥) be free-choice workflow
+nets. (W, W ′′)∈ψ′
+P if (1) ∃W ′=(P ′,T ′,F ′,l′,ps,pe,⊤,⊥)(W, W ′)∈ψP ∧ (W ′, W ′′)∈ψA
+and (2) ∃!p∗∈P ′′({p∗}=P ′\P) ∧ (((T ′′\T ′) × {p∗}) ⊂ F ′′) ∧ ((T ′ × {p∗}) ̸⊂ F ′′).
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+(a)
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+d
+𝑝5
+𝑡3
+(b)
+Fig. 5: (a) ψP adds a place p4. (b) As every transition in •p4 has an arc to every place
+in {p4}, one can directly apply ψA to add p5 and t3.
+Then, we define the set of nets constructed by every possible single application
+of the rules ψA, ψ′
+P , ψT , ψD.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+11
+Definition 18 (Base Candidates Set). Let W=(P, T, F, l, ps, pe, ⊤, ⊥), W ′ =
+(P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be free-choice workflow nets. Let X=(P ′∪T ′)\(P∪T),
+V ⊆P∪T, V ′=(P∪T)\V , and let a ∈ UA be an activity label. The base candidates
+set is base(W, V, a)={W ′|((W, W ′) ∈ (ψA ∪ψT ∪ψ′
+P ∪ψD))∧(∄x∈X(({x}×V ′)∪
+(V ′ × {x})) ⊆ F ′) ∧ (l′=l ∪ ((T ′ \ T) × {a}))}.
+The base candidates set Ci
+base=base(W i, V i, γ(i)) consists of the nets that are
+constructed by every possible single application of the rules ψA, ψ′
+P , ψT , ψD to
+add a transition labeled by γ(i) to W i considering the the constraints on V i.
+Next, we introduce three patterns that make a transition skippable, in a strict
+loop, or in an optional (tau) loop. A transition in a strict loop means that the
+execution of the transition is required, otherwise it is an optional loop.
+Definition 19 (Pattern-Building Functions). Let W=(P, T, F, l, ps, pe, ⊤, ⊥)
+and W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be two free-choice workflow nets. Let a ∈ UA
+be an activity label and ta ∈ T : l(ta) = a be the corresponding transition in W.
+We define the three pattern-building functions2 as
+– skip(W, a) = W ′ such that
+– (W, W ′) ∈ ψT
+– F ′ = F ∪ ({t′} × ta•) ∪ (•ta × {t′}) (where t′ ∈ T ′\T)
+– l′ = l (t′ is a silent transition)
+– loops(LW, a) is defined by two cases:
+1. if ∄t∗∈((ta•)•)(|•t∗| > 1)∧(•t∗\ta• ̸= ∅), then loops(W, a) = W ′ such that
+– (W, W ′) ∈ ψT
+– F ′ = F ∪ (ta• × {t′}) ∪ ({t′} × •ta) (where t′ ∈ T ′\T)
+– l′ = l (t′ is a silent transition)
+2. otherwise, return loops(W ′, a) such that
+– (W, W ′) ∈ ψA
+– (({ta} × (P ′\P)) ∈ F ′) ∧ (({ta} × P) /∈ F ′)
+– l′ = l
+– loopτ(W, a) is defined by two cases:
+1. if ∄t∗∈((ta•)•)(|•t∗| > 1) ∧ (•t∗\ta• ̸= ∅), then loopτ(W, a)=W ′ such that
+– (W, W ′) ∈ ψT
+– F ′ = F ∪ (ta• × {t′}) ∪ ({t′} × •ta) (where t′ ∈ T ′\T)
+– l′ = (l\{(ta, a)}) ∪ {(t′, a)} (the labels of ta and t′ are swapped)
+2. otherwise, return loopτ(W ′, a) such that
+– (W, W ′) ∈ ψA
+– (({ta} × (P ′\P)) ∈ F ′) ∧ (({ta} × P) /∈ F ′)
+– l′ = l
+The second case of the loop functions is there to keep the free-choice property.
+To illustrate the ideas using the running example, consider the net shown in
+Fig. 6a as the input net W and t3 (labeled by d) is the transition for which we
+are going to apply the functions to derive patterns. Fig. 6b shows that function
+2 The input/output nodes notations (•) used in Def. 19 refer to the input net W. We
+drop the superscript for readability.
+
+12
+T. Huang and W. M. P. van der Aalst
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+d
+𝑝5
+𝑡3
+(a) the net (W ). t3 (labeled by d)
+is the target transition.
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+d
+𝑝5
+𝑡3
+𝑡4
+(b) skip(W, d) adds a silent tran-
+sition t4 that makes t3 skippable.
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝6
+d
+𝑝5
+𝑡3
+𝑝4
+𝑡4
+(c) an intermediate net W ′ (be-
+tween (a) and (d)) constructed to
+keep the free-choice property.
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝6
+d
+𝑝5
+𝑡3
+𝑝4
+𝑡4
+𝑡5
+(d)
+the
+resulting
+net
+of
+loops(W, d),
+which
+makes
+t3
+in a loop.
+Fig. 6: Examples showing how the functions are applied to derive patterns.
+skip(W, d) simply adds a silent transition t4 with the same connection as t3 to
+W. Fig. 6c and 6d show an application of loops(W, d) and illustrate the need
+for the two cases for the loop functions. As shown in Fig. 6c, the second case
+of loops is applied since there exists a transition t1 ∈ ((t3•)•) with more than
+one place in its preset (| • t1| > 1) and •t1\t3• ̸= ∅. Therefore, W ′ (Fig. 6c) is
+first constructed by adding p6 and t4. Then, the function returns loops(W ′, d).
+Now, the first case should be applied. In this case, t5 is added with the reverse
+connections of t3. As indicated, the second case in the loop functions helps to
+keep the free-choice property. Imagine a net that is constructed by adding t′ to
+the net in Fig. 6a with connections (p4, t′) and (t′, p5). Such a net makes t3 in a
+loop but it is no longer a free-choice net. The constructs of loops and loopτ are
+almost the same, the difference is that the labels of t3 and the silent transition
+t5 are swapped.
+Finally, to get the set of candidate nets Ci, we apply the three pattern-
+building functions to every net W ∈ Ci
+base. Observe that all the nets in Fig. 6
+are elements of C3.
+4.4
+Selection and Fall-through
+Selection In the last step, we select the next existing net W i+1 from the set
+of candidates Ci evaluated by the projected log Li. The selection is done in a
+stepwise manner. We first try to filter out the candidates that do not reach a
+user-defined replay fitness threshold θ and then select the best net out of the
+rest in terms of F1 score, which is calculated as the harmonic mean of fitness
+and precision. We use alignment-based fitness [4] and precision [5].
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+13
+Fall-through If none of the nets in Ci reach the ���tness threshold θ, we adopt
+a fall-through. This is done by going back to Step 2, where γ(i) is added to
+W i = (P i, T i, F i, li, ps, pe, ⊤, ⊥), but without the constraints of V i. This can
+also be seen as setting V i = P i ∪ T i. In this case, a new place p′ with arcs
+{(⊤, p′), (p′, ⊥)} can be always added by ψP as p is linear dependent on ps and
+pe. Then, the patterns building functions can be applied to ensure that the fitness
+threshold θ is guaranteed in every iteration.
+5
+Evaluation
+In this section, we present the experiments conducted to evaluate our approach.
+The presented approach in this paper is implemented in Python using PM4Py3
+and can be accessed here4. As mentioned, the algorithm takes as inputs a log
+and three parameters including two thresholds θ, c, and the types of ordering
+strategy. Using this implementation, we conduct three experiments to address
+the following questions (1) How effective are the pre-defined patterns? (2) What
+are the effects of the ordering strategy on the model quality and the execution
+time? (3) Can the model quality be improved by the non-block structures?
+5.1
+Experiment Setup
+Dataset: We use four public available real-life event logs, which are BPI20175,
+helpdesk6, hospitalBilling7, and traffic8 respectively. BPI2017 is split into two
+sub logs, BPI2017A and BPI2017O, using the event prefixes. To focus on the
+mainstream behaviors, the logs are filtered to include at least 95% of the cases.
+Experiment 1 (Effectiveness of patterns): The first experiment aims to
+evaluate how effective are the pre-defined patterns. As our approach is based on
+[11], this can be evaluated by comparing the quality of the intermediate models
+of our approach to the ones from ProDiGy [12], which adopts a similar setting.
+To conduct the experiment, we follow the top recommendation of ProDiGy in
+every step to get the intermediate models and compare the models’ quality with
+ours. We use the projected log of every iteration to evaluate the model obtained
+after adding additional activity to the model. To have a fair comparison, we
+force our approach to use the same order of adding activities from ProDiGy.
+Experiment 2 (Effects of Ordering Strategy & Search Space Pruning):
+The order of adding activities to the log is crucial to our approach as model
+quality is highly dependent on the order [11]. Moreover, the order can influence
+the execution time due to its influence on the search space pruning. Therefore, we
+would like to investigate the effects of the ordering strategy on the model quality
+and the execution time. To set up the experiment, we apply the approach to the
+3 https://pm4py.fit.fraunhofer.de/
+4 https://github.com/tsunghao-huang/synthesisRulesMiner
+5 https://doi.org/10.4121/uuid:3926db30-f712-4394-aebc-75976070e91f
+6 https://doi.org/10.4121/uuid:0c60edf1-6f83-4e75-9367-4c63b3e9d5bb
+7 https://doi.org/10.4121/uuid:76c46b83-c930-4798-a1c9-4be94dfeb741
+8 https://doi.org/10.4121/uuid:270fd440-1057-4fb9-89a9-b699b47990f5
+
+14
+T. Huang and W. M. P. van der Aalst
+five event logs using the two different ordering strategies while keeping the other
+two parameters at the same values. We evaluate the model quality in terms of
+fitness, precision, and F1 score. In addition, we keep track of the ratio of the
+reduced nodes, which is calculated by
+|V i|
+|P i∪T i|. This gives us an indication of the
+effectiveness of search space pruning.
+Experiment 3 (Effects of non-block structures): In this experiment, we
+compare our approach to the state-of-the-art: Inductive Miner - Infrequent (IMf)
+[15]. As the models discovered by IMf are guaranteed to be sound free-choice
+workflow net as well, comparing our approach with IMf enables us to see if the
+models can benefit from the non-block structures discovered by our approach.
+For each event log, we apply IMf using five different values ([0.1, 0.2, 0.3, 0.4, 0.5])
+for the filter threshold and choose the best model (by F1 score) to compare the
+quality with the ones discovered by our approach in experiment 2.
+For all the experiments, we use the alignment-based approaches to calculate
+fitness [4] and precision [5]. We also calculate the F1 score as the harmonic mean
+of the fitness and precision.
+5.2
+Results
+Effectiveness of Patterns Fig. 7 shows the result of the comparison. The
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added
+0.4
+0.6
+0.8
+1.0
+Fitness
+our approach
+ProDiGy
+(a) Fitness
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added
+0.4
+0.6
+0.8
+1.0
+Precision
+our approach
+ProDiGy
+(b) Precision
+Fig. 7: Results on fitness and precision comparison for the effectiveness of patterns
+fitness and precision are the average values of the five event logs. As one can
+see from the figures, both approaches can capture the behaviors quite well for
+the first three activities added. When adding more activities to the model,
+our approach has consistently higher values for both fitness and precision than
+ProDiGy. One might think that this is expected as we extend the set of patterns
+used in ProDiGy. However, note that ProDiGy evaluates every possible synthesis
+rules applications while we only focus on a subset of the nodes using log heuris-
+tics. There is a trade-off between optimal solution and time in our approach.
+Nevertheless, the results show that the extended patterns enable us to discover
+models with higher quality compared to the existing approach, ProDiGy, while
+limiting the search space.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+15
+Effects of Ordering Strategy and Search Space Pruning Tab. 1 shows the
+results of experiments 2 and 3. We observe that the BFS-based ordering strat-
+egy performs better than the frequency-based strategy (in terms of F1 score and
+time) for four of the five logs. We further investigate the reason for the shorter
+execution time of BFS-based ordering. As shown in Fig. 8, it turns out that the
+BFS-ordering strategy is more effective (lower
+|V i|
+|P i∪T i|) in reducing the search
+space
+at
+the
+later
+stage
+of
+the
+discovery
+process.
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added
+0.1
+0.3
+0.5
+0.7
+Ratio of nodes reduced 
+|Vi|
+|Pi
+Ti|
+frequency
+BFS
+Fig. 8: Comparison of the to-be-considered
+nodes ratio for each iteration between the
+two ordering strategies.
+As the model grows, checking the pre-
+conditions of an application for the
+linear dependent place or transition
+rule becomes more expensive. Reduc-
+ing the search space more effectively
+at the later stage is more beneficial in
+terms of execution time in most cases.
+BFS-based ordering achieves this by
+considering the closeness of activities
+in the process. In such case, activities
+that are closer together are added first
+and it is more likely for BFS-based
+ordering to focus on a smaller subset
+of nodes on the existing net when pruning the search space compared to the
+frequency-based one.
+Table 1: Results about effects of ordering strategy and comparison to IMf
+Log
+Miner Ordering
+Strategy
+IMf
+filter Fitness Precision
+F1
+Time (s)
+BPI2017A
+ours frequency
+-
+0.970
+0.947
+0.958
+734
+ours
+BFS
+-
+0.989
+0.935
+0.961
+342
+IMf
+-
+0.2
+0.999
+0.936
+0.967
+10
+BPI2017O
+ours frequency
+-
+0.994
+0.962
+0.978
+560
+ours
+BFS
+-
+0.989
+1.000
+0.994
+240
+IMf
+-
+0.2
+0.997
+0.907
+0.950
+7
+helpdesk
+ours frequency
+-
+0.972
+0.984
+0.977
+54
+ours
+BFS
+-
+0.981
+0.976
+0.978
+44
+IMf
+-
+0.2
+0.967
+0.950
+0.958
+1
+hospital
+billing
+ours frequency
+-
+0.961
+0.810
+0.879
+567
+ours
+BFS
+-
+0.989
+0.935
+0.961
+407
+IMf
+-
+0.2
+0.982
+0.906
+0.943
+45
+traffic
+ours frequency
+-
+0.960
+0.930
+0.945
+321
+ours
+BFS
+-
+0.964
+0.720
+0.825
+427
+IMf
+-
+0.4
+0.904
+0.720
+0.801
+28
+Effects of Non-block Structures Table 1 shows that compared to IMf, the
+models discovered by our approach have higher F1 scores for four of the five
+logs. Note that the fitness values of the models discovered by our approach are
+
+16
+T. Huang and W. M. P. van der Aalst
+all higher than the defined threshold 0.95. In general, IMf tends to discover mod-
+els with higher fitness values while our approach discovers models with higher
+precision. In IMf, one can use the filter threshold to balance fitness and precision.
+This is also the case in our approach, the user can set a lower fitness thresh-
+old to include more candidate nets that are less fitting but more precise. Fig. 9
+shows the discovered models from the two approaches for the hospitalBilling
+log. While the overall structure of Fig. 9a is similar to its counterpart in Fig. 9b,
+our approach discovered non-block structures at the later stage of the process.
+Such construct is not possible to model by IMf. The result shows that our ap-
+proach can discover sound free-choice workflow nets with non-block structures
+and produce competitive model quality as the state-of-the-art algorithm.
+(a) The discovered model using our approach. Due to the more flexible structure, one can exe-
+cute EMPTY, BILLED, or REOPEN after CODE NOK while only BILLED or REOPEN are
+executable after CODE OK. The construct is not discoverable by IMf.
+(b) The discovered model using IMf. Note that activity REOPEN is dropped by the filter of IMf.
+Fig. 9: The models discovered by our approach and IMf for the hospitalBilling log.
+6
+Conclusion and Future Work
+In this paper, we present a discovery algorithm that aims to discover sound free-
+choice workflow nets with non-block structures. The algorithm utilizes the syn-
+thesis rules to incrementally add activities with predefined patterns to discover
+models that are guaranteed to be sound and free-choice. Moreover, a certain
+level of replay fitness is guaranteed by a user-defined threshold.
+The approach has been implemented and evaluated using various real-life
+event logs. The results show that the process models discovered by our approach
+have higher model quality (in terms of both replay fitness and precision) than
+the existing approach [12], which also depends on synthesis rules. Moreover, our
+approach produces competitive model quality compared to the state-of-the-art:
+Inductive Miner - infrequent. For future work, we plan to explore more advanced
+ordering strategies and investigate their influences on the model quality and
+computation time. The other direction is to further speed up the approach as
+the long execution time is a clear limitation. This could be done by exploiting
+the log-based heuristics further.
+
+DELETE
+CHANGE
+IAGN
+CODE
+REOPEIDELETE
+CHANGE
+DIAGN
+CODE NOK
+EMPTY
+NEW
+CODE OK
+End
+FIN
+RELEASE
+BLLEDDiscovering Sound Free-choice Workflow Nets With Non-block Structures
+17
+Acknowledgements. We thank the Alexander von Humboldt (AvH) Stiftung
+for supporting our research.
+References
+1. van der Aalst, W.M.P.: The application of Petri nets to workflow management. J.
+Circuits Syst. Comput. 8(1), 21–66 (1998)
+2. van der Aalst, W.M.P.: Process Mining - Data Science in Action, Second Edition.
+Springer (2016)
+3. van der Aalst, W.M.P.: Using free-choice nets for process mining and business
+process management. In: FedCSIS 2021. vol. 25, pp. 9–15 (2021)
+4. van der Aalst, W.M.P., Adriansyah, A., van Dongen, B.F.: Replaying history on
+process models for conformance checking and performance analysis. WIREs Data
+Mining Knowl. Discov. 2(2), 182–192 (2012)
+5. Adriansyah, A., Munoz-Gama, J., Carmona, J., van Dongen, B.F., van der Aalst,
+W.M.P.: Measuring precision of modeled behavior. Inf. Syst. E Bus. Manag. 13(1),
+37–67 (2015)
+6. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Bruno, G.: Automated discovery
+of structured process models from event logs:the discover-and-structure approach.
+Data Knowl. Eng. 117, 373–392 (2018)
+7. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Maggi, F.M., Marrella, A.,
+Mecella, M., Soo, A.: Automated discovery of process models from event logs:
+Review and benchmark. IEEE Trans. Knowl. Data Eng. 31(4), 686–705 (2019)
+8. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Polyvyanyy, A.: Split miner:
+automated discovery of accurate and simple business process models from event
+logs. Knowl. Inf. Syst. 59(2), 251–284 (2019)
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+for discovering process trees. In: CEC 2012. pp. 1–8. IEEE (2012)
+10. Desel, J., Esparza, J.: Free Choice Petri Nets. No. 40, Cambridge university press
+(1995)
+11. Dixit, P.M.: Interactive Process Mining. Ph.D. thesis, Technische Universiteit Eind-
+hoven (2019)
+12. Dixit, P.M., Buijs, J.C.A.M., van der Aalst, W.M.P.: Prodigy : Human-in-the-loop
+process discovery. In: RCIS 2018. pp. 1–12. IEEE (2018)
+13. Dixit, P.M., Verbeek, H.M.W., Buijs, J.C.A.M., van der Aalst, W.M.P.: Interactive
+data-driven process model construction. In: ER 2018. vol. 11157, pp. 251–265.
+Springer (2018)
+14. van Dongen, B.F., de Medeiros, A.K.A., Wen, L.: Process mining: Overview and
+outlook of Petri net discovery algorithms. Trans. Petri Nets Other Model. Concurr.
+2, 225–242 (2009)
+15. Leemans, S.J.J., Fahland, D., van der Aalst, W.M.P.: Scalable process discovery
+and conformance checking. Softw. Syst. Model. 17(2), 599–631 (2018)
+16. Schuster, D., van Zelst, S.J., van der Aalst, W.M.P.: Incremental discovery of
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+

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+arXiv:2301.00094v1  [math.CV]  31 Dec 2022
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL
+SHEAVES ON SINGULAR COMPLEX SPACES AND REGULARITY OF
+WEAK KÄHLER-EINSTEIN METRICS
+ZHENQIAN LI
+Abstract. In this article, we will characterize regular points respectively by the local van-
+ishing, positivity of the Ricci curvature and L2-solvability of the ∂-equation together with
+Skoda’s theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubhar-
+monic (psh) functions on any (reduced) complex space of pure dimension. As a by-product,
+we show that any weak Kähler-Einstein metric on singular Q-Fano/Calabi-Yau/general
+type varieties cannot be smooth, and that in general there exists no singular normal Kähler
+complex space such that the Kähler metric is Kähler-Einstein on the regular locus.
+1. Introduction
+Throughout this note, all complex spaces are always assumed to be reduced and para-
+compact unless otherwise mentioned; we mainly refer to [19, 40] for basic references on
+the theory of complex spaces.
+1.1. Local vanishing for multiplier ideals. The local vanishing theorem for the higher
+direct images of sheaves computing multiplier ideals plays an important role in complex
+geometry and algebraic geometry, by which many local/global properties of multiplier
+ideal could be deduced, e.g., the restriction theorem, Nadel vanishing theorem and Skoda’s
+theorem for multiplier ideals and so on (cf. [10, 28], etc.).
+Let (X, ω) be a Hermitian complex space of pure dimension n and ϕ ∈ QPsh(X) be a
+quasi-psh function on X. Then we can define the Nadel-Lebesgue multiplier ideal sheaf
+INL(ϕ) associated to ϕ on X by the integrability with respect to the Lebesgue measure dVω
+(see Definition 2.2), which coincides with the usual multiplier ideal sheaf I (ϕ) introduced
+by Nadel whenever X is smooth. Let π : �X → X be any log resolution of the Jacobian ideal
+JacX of X, then it follows that the Nadel-Lebesgue multiplier ideal sheaf
+INL(ϕ) = π∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+.
+When X is smooth, the Mather discrepancy divisor �K�X/X is nothing but the relative canon-
+ical divisor K�X/X := K�X − π∗KX of �X over X. Then, we have the following local vanishing
+for multiplier ideals (cf. [28, 37])
+Rqπ∗
+�
+O�X(K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀q ≥ 1.
+Therefore, it is natural to ask whether we could establish a similar local vanishing result
+in the singular setting, i.e.,
+Rqπ∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀q ≥ 1.
+In the present note, one of our goals is to study the local vanishing in the context of Nadel-
+Lebesgue multiplier ideals. In particular, based on Skoda’s division for Nadel-Lebesgue
+Date: January 3, 2023.
+2010 Mathematics Subject Classification. 14F18, 32L20, 32Q20, 32S05, 32U05, 32W05.
+Key words. Multiplier ideal sheaves, plurisubharmonic functions, vanishing theorems, ∂-equations, Skoda’s
+L2 division theorem, Kähler-Einstein metrics.
+E-mail: [email protected].
+1
+
+2
+ZHENQIAN LI
+multiplier ideals, we will prove that such a local vanishing for Nadel-Lebesgue multiplier
+ideals is in fact equivalent to smoothness of the ambient space in some sense; see Theorem
+1.3 for a detailed statement.
+1.2. Skoda’s ideal generation by L2 estimates for the ∂-equation. In the classical works
+[44, 45], relying on the L2 methods due to [1, 25, 26] in several complex variables, Skoda
+established an analytic criterion on the ideal generation by a given collection of holo-
+morphic functions or sections. In the original proof of Skoda’s ideal generation, as well as
+standard techniques in functional analysis for the argument on a priori estimate and solving
+∂-equation with L2 estimates, he also developed special analytic techniques by restricting
+the domain of the ∂-operator to an appropriate subspace of the usual L2 space and inducing
+an L2 estimate on this new operator.
+As applications, Skoda’s theorem is a crucial ingredient in proving the Briançon-Skoda
+theorem in commutative algebra [6, 27, 35] and an effective version of the Nullstellensatz
+in algebraic geometry [13]. Moreover, a special case of Skoda’s ideal generation also
+played key roles in Siu’s works on the deformation invariance of plurigenera [42] and finite
+generation of the canonical ring [43]. The interaction between several complex variables,
+complex algebraic geometry and partial differential equations has been an attractive area
+for the researchers. For the sake of reader’s convenience, we state a version of Skoda’s L2
+division theorem as below.
+Theorem 1.1. ([45], Théorème 2). Let (X, ω) be an n-dimensional weakly pseudoconvex
+Kähler manifold with ϕ ∈ Psh(X), and g : E → Q be a surjective morphism of Hermitian
+holomorphic vector bundles with rE = rank E and rQ = rank Q. Suppose that E is Nakano
+semi-positive on X and L → X is a Hermitian line bundle such that
+√
+−1Θ(L) − ρ
+√
+−1Θ(det Q) ≥ 0
+for ρ = min{n, rE − rQ} + ε and some ε > 0.
+Then, for every f ∈ H0(X, Q ⊗ KX ⊗ L) satisfying
+�
+X
+⟨�
+gg∗ f, f⟩ · (det gg∗)−ρ−1e−2ϕdVω < +∞,
+there exists h ∈ H0(X, E ⊗ KX ⊗ L) such that f = g · h and
+�
+X
+|h|2 · (det gg∗)−ρe−2ϕdVω ≤ ρ
+ε ·
+�
+X
+⟨�
+gg∗ f, f⟩ · (det gg∗)−ρ−1e−2ϕdVω.
+Due to Theorem 1.1, if we consider the trivial bundles E, Q and L on a pseudoconvex
+domain, then by combining with the strong openness of multiplier ideal sheaves established
+by Guan-Zhou [21], we can reformulate Theorem 1.1 in the language of multiplier ideals
+as follows (cf. also Remark A.3):
+Theorem 1.2. Let X be an n-dimensional complex manifold with ϕ ∈ QPsh(X) a quasi-psh
+function and a ⊂ OX a nonzero ideal sheaf with r (local) generators. Then, it follows that
+I �ϕ + kϕa
+� = a · I �ϕ + (k − 1)ϕa
+�, ∀k ≥ min{n, r},
+where ϕa := 1
+2 log(�
+i |gi|2) and (gi) is any local system of generators of a.
+Motivated by the above reformulation of Theorem 1.1, it is interesting for us to explore
+an analogue to Theorem 1.2 for Nadel-Lebesgue multiplier ideals in the singular setting.
+In order to achieve such a goal, a natural idea is to generalize Skoda’s L2 methods to the
+singular case, i.e., creating an appropriate L2 theory for the ∂-operator on singular complex
+spaces. However, as presented in [15, 16], it seems not to be possible to establish a general
+theory as in the smooth setting to solve the ∂-equation with L2 estimates on complex spaces
+with singularities; one can refer to [17, 38, 39, 41] for some partial results on the related
+topics.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+3
+On the other hand, we can also consider to apply Theorem 1.1 near the singularities
+under some reasonable assumptions on the positivity of curvatures. Fortunately, we could
+show that positivity of the Ricci curvature on the regular locus is in fact equivalent to the
+desired Skoda’s ideal generation and L2-solvability of the ∂-equation. More precisely, we
+state our main result in the following:
+Theorem 1.3. Let X be a (Hermitian) complex space of pure dimension n with x ∈ X a
+normal point and π : �X → X a log resolution of the Jacobian ideal JacX of X. Then, the
+following statements are equivalent:
+(1) For each quasi-psh function ϕ near the point x ∈ X, we have
+Rqπ∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀q ≥ 1.
+(2) For each quasi-psh function ϕ near the point x ∈ X, we have
+Rqπ∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀1 ≤ q < n.
+(3) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω on Ω such that the Ricci curvature Ric(ω) ≥ 0 on the regular locus Ωreg of Ω.
+(4) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω and a C ∞ differentiable real function ψ on Ω such that Ric(ω) +
+√
+−1∂∂ψ ≥ 0
+on the regular locus Ωreg of Ω.
+(5) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω and a Hermitian line bundle L on Ω such that for any smooth ϕ ∈ SPsh(Ω) and
+v ∈ L2
+0,q(Ωreg, L) satisfying ∂v = 0 and
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞
+with the curvature operator Aϕ = [
+√
+−1∂∂ϕ, Λω] on Ωreg, we have u ∈ L2
+0,q−1(Ωreg, L)
+such that ∂u = v and
+�
+Ωreg
+|u|2e−2ϕdVω ≤
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω.
+(6) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω and a Hermitian line bundle L on Ω such that for any smooth ϕ ∈ SPsh(Ω) and
+v ∈ L2
+0,1(Ωreg, L) satisfying ∂v = 0 and
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞,
+we have u ∈ L2(Ωreg, L) such that ∂u = v and
+�
+Ωreg
+|u|2e−2ϕdVω ≤
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω.
+(7) The Skoda’s theorem holds for Nadel-Lebesgue multiplier ideals, i.e., for any
+nonzero ideal sheaf a with r generators and quasi-psh function ϕ near the point
+x ∈ X, it holds that
+INL
+�ϕ + kϕa
+� = a · INL
+�ϕ + (k − 1)ϕa
+�, ∀k ≥ min{n, r}.
+(8) For any nonzero ideal sheaf a near the point x ∈ X, it holds that
+INL
+�nϕa
+� = a · INL
+�(n − 1)ϕa
+�.
+(9) For any nonzero ideal sheaf a near the point x ∈ X, it holds that
+INL
+�nϕa
+� ⊂ a.
+(10) The point x ∈ X is a regular point of X.
+
+4
+ZHENQIAN LI
+In the above result, the most interesting and amazing point is that it presents several
+characterizations of regular points by various statements involved Nadel-Lebesgue mul-
+tiplier ideals, which look like almost irrelevant; e.g., (1, 2) in algebraic geometry, (3, 4)
+in differential geometry and (5, 6) in partial differential equations together with (7—9) in
+commutative algebra. The core idea of all arguments originates from the Skoda’s ideal
+generation by the L2 approaches in several complex variables.
+Remark 1.4. Simple examples show that the assumption that x is a normal point of X
+cannot be removed in Theorem 1.3; in particular, any of the statements (1, 2, 7, 8) will not
+imply (10) in that case.
+As a straightforward consequence of Theorem 1.3, we have
+Corollary 1.5. Any normal Kähler space with nonnegative Ricci curvature on the regular
+locus must be non-singular.
+1.3. Kähler-Einstein metrics on singular varieties. Let X be a normal Q-Gorenstein
+Kähler space, that is, a normal Kähler space whose canonical class KX defines a Q-line
+bundle on X. A Kähler current ω ∈ c1(±KX) is called a weak (or singular) Kähler-Einstein
+metric on X if ω has bounded local potentials and is a genuine Kähler-Einstein metric on
+the regular locus Xreg of X (cf. [3, 4, 5, 14], etc.). A weak Kähler-Einstein metric ω on
+X is called a Kähler-Einstein metric if ω is a Kähler metric on X, i.e., ω has smooth local
+potentials. For general expositions on the topic of Kähler-Einstein metrics one can refer to
+[2, 23, 47, 48, 49, 50] and the references therein. In particular, we state some recent results
+as follows.
+Theorem 1.6. ([3, 5, 14, 30, 31, 32, 36], etc.). Let X be a normal Q-Gorenstein complex
+projective variety. Then:
+(1) If X is a Q-Calabi-Yau variety with only log terminal singularities, then X admits
+a weak Kähler-Einstein metric.
+(2) If KX is ample, then X admits a weak Kähler-Einstein metric if and only if X is
+K-stable.
+(3) If −KX is ample, then X admits a weak Kähler-Einstein metric if and only if X is
+K-polystable.
+A basic and widely open problem in Kähler geometry/geometric analysis is understand-
+ing the geometric asymptotic behavior of the weak Kähler-Einstein metric near the singular
+locus Xsing of X. In [24], the authors made a breakthrough with a very precise descrip-
+tion for a class of Calabi-Yau varieties with smoothable isolated singularities, which are
+in further required to be isomorphic to a neighborhood of the vertex in a strongly regular
+Calabi-Yau cone; see also [7, 8, 18] for some recent progress in this direction. In more gen-
+eral situations, by using deep tools in the theory of degenerate complex Monge-Ampère
+equations on singular complex spaces, the continuity of local potentials of weak Kähler-
+Einstein metrics is established for all Q-Fano/Calabi-Yau varieties in [4, 22], but so far
+little is known for the higher order regularity in general and it is desirable to establish one
+for weak Kähler-Einstein potentials. However, relying on Theorem 1.3, we will see that
+too much regularity cannot be expected and in fact any weak Kähler-Einstein potential is at
+most C α (α < 2) differentiable near the singularities. In particular, we obtain the following
+Theorem 1.7. Let X be a normal Q-Gorenstein Kähler space admitting a weak Kähler-
+Einstein metric ω. Then, ω is smooth on X if and only if X is non-singular.
+2. Preliminaries
+Firstly, we introduce the notion of Nadel-Lebesgue multiplier ideal sheaf on any com-
+plex space of pure dimension and then present some useful facts used throughout this note.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+5
+Definition 2.1. Let X be a complex space of pure dimension and ϕ ∈ L1
+loc(Xreg) with
+respect to the Lebesgue measure. Then, the complex space X is said to be a Hermitian
+complex space if there is a Hermitian metric ω on the regular part (may be disconnected)
+Xreg of X such that ω is locally the restriction of a Hermitian metric on some CN for a local
+embedding of X. It follows from the differentiable partition of unity that every complex
+space is a Hermitian complex space as in the smooth case.
+The complex space X is called to be a Kähler space if there is a Hermitian metric ω on X
+such that ω is locally the restriction of a Kähler metric on some CN for a local embedding
+of X. In particular, it admits smooth strictly psh functions as local potentials.
+We say that the function ϕ is quasi-plurisubharmonic (quasi-psh for short) on X if it is
+locally equal to the sum of a psh function and of a smooth function on X. The set of quasi-
+psh (resp. psh and strictly psh) functions on X is denoted by QPsh(X) (resp. Psh(X) and
+SPsh(X)). A quasi-psh function ϕ ∈ QPsh(X) will be said to have analytic singularities on
+X if ϕ can be written locally as
+ϕ = c
+2 log(|f1|2 + · · · + |fN0|2) + O(1),
+where c ∈ R≥0 and (fi) are holomorphic functions.
+Definition 2.2. Let (X, ω) be a Hermitian complex space of pure dimension and ϕ ∈
+L1
+loc(Xreg) with respect to the Lebesgue measure.
+The Nadel-Lebesgue multiplier ideal sheaf associated to ϕ on X is defined to be the
+OX-submodule INL(ϕ) ⊂ MX of germs of meromorphic functions f ∈ MX,x such that
+|f|2e−2ϕ is integrable with respect to the Lebesgue measure dVω near the point x ∈ X. One
+can check that INL(ϕ) is independent of the choice of Hermitian metric ω on X.
+The log canonical threshold (or complex singularity exponent) LCTx(ϕ) of ϕ at a point
+x ∈ X is defined to be
+LCTx(ϕ) := sup �c ≥ 0 | OX,x ⊂ INL(cϕ)x
+� .
+It is convenient to put LCTx(−∞) = 0.
+It is easy to see that INL(ϕ) ⊂ OX is an ideal sheaf when X is a normal complex space
+and ϕ is locally bounded from above on X. In addition, if X is smooth and ϕ ∈ QPsh(X),
+then INL(ϕ) is nothing but the usual multiplier ideal sheaf I (ϕ) introduced by Nadel (see
+[10]).
+Remark 2.3. Since the definition of Nadel-Lebesgue multiplier ideals is local, we can com-
+pute the multiplier ideals by choosing a special Hermitian metric ω for a local embedding
+of X. In particular, if X is an n-dimensional complex subspace of some domain in CN,
+we can take Hermitian metric ω on X to be the inherited standard Kähler metric from CN.
+Then, we have dVω = 1
+n!υn|Xreg, where υ =
+√
+−1
+2
+N�
+k=1
+dzk ∧ d¯zk.
+For the sake of reader’s convenience, we state a basic estimate related to local volume
+of an analytic subset as follows.
+Lemma 2.4. ([20], Lemma 2.3). Let X be a pure n-dimensional analytic subset through
+the origin 0 of some domain in CN (N ≥ 2). Then, there is a Stein neighborhood U ⊂⊂ CN
+of the origin 0 such that for any 0 ≤ ε < 1, we have
+�
+U∩X
+1
+(|z1|2 + · · · + |zN|2)n−1+ε dVω < +∞,
+where dVω = 1
+n!υn|Xreg and υ =
+√
+−1
+2
+N�
+k=1
+dzk∧d¯zk.
+Analogous to the Nadel-Ohsawa multiplier ideal sheaves introduced in [33, 34] (see also
+[9, 12] for the algebro-geometric counterpart), we state some related properties as follows.
+
+6
+ZHENQIAN LI
+Proposition 2.5. (1) Let π : �X → X be a log resolution of the Jacobian ideal JacX of X
+and �K�X/X be the Mather discrepancy divisor. Then, we have the image
+Im
+�
+π∗Ωn
+X ֒→ Ωn
+�X
+�
+= O�X(−�K�X/X) · Ωn
+�X,
+and
+INL(ϕ) = π∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+.
+Furthermore, we can deduce that INL(ϕ + log |JacX|) = INO(ϕ), the Nadel-Ohsawa
+multiplier ideal sheaf associated to ϕ on X.
+(2) When X is normal and ϕ has analytic singularities, INL(ϕ) coincides with the
+Mather multiplier ideal sheaf defined in [9].
+(3) For any ϕ ∈ QPsh(X), it follows that INL(ϕ) ⊂ MX is a coherent fractional ideal
+sheaf and satisfies the strong openness, i.e., INL(ϕ) = �
+ε>0
+INL
+�(1 + ε)ϕ�.
+For our proof of Theorem 1.3, we need the following L2 estimates for the ∂-equation
+and relative version of Grauert-Riemenschneider vanishing theorem for the higher direct
+images.
+Theorem 2.6. (cf. [10], Theorem 5.2). Let (X, ω) be an n-dimensional Kähler manifold,
+which contains a weakly pseudoconvex Zariski open subset. Let L be a Hermitian line
+bundle on X such that
+√
+−1Θ(L) + Ric(ω) > 0.
+Then, for every smooth ϕ ∈ Psh(X) and v ∈ L2
+0,q(X, L) satisfying ∂v = 0 and
+�
+X
+⟨A−1v, v⟩ e−2ϕdVω < +∞
+with the curvature operator A = [
+√
+−1Θ(L) + Ric(ω) +
+√
+−1∂∂ϕ, Λω] on X, there exists
+u ∈ L2
+0,q−1(X, L) such that ∂u = v and
+�
+X
+|u|2e−2ϕdVω ≤
+�
+X
+⟨A−1v, v⟩ e−2ϕdVω.
+Theorem 2.7. ([11], Theorem 1.1). Let (X, ω) be an n-dimensional Kähler manifold which
+is a Zariski open subset of some Stein space X∗, and L be a Hermitian line bundle on X.
+If for any smooth ϕ ∈ SPsh(X∗) and v ∈ L2
+0,1(X, L) satisfying ∂v = 0 and
+�
+X
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞
+with the curvature operator Aϕ = [
+√
+−1∂∂ϕ, Λω] on X, there exists u ∈ L2(X, L) such that
+∂u = v and
+�
+X
+|u|2e−2ϕdVω ≤
+�
+X
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω,
+then it follows that L ⊗ K−1
+X is Nakano semi-positive on X.
+Theorem 2.8. ([37], Corollary 1.5). Let π : X → Y be a surjective proper (locally) Kähler
+morphism from a complex manifold X to a complex space Y, and (L, e−ϕL) be a (possibly
+singular) Hermitian line bundle on X with semi-positive curvature. Then, the higher direct
+image sheaf
+Rqπ∗
+�
+KX ⊗ L ⊗ I (ϕL)
+�
+= 0,
+for every q > dim X − dim Y.
+Remark 2.9. Any log resolution π : �X → X of a coherent ideal sheaf I on a complex space
+X is a locally Kähler (proper modification), which is locally a finite sequence of blow-ups
+with smooth centers. Besides, any finite holomorphic mapping between complex spaces is
+(locally) proper Kähler.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+7
+In the remainder of this section, we recall some algebraic properties on the integral
+closure of ideals.
+Definition 2.10. ([46]). Let R be a commutative ring and I an ideal of R. An element
+f ∈ R is said to be integrally dependent on I if it satisfies a relation
+f d + a1 f d−1 + · · · + ad = 0
+(ak ∈ Ik, 1 ≤ k ≤ d).
+The set I consisting of all elements in R which are integrally dependent on I is called
+the integral closure of I in R. I is called integrally closed if I = I. One can prove that I is
+an ideal of R, which is the smallest integrally closed ideal in R containing I.
+Definition 2.11. ([46]). Let R be a commutative ring with identity and let J ⊂ I be ideals in
+R. J is said to be a reduction of I if there exists a nonnegative integer n such that In+1 = JIn.
+A reduction J of I is called minimal if no ideal strictly contained in J is a reduction of
+I. An ideal that has no reduction other than itself is called basic.
+One can prove that minimal reductions do exist in Noetherian local rings and an ideal
+which is a minimal reduction of a given ideal is necessarily basic. Moreover, if R is a
+Noetherian ring, J ⊂ I is a reduction of I if and only if J = I.
+In the analytic setting, we have the following characterization on integral closure and
+reduction of ideals.
+Theorem 2.12. (cf. [29], Théorème 2.1). Let X be a complex space and Y ⊂ X be a
+proper closed complex subspace (may be non-reduced) defined by a coherent OX-ideal I
+with x ∈ Y a point. Let J ⊂ OX be a coherent OX-ideal and I (resp. J) be the germ of
+I (resp. J ) at x. Then, the following conditions are equivalent:
+(1) J ⊂ I.
+(2) For every morphism π : �X → X satisfying: (i) π is a proper and surjective, (ii) �X is
+a normal complex space and (iii) I · O�X is an invertible OX-module, there exists
+an open neighborhood U of x in X such that
+J · O�X|π−1(U) ⊂ I · O�X|π−1(U).
+(3) If V is an open neighborhood of x on which I and J are generated by their
+global sections, then for every system of generators g1, ..., gr ∈ Γ(V, I ) and every
+f ∈ Γ(V, J ), one can find an open neighborhood V′ of x and a constant C > 0
+such that
+|f(y)| ≤ C · sup
+k
+|gk(y)|, ∀y ∈ V′.
+Remark 2.13. Let X be a normal complex space and I ⊂ OX a coherent ideal sheaf. Let
+π : �X → X be any proper modification from a normal complex space �X onto X such that
+I · O�X = O�X(−D) for some effective Cartier divisor D on �X. Then, we have π∗O�X(−D) =
+I , the integral closure of I in OX.
+Lemma 2.14. (cf. Example 9.6.19 in [28]; see also [10], Lemma 11.16). Let X be a
+normal complex space of dimension n and a ⊂ OX a nonzero ideal. Then, there exists
+an open covering {Uα}α∈N of X such that a|Uα has a reduction bα generated by at most n
+elements.
+3. Proofs of the main results
+3.1. Proof of Theorem 1.3. Since all of the statements are local, without loss of gener-
+ality, we may assume that X is an n≥2-dimensional normal (Hermitian) complex subspace
+of some domain in CN with ϕ ∈ QPsh(X) and a = (g1, . . ., gr) · OX an ideal sheaf gen-
+erated by holomorphic functions g1, . . . , gr on X. Moreover, we may also assume that ϕ
+is (locally) a strictly psh function on X if necessary, by adding some smooth strictly psh
+
+8
+ZHENQIAN LI
+function. It is easy to see that the implications (1) =⇒ (2), (3) =⇒ (4), (5) =⇒ (6) and
+(7) =⇒ (8) =⇒ (9) are trivial; in particular, we will present a proof in the following order:
+(1)
+� (2)
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+(10)
+�④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+�
+(9)
+�
+(8)
+�
+(7)
+�
+(3)
+�❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+� (4)
+�
+(5)
+� (6)
+�⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+“(2) =⇒ (7)”. By the definition of Nadel-Lebesgue multiplier ideal sheaf, it follows
+that
+a · INL
+�ϕ + (k − 1)ϕa
+� ⊂ INL
+�ϕ + kϕa
+�,
+and so it is sufficient to show the reverse inclusion.
+Case (i). When r ≤ n.
+Let π : �X → X be a common log resolution of JacX and a such that a · O�X = O�X(−F)
+for some effective divisors F on �X. Denote by
+Am := O�X(�K�X/X) ⊗ I (ϕ ◦ π + mϕa ◦ π)
+= O�X(�K�X/X) ⊗ I (ϕ ◦ π) ⊗ O�X(−mF)
+for any m ∈ N, and consider the Koszul complex determined by g1, . . ., gr:
+0 → ΛrV ⊗ O�X(rF) → · · · → Λ2V ⊗ O�X(2F) → V ⊗ O�X(F) → O�X → 0,
+where V is the vector space spanned by g1, . . . , gr. Note that the Koszul complex is locally
+split and its syzygies are locally free, so twisting through by any coherent sheaf will pre-
+serve the exactness. Then, by twisting with Ak (k ≥ r), we obtain the following long exact
+sequence
+0 → ΛrV ⊗ Ak−r → · · · → Λ2V ⊗ Ak−2 → V ⊗ Ak−1 → Ak → 0.
+(⋆)
+On the other hand, for any m ∈ N, by (2) we have the local vanishing of the higher direct
+images Rqπ∗Am = 0 (1 ≤ q < n). Note that
+INL
+�ϕ + mϕa
+� = π∗Am
+by the functoriality property with respect to direct images of sheaves by modifications, and
+then by taking direct images of (⋆) we will deduce the following so-called exact Skoda
+complex (cf. [28], p. 228):
+0 → ΛrV ⊗ INL
+�ϕ + (k − r)ϕa
+� → · · · → V ⊗ INL
+�ϕ + (k − 1)ϕa
+� → INL
+�ϕ + kϕa
+� → 0.
+In particular, the map V ⊗ INL
+�ϕ + (k − 1)ϕa
+� → INL
+�ϕ + kϕa
+� is surjective, by which we
+can infer that INL
+�ϕ + kϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+� for any k ≥ r.
+Case (ii). When r > n.
+As the statement is local, then by Lemma 2.14 we may assume that b is a reduction of
+a generated by n elements �g1, ...,�gn. Consider a common log resolution π : �X → X of
+JacX, a and b such that a · O�X = b · O�X = O�X(−F) for some effective divisors F on �X.
+Then, by the same argument as above, we can deduce the following exact Skoda complex:
+0 → ΛnV ⊗ INL
+�ϕ + (k − n)ϕa
+� → · · · → V ⊗ INL
+�ϕ + (k − 1)ϕa
+� → INL
+�ϕ + kϕa
+� → 0.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+9
+for any k ≥ n, where V is the vector space spanned by �g1, ...,�gn. Therefore, it follows that
+INL
+�ϕ + kϕa
+� ⊂ b · INL
+�ϕ + (k − 1)ϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+�.
+“(3) =⇒ (5)”. It follows from the assumption that we have a Stein neighborhood Ω ⊂⊂
+X of the point x with a Kähler metric ω such that Ric(ω) ≥ 0 on Ωreg. Let ϕ ∈ SPsh(Ω) be
+any smooth strictly psh function on Ω and L = Ω × C be a trivial bundle equipped with the
+trivial Hermitian metric, which implies that
+√
+−1Θ(L) + Ric(ω) +
+√
+−1∂∂ϕ ≥
+√
+−1∂∂ϕ > 0
+on Ωreg.
+Since Ω is a Stein space, we are able to choose a complex hypersurface Z ⊂ Ω which
+contains the singular locus Ωsing of Ω such that Ω − Z ⊂ Ωreg is a Stein manifold. Then, by
+Theorem 2.6 we obtain that, for any smooth ϕ ∈ SPsh(Ω) and v ∈ L2
+0,q(Ωreg, L) satisfying
+∂v = 0 and
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞,
+we can find u ∈ L2
+0,q−1(Ωreg, L) such that ∂u = v and
+�
+Ωreg
+|u|2e−2ϕdVω ≤
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω.
+“(6) =⇒ (4)”. As a straightforward application of Theorem 2.7 on Ωreg, it yields that
+√
+−1Θ(L) + Ric(ω) ≥ 0 on Ωreg. Let Ω′ ⊂ Ω be a small Stein neighborhood of the point
+x such that the Hermitian line bundle L has a smooth potential ψ on Ω′. Therefore, we
+deduce that
+√
+−1∂∂ψ + Ric(ω) =
+√
+−1Θ(L) + Ric(ω) ≥ 0
+on Ω′
+reg.
+“(4) =⇒ (7)”. Due to the definition and Lemma 2.14, it is sufficient to prove
+INL
+�ϕ + kϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+�
+for the case r ≤ n near the point x ∈ X. Let f ∈ INL
+�ϕ + kϕa
+�
+x with k ≥ min{n, r} = r,
+then by the strong openness of multiplier ideals there exists small enough ε > 0 such that
+f ∈ INL
+�ϕ + (k + ε)ϕa
+�
+x.
+By the assumption of (4), we let Ω ⊂⊂ X be a Stein neighborhood of the point x with
+a Kähler metric ω and a smooth real function ψ on Ω such that Ric(ω) +
+√
+−1∂∂ψ ≥ 0 on
+Ωreg. After shrinking Ω if necessary, we may assume that the function ψ is bounded on Ω
+and f is holomorphic on Ω such that
+�
+Ω
+|f|2 · |g|−2(r+ε)e−2(ϕ+(k−r)ϕa)dVω < +∞.
+In addition, we also choose a complex hypersurface Z ⊂ Ω which contains the singular
+locus Ωsing of Ω and the common zero-set of holomorphic functions g1, ..., gr such that
+Ω′ := Ω − Z is a Stein manifold.
+Let E = Ω′ × Cr and Q = Ω′ × C be the trivial bundles on Ω′ and L = K−1
+Ω′ be the
+anti-canonical line bundle with the induced metric twisted by a weight e−ψ. The morphism
+g : E → Q determined by holomorphic functions g1, ..., gr is given by
+(h1, ..., hr) �→
+r�
+m=1
+gm · hm = g · h.
+Note that �
+gg∗ = IdQ when rank Q = 1, and on Ω′ we have
+√
+−1Θ(L) − (r − 1 + ε)
+√
+−1Θ(det Q) = Ric(ω) +
+√
+−1∂∂ψ ≥ 0.
+
+10
+ZHENQIAN LI
+Thus, we can apply Theorem 1.1 on Ω′ and then obtain an r-tuple (h1, ..., hr) of holomor-
+phic functions on Ω′ such that f = g · h on Ω′ and
+�
+Ω′ |h|2 · |g|−2(r−1+ε)e−2(ϕ+(k−r)ϕa)dVω =
+�
+Ω′ |h|2e−2(ϕ+(k−1+ε)ϕa)dVω < +∞.
+We can now extend every hm to be a holomorphic function on Ω from the L2 estimate above
+and normality of X, which implies that
+INL
+�ϕ + kϕa
+� ⊂ a · INL
+�ϕ + (k − 1 + ε)ϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+�
+on Ω; we finish the argument.
+“(9) =⇒ (10)”. By the assumption, we have INL
+�nϕa
+� ⊂ a. Suppose that x ∈ X
+is a singular point. Then, by the local parametrization for analytic sets, we can find a
+local coordinate system (z′; z′′) = (z1, ..., zn; zn+1, ..., zN) near x such that for some constant
+C > 0, we have |z′′| ≤ C · |z′| for any point z ∈ X near x.
+Let a ⊂ OX be the ideal sheaf generated by holomorphic functions �z1, ...,�zn ∈ OX
+(shrinking X if necessary), where �zk are the residue classes of zk in OX. From the non-
+smoothness of X at the point x, we deduce that the embedding dimension dimC(mX,x/m2
+X,x) ≥
+n + 1 of X at x, which implies that there exists k0 (n + 1 ≤ k0 ≤ N) such that�zk0 � a.
+On the other hand, after shrinking X again, it follows that
+�
+X
+|zk0|2
+|z′|2n dVω ≤ C2(1 + C2)n−1 ·
+�
+X
+|z|−2(n−1)dVω < +∞,
+where the finiteness of the integration follows from Lemma 2.4. Then, we infer that�zk0 ∈
+INL
+�nϕa
+�, but�zk0 � a, which contradicts to the assumption INL
+�nϕa
+� ⊂ a. Thus, we obtain
+that x ∈ X is a regular point.
+“(10) =⇒ (1)”. It is a straightforward consequence of Theorem 2.8.
+“(10) =⇒ (3)”. Since x is a regular point of X, after choosing an appropriate coordinate
+neighborhood of x, we may assume that Ω ∋ x is a Stein domain in Cn. Therefore, we can
+take ω =
+√
+−1
+2
+n�
+k=1
+dzk ∧ d¯zk to be the standard Euclidean metric on Cn and then we have
+Ric(ω) = 0 on Ω; the proof of Theorem 1.3 is concluded.
+□
+Remark 3.1. In addition, we can deduce from the proof of Theorem 1.3 that
+(i) if (1) or (2) holds for each quasi-psh function ϕ with analytic singularities, then x ∈ X
+is a regular point;
+(ii) both of the statements (3) and (4) could be respectively modified to be Ric(ω) ≥ 0
+and Ric(ω) +
+√
+−1∂∂ψ ≥ 0 on a Zariski open subset of Ω contained in Ωreg.
+3.2. Proof of Theorem 1.7. It is sufficient to prove the necessity.
+Let x ∈ X be any point. Since ω is a smooth Kähler metric on X, then ω has smooth
+local potentials, i.e., there exists a Stein neighborhood Ω ⊂ X of x and a smooth strictly psh
+functions ψ on Ω such that ω =
+√
+−1∂∂ψ on Ωreg, which implies that Ric(ω)+
+√
+−1∂∂ψ ≥ 0
+on Ωreg whenever Ric(ω) = ±ω, 0. Thus, it follows from (4) in Theorem 1.3 that x ∈ X is
+a regular point.
+□
+Remark 3.2. The same arguments as in the proof of Theorem 1.3 and 1.7 also imply that
+each local potential of weak Kähler-Einstein metric ω is C 2 differentiable on X if and only
+if X is non-singular, and that there exists no singular normal Kähler space such that the
+Kähler metric is Kähler-Einstein on the regular locus.
+In fact, our method is still available when the weak Kähler-Einstein metric is (locally)
+equivalent to the standard induced Kähler metric by restriction near the singularities; for
+instance, when the weak Kähler-Einstein metric is of locally bounded coefficients.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+11
+Appendix A. Uniform bounds of powers associated to an L2 division problem
+The ideal membership is an important object to study in commutative algebra, algebraic
+geometry and several complex variables, e.g., the famous Hilbert’s Nullstellensatz and
+Briançon-Skoda theorem and so on. In this part, we are mainly interested in the uniform
+bounds of powers associated to an L2 division problem, a kind of special ideal membership.
+Let X be a Stein manifold of dimension n and a = (g1, . . . , gr)·OX an ideal sheaf generated
+by holomorphic functions g1, . . . , gr on X. In general, the division problem states that,
+given a positive integer k ∈ N and holomorphic function f on X, we wish to determine
+when f is generated by holomorphic functions g1, . . ., gr; more precisely, when we can
+find holomorphic functions h1, . . . , hr ∈ ak−1 on X such that
+f =
+r�
+m=1
+gm · hm.
+Thanks to the Oka-Cartan’s theory on Stein manifolds, the division problem is solvable if
+and only if f ∈ ak.
+Note that the condition f ∈ ak is purely algebraic, and so it is natural to ask whether we
+could find an analytic condition to replace the algebraic one. It is easy to see that f ∈ ak
+implies that |f|e−ϕk is locally bounded on X, or L2
+loc more generally; where ϕk := k log |g|
+and |g|2 := |g1|2 + · · · + |gr|2. On the other hand, local boundedness of |f|e−ϕk is equivalent
+to the fact that f ∈ ak, the integral closure of ak in OX (see Theorem 2.12). Thus, it is
+an interesting question whether we could establish solvability of an L2 analogue of the
+division problem.
+Let ϕ ∈ Psh(X) be a psh function on X and denote by
+A2
+loc(X, ϕ) :=
+�
+f ∈ OX(X)
+��� |f|2e−2ϕ is locally integrable on X
+�
+.
+Then, we raise the following L2 division problem:
+Question A.1. Let X be an n-dimensional Stein manifold with a psh function ϕ ∈ Psh(X),
+and a = (g1, . . . , gr) · OX an ideal sheaf generated by holomorphic functions g1, . . ., gr on
+X. Given positive integer k ∈ N and f ∈ A2
+loc(X, ϕ + ϕk), are there holomorphic functions
+h1, . . ., hr ∈ A2
+loc(X, ϕ + ϕk−1) such that
+f =
+r�
+m=1
+gm · hm?
+A.1. A solution to Question A.1. Unfortunately, the answer of Question A.1 is negative
+for general k (see Example A.2). Motivated by the Skoda’s L2 division theorem (cf. The-
+orem 1.1), it seems to be reasonable to find a uniform integer k0, depending only on n,
+such that Question A.1 is solvable for any k ≥ k0. The goal of this part is to present an
+optimal uniform lower bounds of powers associated to Question A.1. In particular, we will
+establish the following
+Theorem A.1. There exists a uniform integer k0 = min{n, r} such that the solution to
+Question A.1 is positive for any k ≥ k0. In further, the uniform lower bound k0 = min{n, r}
+is optimal.
+In fact, the optimality of uniform integer k0 = min{n, r} is straightforward by the fol-
+lowing:
+Example A.2. Let Bn(0) be the unit ball centered at the origin 0 = (0′, 0′′) in Cr×Cn−r (1 ≤
+r ≤ n) and take g1 = z1, ..., gr = zr, f ≡ 1, ϕ ��� 0 on Bn(0). Then, for every k < k0 = r, the
+answer of Question A.1 is negative.
+Indeed, by the fact that the log canonical threshold LCT(0′,z′′)(ϕk) = r
+k > 1 of ϕk at any
+point (0′, z′′), one can derive that f ∈ A2
+loc(Bn(0), ϕ + ϕk). Then, we infer from the fact that
+
+12
+ZHENQIAN LI
+f has no zeros in Bn(0) that there exist no holomorphic functions h1, . . . , hr on Bn(0) such
+that f =
+r�
+m=1
+gm · hm.
+Proof of Theorem A.1. It follows from the local vanishing (Theorem 2.8) and the argu-
+ments as in the proof of Theorem 1.3 that for any k ≥ min{n, r}, we have
+I �ϕ + ϕk
+� = a · I �ϕ + ϕk−1
+�.
+Let
+τ : I �ϕ + ϕk−1
+�⊕r −→ I �ϕ + ϕk
+�
+be the sheaf homomorphism defined by
+τ(h1,x, . . . , hr,x) =
+r�
+m=1
+gm · hm,x
+for any germs hm,x ∈ I �ϕ + ϕk−1
+�
+x. Then, we have an exact sequence of sheaves
+I �ϕ + ϕk−1
+�⊕r
+τ
+−→ I �ϕ + ϕk
+� −→ 0.
+It follows from the Oka-Cartan theory on Stein manifolds that the induced sequence of
+sections
+Γ
+�
+X, I �ϕ + ϕk−1
+�⊕r�
+τ∗
+−→ Γ
+�
+X, I �ϕ + ϕk
+��
+−→ 0
+is also exact, which implies that any section f ∈ Γ
+�
+X, I �ϕ + ϕk
+��
+can be written as the
+image f =
+r�
+m=1
+gm · hm for some sections hm ∈ Γ
+�
+X, I �ϕ + ϕk−1
+��
+.
+□
+Remark A.3. (An alternative argument on Theorem A.1). In fact, we could also give an-
+other argument on the proof of Theorem A.1 depending on the strong openness of multi-
+plier ideals established by Guan-Zhou [21] and the Skoda’s L2 division theorem for holo-
+morphic functions (see Theorem 1.1).
+Since the statement is local, it follows from Lemma 2.14 that it is sufficient to prove
+I �ϕ+ϕk
+� ⊂ a·I �ϕ+ϕk−1
+� for the case r ≤ n. Given f ∈ Γ
+�
+X, I �ϕ+ϕk
+��
+, after shrinking
+X, we may assume that X is the unit ball in Cn and
+�
+X
+|f|2e−2(ϕ+ϕk)dλn =
+�
+X
+|f|2 · |g|−2ke−2ϕdλn < +∞.
+Then, for each k ≥ r, by the strong openness of multiplier ideals there exists sufficiently
+small ε > 0 such that
+�
+X
+|f|2e−2(ϕ+(1+ε)ϕk)dλn =
+�
+X
+|f|2 · |g|−2(1+ε)ke−2ϕdλn < +∞,
+shrinking X if necessary. Finally, combining with Theorem 1.1, we deduce the desired
+result.
+A.2. A global L2 version of Question A.1. Let (X, ω) be an n-dimensional Stein manifold
+with a Kähler form ω. Let ϕ ∈ Psh(X) and I = (g1, . . . , gr) · OX an ideal sheaf generated
+by holomorphic functions g1, . . . , gr on X. Denote by
+A2(X, ϕ) :=
+�
+f ∈ OX(X)
+�����
+�
+X
+|f|2e−2ϕdVω < +∞
+�
+.
+Then, we have the following global analogue of Question A.1:
+Question A.2. Can we find a uniform integer k0 such that for each k ≥ k0 and f ∈ A2(X, ϕ+
+ϕk), there exist h1, . . ., hr ∈ A2(X, ϕ + ϕk−1) satisfying
+f =
+r�
+m=1
+gm · hm?
+As an immediate consequence of Theorem 1.1, we obtain the following
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+13
+Theorem A.4. Let X be a pseudoconvex domain in Cn. Then, there exists a uniform integer
+k0 = min{n + 2, r + 1} such that the solution to Question A.2 is positive.
+Remark A.5. (1) More generally, Theorem A.4 also holds for any complete Kähler domain
+in Cn with smooth psh function ϕ ∈ Psh(X).
+(2) In this case, combining with the Example A.2, it follows that the optimal uniform
+lower bound k0 is at least min{n, r}, and at most min{n + 2, r + 1}.
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+

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+Fighting the sign problem in a chiral random matrix model with contour deformations
+Matteo Giordano,1 Attila Pásztor,1 Dávid Pesznyák,1 and Zoltán Tulipánt1
+1ELTE Eötvös Loránd University, Institute for Theoretical Physics,
+Pázmány Péter sétány 1/A, H-1117, Budapest, Hungary
+We studied integration contour deformations in the chiral random matrix theory of Stephanov [1]
+with the goal of alleviating the finite-density sign problem. We considered simple ansätze for the
+deformed integration contours, and optimized their parameters.
+We find that optimization of a
+single parameter manages to considerably improve on the severity of the sign problem. We show
+numerical evidence that the improvement achieved is exponential in the degrees of freedom of the
+system, i.e., the size of the random matrix. We also compare the optimization method with contour
+deformations coming from the holomorphic flow equations.
+I.
+INTRODUCTION
+Euclidean quantum field theories at non-zero particle
+density (or chemical potential) generally suffer from a
+complex action problem: the weights in the path integral
+representation are complex, and thus cannot be inter-
+preted as a joint probability density function on the space
+of field configurations (up to a proportionality factor).
+This prevents the use of importance sampling methods
+for the direct simulation of these theories. In QCD, this
+complex action problem severely hampers first-principles
+studies of dense matter in the core of neutron stars, neu-
+tron star mergers, core collapse supernovae, as well as in
+heavy ion collisions at certain collision energies.
+In the presence of a complex action problem one can
+still (in principle) simulate a modified theory with real
+and positive weights, and then use reweighting methods
+to calculate observables in the theory of interest. If the
+target theory has field variables φ, path integral weights
+wt(φ), and partition function Zt =
+�
+Dφ wt(φ), and the
+simulated theory has the same field variables, but dif-
+ferent – real and positive – path integral weights ws(φ)
+and partition function Zs =
+�
+Dφ ws(φ), we can obtain
+expectation values in the target theory via the formula
+⟨O⟩t =
+�
+wt
+ws O
+�
+s
+�
+wt
+ws
+�
+s
+,
+⟨O⟩x = 1
+Zx
+�
+Dφ wx(φ)O(φ) , (1)
+where x may stand for t or s and O(φ) is some phys-
+ical observable of interest. The denominator in Eq. (1)
+gives the ratio of the partition functions in the target and
+simulated theories, i.e.,
+� wt
+ws
+�
+s
+= Zt
+Zs
+.
+(2)
+This ratio is typically exponentially small in the physical
+volume, with the exponent given by the free energy dif-
+ference between the target and simulated theories. This
+ratio is also a rough measure of the numerical difficulty
+of a given reweighting scheme, with a given simulated
+and target theory. In order for reweighting to be effec-
+tive, one wants the target and simulated theories to be as
+close to each other as possible. Ideally, one should find a
+simulated theory with Zs ≈ Zt.
+Two simple choices of a simulated theory are the phase-
+quenched (PQ) theory, with simulated weights propor-
+tional to
+wPQ
+s
+≡ |wt(φ)| ,
+(3)
+or – assuming that the partition function Zt is real –
+the sign-quenched (SQ) theory, with simulated weights
+proportional to
+wSQ
+s
+≡ |Re wt(φ)| .
+(4)
+For the first case (phase reweighting) the reweighting
+factors wt/wPQ
+s
+≡ eiθ are pure phases.
+For the sec-
+ond case (sign reweighting) the reweighting factors are
+wt/wSQ
+s
+= eiθ/ |cos θ|. For certain observables, such as
+manifestly real observables or observables with a con-
+jugation (φ → φ) symmetry, one can substitute wt/wPQ
+s
+with cos θ and wt/wSQ
+s
+with a pure sign cos θ/ |cos θ|. For
+phase or sign reweighting, we can then say that the com-
+plex action problem becomes a sign problem: the cancel-
+lations between contributions with different signs of cosθ
+lead to a small Zt
+Zs ratio, and in turn to small signal-to-
+noise ratios in the expectation values of observables.
+The sign-quenched ensemble always has a less severe
+sign problem, due to the inequality Zt < ZSQ
+s
+< ZPQ
+s
+,
+which is a consequence of cos θ ≤ | cos θ| ≤ 1. However, in
+the limit of a severe sign problem – i.e., as the distribution
+of the argument θ tends to to a uniform distribution on
+[−π, π) – the severity of the sign problem for these two
+reweighting schemes only differs by a constant factor [2],
+given by
+�
+ZPQ
+s
+/ZSQ
+s
+�2 → (π/2)2.
+In QCD and in other (more or less) QCD-like models,
+describing the interactions of several “flavors” of fermions,
+the path integral weights can be written schematically as
+wt(φ) = det M1(φ, µ1) . . . det MNf (φ, µNf )e−SB(φ), (5)
+where the fields φ are real bosonic variables and SB
+is the corresponding bosonic part of the action, Nf is
+the number of fermion flavors in the model, det Mk is
+the fermionic determinant of the kth flavor and µk is
+the corresponding chemical potential, for k = 1, . . . , Nf.
+arXiv:2301.12947v1  [hep-lat]  30 Jan 2023
+
+2
+The source of the sign problem is the fermionic determi-
+nant, which at non-zero µ is generally a complex number.
+Moreover, an important feature of the sign problem in
+QCD and QCD-like theories is that it tends to get much
+worse in the ranges of µ where zeros of the determinant
+in the complex µ plane become dense [3].
+Nonetheless, reweighting from the phase- and sign-
+quenched theories is starting to become feasible even in
+full QCD [2, 4], which has recently led to the calculation
+of the equation of state of a hot-and-dense quark-gluon
+plasma in the region of chemical potentials covered by
+the RHIC Beam Energy Scan [5]. However, the range of
+practical applicability of such an approach is limited both
+in volume and chemical potential by the smallness of the
+ratio Zt/Zs. Lacking a solution of the sign problem, it is
+then desirable to develop methods that at least alleviate
+it, to extend the range of parameters that reweighting
+methods can practically reach.
+One possible route to do this is the use of contour
+deformations in the path integral (see Ref. [6] for a re-
+cent review). If the path integral weights wt(φ) are holo-
+morphic functions of the field variables,1 the multivariate
+Cauchy theorem guarantees that complexified integration
+manifolds in the same homology class as the original one
+yield the same partition function. However, the phase-
+and sign-quenched integrands are not holomorphic, and
+therefore the phase- and sign-quenched partition func-
+tions are not invariant under such deformations. It may
+then be possible to bring the ratios Zt/Zs closer to unity,
+thus making reweighting more effective.
+There are different ways to deform integration con-
+tours. Historically, methods based on Lefschetz thimbles
+appeared first [6, 9–14]. Lefschetz thimbles are the dis-
+joint components of the integration contour defined by
+requiring that the imaginary part of the classical action
+is constant in each component. The thimble structure
+of theories with a fermionic determinant is usually quite
+complicated [15–19]. Simple toy models reveal the follow-
+ing features: i) cancellations between competing thimbles
+are very important for getting the correct results, and
+ii) the thimbles themselves are not smooth at the zeros
+of the fermionic determinant.
+Thus, the use of thim-
+bles might be impractical for such theories.
+However,
+Lefschetz thimbles are, in general, not the numerically
+optimal integration contours [20], i.e., they are not nec-
+essarily the contours with the largest Zt/Zs, so there is
+no need to concentrate solely on them.
+A second class of methods is based on numerical op-
+timization.
+The main idea here is to parametrize the
+integration manifold by a finite number of parameters,
+which are then optimized to make the sign problem
+as mild as possible.
+Such methods were applied to
+a one-dimensional integral [21], the 0+1D scalar the-
+1 A notable exception is lattice QCD with rooted staggered
+fermions [7, 8].
+ory [22], the 0+1D Polyakov-improved Nambu-Jona-
+Lasinio model [23], 0+1D QCD [24], 1+1D scalar field
+theory [25], the 1+1D Thirring model [26], the 2+1D
+Thirring model [27], Bose gases of several dimensions [28],
+1+1D U(1) gauge theory with a complex coupling con-
+stant [29] and the 2+1D XY model at finite density [30].
+Here, we apply contour optimization methods to a
+fermionic toy model that shares relevant technical fea-
+tures with finite chemical potential QCD: the chiral ran-
+dom matrix model proposed by Stephanov in Ref. [1].
+Since it is an exactly solvable model with a sign prob-
+lem, the Stephanov model is a very useful testbed for
+methods aimed at solving or alleviating the sign prob-
+lem.
+This model has been studied with the complex
+Langevin approach [31–33], which fails for this particu-
+lar model [34] even with the introduction of gauge cool-
+ing [33]. There are also preliminary results for this model
+with the tempered Lefschetz thimble method [35] which
+is based on parallel tempering [36] in the flow time of
+the holomorphic flow [11, 37]. This method – similarly
+to other flow-based methods — produces a weaker sign
+problem, albeit at the cost of substantially increasing the
+per-configuration-cost of generating the ensemble com-
+pared to ordinary phase reweighting.
+In this paper we study the Stephanov model with op-
+timization methods. There are, roughly speaking, two
+approaches to such an optimization: one can look for the
+optimum using either a very general ansatz with a large
+number of parameters, or a very specific ansatz tailored
+for the model at hand, and with a small number of pa-
+rameters. The first approach has clearly the potential to
+find a good optimum, e.g., using machine learning tech-
+niques, but it also has some disadvantages. In fact, for
+such a general approach the number of optimization pa-
+rameters has to be increased as one increases the number
+of degrees of freedom of the system. This means that the
+cost of finding good contours might turn out to be pro-
+hibitive, similarly to what happens with methods based
+on Lefschetz thimbles. In this exploratory study we fol-
+low the second, ad hoc approach, and optimize ansätze
+with only few parameters. Moreover, the number of these
+parameters is kept independent of the number of degrees
+of freedom of the system.
+We can then be sure that
+the optimization itself is numerically cheap, and that the
+per-configuration cost of generating the ensembles is es-
+sentially as low as on the original contours. Obviously,
+the drawback of this approach is that to write down an
+ansatz with only a few parameters that produces a sub-
+stantial improvement in the severity of the sign problem,
+some physical or mathematical insight is needed.
+For the toy model studied in this paper, the insight
+required to use the ad hoc approach is available, and so
+we can write down appropriate ansätze.
+We will then
+show that a quite cheap numerical optimization proce-
+dure leads one to contours with a reduced sign problem.
+We will also present numerical evidence that the reduc-
+tion in the severity of the sign problem is exponential:
+while the sign problem on the optimized contours is still
+
+3
+exponential in the number of degrees of freedom, the cor-
+responding exponent is reduced. This conclusion is simi-
+lar to what some of us have shown in Ref. [30] for a purely
+bosonic model (the 2+1 dimensional XY model at non-
+zero chemical potential). Notably, such an exponential
+reduction can be achieved without changing the number
+of optimization parameters with the system size.
+In this work we will only consider phase-quenched sim-
+ulations, for simplicity.
+Similar arguments and meth-
+ods should, however, also apply to the sign-quenched
+case [30].
+The plan of the paper is the following: In Section II
+we introduce the model discussed in this work. In Sec-
+tion III we provide details on the different contour de-
+formation procedures we tested. In Section IV we illus-
+trate the chemical potential and volume dependence of
+the achieved improvement and also compare our results
+with a method based on Lefschetz thimbles: the holo-
+morphic flow of Ref. [11]. We summarize our conclusions
+in Section V.
+II.
+THE CHIRAL RANDOM MATRIX MODEL
+Throughout this paper we will only consider Nf = 2
+with µ1 = µ2 ≡ µ for simplicity.
+The random ma-
+trix model of Stephanov [1] for Nf degenerate flavors of
+quarks is then defined by the partition function
+ZNf
+N
+= eNµ2 �
+dWdW † (det(D + m))Nf e−NTrW W †,
+(6)
+where the massless Dirac matrix is
+D =
+�
+0
+iW + µ
+iW † + µ
+0
+�
+,
+(7)
+m is the quark mass and W is a general N × N complex
+matrix. The model has no concept of physical volume.
+The number of degrees of freedom of the model scales
+with N 2.
+The two observables we will study in this paper are the
+chiral condensate:
+Σ =
+1
+2N
+∂ log ZNf
+N
+∂m
+,
+(8)
+and the quark density
+n =
+1
+2N
+∂ log ZNf
+N
+∂µ
+.
+(9)
+An important feature of the model is that it can be solved
+analytically, both in the N → ∞ limit where the integral
+is dominated by a saddle point, and at finite N where it
+reduces to the calculation of moments of Gaussian inte-
+grals. Thus, in this particular model we will be able to
+compare numerical results with exact analytic solutions.
+The model shares with QCD the feature that the
+phase-quenched theory corresponds to an isospin chemi-
+cal potential, and has an analogue of the pion condensa-
+tion transition at some µ = µPQ
+c
+. For chemical potentials
+exceeding µPQ
+c
+the sign problem of the model is severe.
+From the point of view of the Dirac spectrum, for µ = 0
+the eigenvalues are purely imaginary, while for µ ̸= 0 the
+eigenvalues of D acquire a real part, and are distributed
+inside a strip of width µ2 in the real direction. When the
+quark mass is inside this strip, the model has a severe sign
+problem. This roughly corresponds to the analogue of the
+pion condensed phase in the phase-quenched theory. Due
+to these similarities, this model has been considered sev-
+eral times in the literature as a good toy model for the
+sign problem in QCD [34, 35].
+We will consider the model for Nf = 2 and use the
+same quark chemical potential for both fermion flavors.
+In this model, unlike in QCD, the expectation value
+of the average phase does not always tend to zero in the
+limit of an infinite system. Rather, it only goes to zero
+in a given range of chemical potentials bounded by the
+solutions to the equation [38]:
+0 = 1 − µ2 +
+m2
+µ2 − m2 −
+m2
+4(µ2 − m2)2 .
+(10)
+Using a quark mass of m = 0.2, the two solutions of this
+equation are µ = 0.35 = µPQ
+c
+and 1.02. This is the regime
+where the sign problem in the model is strongest.
+III.
+CONTOUR DEFORMATION METHODS
+A.
+Optimization method
+We will restrict ourselves to ansätze with simple, ana-
+lytically calculable Jacobians with O(N 0) computational
+cost and a small number of parameters, independent of
+the number of degrees of freedom.
+Let A = Re W and B = Im W. These two real matrices
+will be deformed to complex matrices α and β. Thus,
+W = A + iB → X = α + iβ,
+W † = AT − iBT → Y = αT − iβT.
+(11)
+After applying such a deformation X† ̸= Y . After the
+deformation, the severity of the sign problem is given by:
+⟨eiθ⟩ =
+�� det(D + m)detJ
+|det(D + m)detJ |
+�Nf
+e−iNImTrXY
+�
+,
+(12)
+where the Jacobian determinant is
+detJ =
+����
+∂(α, β)
+∂(A, B)
+����.
+(13)
+B.
+Holomorphic flow
+Using the holomorphic flow (or generalized thimble
+method) of Ref. [11] for the complexified action of the
+
+4
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+k2
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+k1
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+�
+exp(iθ)
+�
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+p2
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+p1
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+�
+exp(iθ)
+�
+FIG. 1: Left: the average phase with Ansatz-1 as a function of k1 and k2. There is a local minimum at k2 ≈ 0 and k1 > 0.
+Right: the average phase with Ansatz-2 as a function of p1 and p2. There is an apparent saddle parallel to the p1 = 0 line at
+p1 = k1 > 0.
+Stephanov model,
+S = −Nµ2 − Nf log det(D + m) + NTr(XY ),
+(14)
+we deform the integration manifold by evolving the orig-
+inal one with the differential equation
+dYij
+dt
+= ∂S
+∂Yij
+= NXji − Nf[(XG)ji + iµGji],
+(15)
+where the overbar denotes complex conjugation, t is the
+flow parameter and
+G =
+�
+m2 − µ2 − iµ(X + Y ) + Y X
+�−1
+.
+(16)
+Solving this system of equations with initial conditions
+X0 = W, Y0 = W † for a fixed flow time tf we obtain
+a deformed manifold Mtf. We parametrize each point
+on the flowed manifold by the real matrices A and B.
+I.e., we parametrize the flowed manifold by the initial
+conditions of the flow equation.
+The computation of expectation values requires the Ja-
+cobian of the holomorphic flow,
+det J =
+����
+∂(X, Y )
+∂(A, B)
+���� ,
+(17)
+as well. Denoting the Hessian with H, the Jacobian ma-
+trix J is obtained as the solution of the equation
+dJ
+dt = H J,
+(18)
+with initial conditions
+JXij,Aij = 1, JXij,Bij = i, JYij,Aji = 1, JYij,Bji = −i.
+(19)
+Computing the Jacobian directly is numerically expen-
+sive, so we estimate it [39] with
+W = exp
+� � T
+0
+dt Tr H(t)
+�
+.
+(20)
+The difference between W and det J is taken into account
+by reweighting when computing observables,
+⟨O⟩ =
+⟨Oe−∆S⟩S′
+eff
+⟨e−∆S⟩S′
+eff
+,
+(21)
+where S′
+eff = S − ln W, ∆S = Seff − ReS′
+eff and ⟨.⟩S′
+eff is
+the average with respect to e−ReS′
+eff. This way, we needed
+to compute det J exactly only for the configurations used
+for measurements.
+In the large flow time limit, the flowed manifold tends
+towards the Lefschetz thimbles. At smaller flow times,
+it still reduces the sign problem, although less than a
+complete thimble decomposition would.
+IV.
+NUMERICAL RESULTS
+A.
+Simple ansätze
+As a rule, all of our ansätze have been parametrized
+such that the undeformed integration manifold is at value
+zero for all optimizable parameters.
+
+5
+0
+200
+400
+600
+800
+1000
+Nopt
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Contour deformation parameters
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+Re(a)
+Im(a)
+Re(b)
+Im(b)
+Re(c)
+Im(c)
+Re(d)
+Im(d)
+Re(e)
+Im(e)
+Re(f)
+Im(f)
+Re(g)
+Im(g)
+Re(h)
+Im(h)
+Re(j)
+Im(j)
+Re(k)
+Im(k)
+0
+200
+400
+600
+800
+1000
+Nopt
+0.25
+0.30
+0.35
+0.40
+0.45
+�
+exp(iθ)
+�
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+FIG. 2: Left: parameters as a function of the optimization step for Ansatz-3. Right: the average phase as a function of the
+optimization step for Ansatz-3.
+Ansatz-1
+From the definition in Eq. (7) it is easy to see that
+the sign problem can be removed from the quark de-
+terminant by a simple shift of the form α = A + iµ1.
+This, however, introduces a sign problem in the Gaus-
+sian term e−N Tr(XY ). By finding a trade-off between the
+two terms, the severity of the sign problem may be op-
+timized. This motivates our first ansatz, with two real
+parameters k1 and k2 defined by
+α = A + ik11
+(22)
+β = B + ik21.
+(23)
+The Jacobian determinant for this ansatz is simply unity.
+The parameter k2 is introduced on a whim, as the ma-
+trices A and B do not have to be treated symmetrically.
+The results for the average phase in a scan in these two
+parameters for N = 2, m = 0.2 and µ = 1.0 is shown in
+Fig. 1 (left). While there is a clearly non-zero optimal
+value for k1, the optimal value of the k2 parameter is near
+zero. This remains true for all values of the parameters
+N, µ and m we simulated.
+Ansatz-2
+When we introduce a shift A → A + ik1 the argument
+of the Gaussian term changes according to
+Tr(XY ) = Tr(AAT + BBT) − Nk2 + 2ikTrA.
+(24)
+This motivates our second ansatz, with two real param-
+eters p1 and p2 defined by
+α = A + ip11 + p2TrA1,
+(25)
+β = B.
+(26)
+The p1 parameter of this ansatz is identical to the k1
+parameter of the previous ansatz. The Jacobian deter-
+minant for this ansatz is simply detJ = 1 + Np2, i.e.,
+configuration-independent, and can be ignored. The re-
+sults for the average phase in a scan in these two param-
+eters for N = 2, m = 0.2 and µ = 1.0 can be seen in
+Fig. 1 (right). While there is a clearly non-zero optimal
+value for p1 = k1, the p2 parameter only appears to move
+on a saddle.
+Ansatz-3
+We now move on to a more complicated ansatz with
+10 complex (or 20 real) parameters a, b, c, d, e, f, g, h, j, k
+defined by
+α = (a + bTrA + cTrB)1 + (1 + d)A + eB
+(27)
+β = (f + gTrA + hTrB)1 + jA + (1 + k)B
+(28)
+The Jacobian determinant for this ansatz is
+detJ =
+�
+(1 + d)(1 + k) − ej
+�N 2−1×
+��
+(1 + d) + Nb
+��
+(1 + k) + Nh
+�
+−(e + Nc)(j + Ng)
+�
+.
+(29)
+The severity of the sign problem was then optimized via
+the AdaDelta method [40], with the objective function
+− log⟨eiθ⟩ = − log
+Z
+ZPQ
+= − log Z + log ZPQ,
+(30)
+where we suppressed the N and Nf indices for the parti-
+tion function. The gradient with right to the deformation
+parameters is given by
+∇ log ZPQ = −⟨∇SA
+eff⟩,
+(31)
+where
+Sa
+eff = NReTrXY − Nf log |detM| − log |detJ |
+(32)
+with gradient
+∇Sa
+eff =NReTr
+�
+(∇X)Y + X(∇Y )
+�
+− Nf
+2 Tr
+�
+M −1(∇M) + M
+−1(∇M)
+�
+− Re
+�∇detJ
+detJ
+�
+.
+(33)
+Note that for Ansatz-3 the Jacobian is independent of
+the configuration, and the last term can be dropped from
+Eq. (33). For Ansatz-4, to be discused below, the Jaco-
+bian will depend on the configuration, and thus the last
+term is needed. An example of such an optimization run
+is shown in Fig. 2. As with the previous two ansätze,
+only a single parameter emerges k1 = p1 = Ima.
+
+6
+Ansatz-4
+Experiments with the first three ansätze revealed only
+one parameter of interest, which can be thought of as
+a simple one-parameter imaginary shift of the trace of
+the matrix A. One might wonder whether more general
+deformations of the trace could lead to a better improve-
+ment. Thus we look at non-linear deformations of the
+trace τ = TrA of the matrix A with an undeformed B
+matrix. The integral measure is given by
+N
+�
+i,j=1
+dAij = dτ
+N
+�
+i,j=1
+(i,j)̸=(N,N)
+dAij
+= dτ
+N
+�
+i,j=1
+i̸=j
+dAij
+N
+�
+k=1
+d
+�
+Akk − τ
+N
+�
+.
+(34)
+The deformed matrix α is obtained from A as
+A = τ
+N 1 +
+�
+A − τ
+N 1
+�
+= τ
+N 1 + ˜A
+→
+α = τ
+N 1 + ˜A,
+(35)
+where Tr ˜A = 0 and
+τ = t + if(τ; . . . ),
+(36)
+for some function f that depends on τ and possibly other
+parameters. For simplicity, we choose f to be piecewise
+linear,
+f(τ; xk(τ), xk(τ)+1, yk(τ), yk(τ)+1) =
+yk(τ)(xk(τ)+1 − τ)
+xk(τ)+1 − xk(τ)
++ yk(τ)+1(τ − xk(τ))
+xk(τ)+1 − xk(τ)
+.
+(37)
+The parameters to optimize are the yi, while the node
+points xi of the linear interpolation are fixed parameters,
+and chosen with regular spacing, xl+1 − xl = ∆ for all l,
+and
+k(τ) = floor
+�τ − x0
+∆
+�
+.
+(38)
+By numerical experimentation we have found that the
+choice of the node points is not important, as long as the
+full interpolation range is large enough to cover the most
+probable values of TrA on the original contours and ∆ is
+small enough. If these conditions are met, optimal con-
+tours with ansätze with different node points appear to
+be piecewise approximations of the same smooth curve.
+The Jacobian is
+detJ = 1 + iyk(τ)+1 − yk(τ)
+∆
+.
+(39)
+The parameters are then optimized as with Ansatz-3.
+A comparison of the results from this ansatz with the
+6
+4
+2
+0
+2
+4
+6
+ReTrA
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+ImTrA
+piecewise
+α = A + ik  ansatz
+N = 2, Nf = 2, m = 0.2, µ = 1.0
+FIG. 3:
+Ansatz-4 (piecewise optimization of the trace) com-
+pared to Ansatz-1 (imaginary constant shift of A proportional
+to the unit matrix). The two procedures find essentially the
+same contour, as the differing tails are at large values of |TrA|,
+and have small statistical weight.
+constant shift found using ansätze 1 to 3 is shown in
+Fig. 3. For highly probable values of Tr A the two ansätze
+agree, while for the highly improbably values of Tr A,
+the optimization does not move the ansatz away from
+the original contour, as there are no configuration to use
+for the optimization of that part of the contour. These
+two asymptotic regimes are smoothly connected.
+The
+measured sign problem on this contour is identical to the
+one measured with ansätze 1 to 3, up to statistical errors
+– not surprisingly since deviations of f from a constant
+happen on unimportant configurations.
+B.
+Chemical potential and matrix size dependence
+Now that we have discovered a good contour defor-
+mation parameter, let us look at what kind of improve-
+ments can be achieved by such a 1-parameter deforma-
+tion. From here on out we show results with Ansatz-1,
+with k2 set to zero.
+The volume and chemical potential dependence of the
+average phase for the original and optimized contours
+is shown in Fig. 4. The "volume", i.e., matrix size de-
+pendence at a fixed chemical potential in the left panel
+reveals an improvement on the sign problem that is expo-
+nential in the matrix size: while the severity of the sign
+problem is roughly linear on a logarithmic plot for both
+the original and optimized contours, the slopes are quite
+different. The right panel shows the chemical potential
+dependence for several values of N. Apparently, contour
+optimization improves the most on the sign problem in
+the regime where it is the most severe.
+The statistical improvement factor, defined as the
+square of the ratio of the average phase on the deformed
+vs the original contours,
+��
+eiθ�
+orig /
+�
+eiθ�
+def
+�2
+, is shown
+on the left panel of Fig. 5 for N = 2, 4 and 6. For larger
+matrices,
+�
+eiθ�
+was zero within statistical errors on the
+original contours, and this ratio could not be calculated.
+We see that the ratio monotonically increases with N,
+and as a function of µ it is maximal close to the value of
+µ where the sign problem is the strongest. The optimal
+
+7
+2
+4
+6
+8
+10
+N
+10
+-3
+10
+-2
+10
+-1
+�
+exp(iθ)
+�
+m = 0.2, µ = 0.9, Nf = 2
+deformed
+original
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+10
+-3
+10
+-2
+10
+-1
+10
+0
+�
+exp(iθ)
+�
+m = 0.2, Nf = 2
+original, N = 2
+deformed
+original, N = 4
+deformed
+original, N = 6
+deformed
+original, N = 8
+deformed
+FIG. 4: Left: dependence of the average phase on the size of the random matrix for the original and optimized contours. Right:
+dependence of the average phase on the chemical potential for the original and optimized contours.
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0
+10
+20
+30
+40
+50
+60
+70
+improvement factor
+m = 0.2, Nf = 2
+N = 2
+N = 4
+N = 6
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Im(a) = k1 = p1
+m = 0.2, Nf = 2
+N = 2
+N = 4
+N = 6
+N = 8
+N = 10
+FIG. 5: Left: dependence of the statistical improvement (calculated as the square of the ratio of the average phases on
+the optimized and original contours) achieved by contour optimization as a function of µ for different matrix sizes. Right:
+dependence of the optimal contour parameter k1 = p1 = Ima on µ for different matrix sizes.
+values for the deformation parameter k1 = p1 = Ima for
+different values of µ and N are shown in the right panel
+of Fig. 5.
+As a sanity check, we also calculated the expectation
+value of the chiral condensate and the quark number on
+both the original and the optimized contours, and com-
+pared them to the analytic results, see Fig. 6. They both
+show excellent agreement, but the optimized contours
+have significantly smaller error bars.
+C.
+Comparison with the holomorphic flow
+As experiments with simple ansätze so far revealed
+only a single important contour deformation parameter,
+it is a natural question to ask whether Lefschetz-thimble
+based methods also “find” this deformation or not, and
+whether by utilizing such methods it is possible to im-
+prove the sign problem further compared to such a 1-
+parameter deformation. For this reason, we performed
+the holomorphic flow on our N = 2 random matrices, and
+obtained an estimate of the k1 parameter from the flowed
+variables via: kflow
+1
+= Im ⟨Tr(α(tf) − A)⟩ /N.
+This k1
+can then be substituted back to the 1-parameter ansatz
+α = A + ik11 and the severity of the sign problem can
+be compared with the properly flowed manifold.
+The sign problem as a function of µ is shown on the
+original contour, the optimized contour, the flowed con-
+tour, and on the contour with k1 extracted from the flow
+in Fig. 7. A few observations can be drawn from this
+figure. For small chemical potentials, the flow performs
+better than the optimization, which does not noticeably
+improve the sign problem. For larger chemical potentials,
+optimization vastly outperforms the flow. Of course, this
+is only compared at a fixed flow time, and we do not
+know where the severity of the sign problem would end
+up at infinite flow time (on the thimbles). However, go-
+ing to large flow times gets very expensive already for
+small systems.
+For larger chemical potentials, the 1-parameter ansatz
+with k1 = kflow
+1
+extracted from the flow gives very similar
+results as the full flow. This may be a hint for the possi-
+bility that at larger chemical potentials most of the im-
+provement from the flow comes from this simple deforma-
+tion. Interestingly, while the full flow at small chemical
+potentials gives a slightly weaker sign problem compared
+to the ansatz with kflow
+1
+, at larger chemical potentials the
+situation is reversed: the sign problem is slightly weaker
+with kflow
+1
+than with the solution of the full flow equation.
+While this may be somewhat surprising at first, it is not
+
+8
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Σ
+original, N = 2
+optimized
+analytic
+original, N = 4
+optimized
+analytic
+original, N = 6
+optimized
+analytic
+original, N = 8
+optimized
+analytic
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+1
+0
+1
+2
+3
+4
+5
+n
+original, N = 2
+optimized
+analytic
+original, N = 4
+optimized
+analytic
+original, N = 6
+optimized
+analytic
+original, N = 8
+optimized
+analytic
+FIG. 6: The chiral condensate (left) and the quark number (right) as a function of µ for several values of the matrix size N.
+Analytic results are compared with results from simulations on the original and on the improved contours.
+in contradiction with what we already now about contour
+deformations. The flow goes towards the Lefschetz thim-
+bles, which are not the numerically optimal contours, and
+thus there is no reason for the full flow curve to be always
+above the curve with the simple ansatz with kflow
+1
+.
+V.
+SUMMARY AND DISCUSSION
+We have discussed contour deformations in the chiral
+random matrix model of Stephanov as a way to alleviate
+its sign problem. Using simple ad-hoc ansätze we iden-
+tified a single important deformation parameter, which
+allowed for an exponential reduction in the severity of
+the sign problem as a function of the matrix size.
+Our results are quite encouraging, as they show that a
+simple one-parameter optimization can lead to exponen-
+tially alleviating the sign problem even in a fermionic the-
+ory, where the thimble decomposition is complicated and
+contour deformation approaches based on them might
+not be numerically effective.
+The fermionic nature of
+the matter fields does not appear to be a fundamental
+obstruction in the construction of exponentially better
+contours.
+Furthermore, the phase diagram of the random matrix
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+�
+exp(iθ)
+�
+m = 0.2, Nf = 2, N = 2
+original
+deformed: ansatz
+deformed: flow (tf = 0.04)
+deformed: ansatz with k1 from flow
+FIG. 7:
+The severity of the sign problem for N = 2 and m =
+0.2 as a function of µ on the original contours, the optimized
+contours, the flowed contours and the contours where the k1
+parameter of the ansatz is extracted from the flow.
+model is similar to what we expect in full QCD: the chi-
+ral phase transition is “hidden behind” the pion conden-
+sation phase in the phase-quenched theory. Hence, this
+bulk thermodynamic feature – the existence of a phase
+transition in the phase-quenched theory – also does not
+appear to be a fundamental obstruction.
+The results and the ansätze in this paper, however,
+cannot be used directly to construct a good optimization
+ansatz in full QCD, as the toy model studied here and
+QCD differ on an important technical aspect. Concretely,
+in the Stephanov model there are contour deformations
+that can remove the sign problem from the fermion deter-
+minant for a single flavor (so from the full determinant
+when all chemical potentials are equal) – albeit at the
+cost of reintroducing it somewhere else in the Boltzmann
+weights. There are no such deformations in full QCD.
+The complexification of the SU(3) gauge group is the
+SL(3, C) group, which still requires a unit determinant.
+To remove the chemical potential from a single quark de-
+terminant the time-like links would have to be deformed
+to GL(3, C) matrices, with non-unit determinant, which
+lie outside the complexified gauge group.
+Comparison with the holomorphic flow method shows
+that as one goes near the Lefschetz thimbles in this
+model, the bulk (but not all) of the improvement on the
+severity of the sign problem is captured by these types of
+deformations – which have no direct analogue in QCD.
+In the future it will therefore be important to work with
+more realistic toy models of QCD or even full QCD it-
+self, as the choice of a suitable sign-problem improving
+ansatz appears to be strongly dependent on the exact
+symmetries and exact matter content of a given theory.
+Acknowledgements
+This work was supported by the NKFIH grant KKP-
+126769. D.P. is supported by the ÚNKP-22-3 New Na-
+tional Excellence Program of the Ministry for Culture
+and Innovation from the source of the National Research,
+Development and Innovation Fund.
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UNAyT4oBgHgl3EQfVvd8/content/tmp_files/2301.00149v1.pdf.txt

@@ -0,0 +1,2351 @@
+Rethinking Rotation Invariance with Point Cloud Registration
+Jianhui Yu, Chaoyi Zhang, Weidong Cai
+School of Computer Science, University of Sydney, Australia
+{jianhui.yu, chaoyi.zhang, tom.cai}@sydney.edu.au
+Abstract
+Recent investigations on rotation invariance for 3D point
+clouds have been devoted to devising rotation-invariant fea-
+ture descriptors or learning canonical spaces where objects
+are semantically aligned. Examinations of learning frame-
+works for invariance have seldom been looked into. In this
+work, we review rotation invariance in terms of point cloud
+registration and propose an effective framework for rota-
+tion invariance learning via three sequential stages, namely
+rotation-invariant shape encoding, aligned feature integration,
+and deep feature registration. We first encode shape descrip-
+tors constructed with respect to reference frames defined over
+different scales, e.g., local patches and global topology, to
+generate rotation-invariant latent shape codes. Within the in-
+tegration stage, we propose Aligned Integration Transformer
+to produce a discriminative feature representation by inte-
+grating point-wise self- and cross-relations established within
+the shape codes. Meanwhile, we adopt rigid transformations
+between reference frames to align the shape codes for fea-
+ture consistency across different scales. Finally, the deep in-
+tegrated feature is registered to both rotation-invariant shape
+codes to maximize feature similarities, such that rotation in-
+variance of the integrated feature is preserved and shared se-
+mantic information is implicitly extracted from shape codes.
+Experimental results on 3D shape classification, part segmen-
+tation, and retrieval tasks prove the feasibility of our work.
+Our project page is released at: https://rotation3d.github.io/.
+1
+Introduction
+Point cloud analysis has recently drawn much interest from
+researchers. As a common form of 3D representation, the
+growing presence of point cloud data is encouraging the de-
+velopment of many deep learning methods (Qi et al. 2017a;
+Guo et al. 2021; Zhang et al. 2021), showing great success
+for well-aligned point clouds on different tasks. However, it
+is difficult to directly apply 3D models to real data as raw
+3D objects are normally captured at different viewing an-
+gles, resulting in unaligned data samples, which inevitably
+impact the deep learning models which are sensitive to rota-
+tions. Therefore, rotation invariance becomes an important
+research topic in the 3D domain.
+To achieve rotation invariance, a straightforward way is
+to augment training data with massive rotations which, how-
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Aligned Integration
+Transformer
+Registration
+Registered U
+(a)
+(b)
+(c)
+Local & Global Descriptors
+Correspondence
+Mapping
+Local Patches Pℓ Global Shape P"
+Correspondence
+Mapping
+Fℓ
+F"
+U
+RI
+T
+Correspondence
+Mapping
+Shape Descriptors
+(d)
+(e)
+Source Points P# Target Points P$
+T
+Registration
+F#
+F$
+TI
+Encoding
+Integration
+Registration
+Registered P!
+Ours
+PCR
+Figure 1: Frameworks of our design (left) and robust point
+cloud registration (right), where TI and RI are transforma-
+tion invariance and rotation invariance, and T is the rigid
+transformation. The dotted line indicates the computation of
+T between reference frames.
+ever, requires a large memory capacity and exhibits limited
+generalization ability to unseen data (Kim, Park, and Han
+2020). There are attempts to align 3D inputs to a conical
+pose (Jaderberg et al. 2015; Cohen et al. 2018), or to learn
+rotation robust features via equivariance (Deng et al. 2021;
+Luo et al. 2022), while these methods are not rigorously
+rotation-invariant and present noncompetitive performance
+on 3D shape analysis. To maintain consistent model be-
+havior under random rotations, some methods (Zhang et al.
+2019; Chen et al. 2019; Xu et al. 2021) follow Drost et al.
+(2010) to handcraft rotation-invariant point-pair features.
+Others (Zhang et al. 2020; Li et al. 2021a; Zhao et al. 2022)
+design robust features from equivariant orthonormal bases.
+Most of the mentioned works either manipulate model in-
+puts or generate canonical spaces to achieve rotation invari-
+ance (RI). In this work, we review the problem of RI from
+a different aspect: robust point cloud registration (PCR). We
+find that PCR and RI share the same goal: PCR aligns low-
+dimensional point cloud features (e.g., xyz) from the source
+domain to the target domain regardless of transformations,
+arXiv:2301.00149v1  [cs.CV]  31 Dec 2022
+
+while RI can be considered to align high-dimensional la-
+tent features to rotation-invariant features. Specifically, the
+goal of PCR is to explicitly align the source point cloud to
+the target, both representing the same 3D object, and for RI
+learning, we implicitly align the final feature representation
+of a 3D shape to a hidden feature of the same shape, which
+is universally rotation-invariant to any rotations.
+Motivated by this finding, we propose our learning frame-
+work in Fig. 1 with three sequential stages, namely rotation-
+invariant shape encoding, aligned feature integration, and
+deep feature registration. Firstly, we (a) construct and feed
+point pairs with different scales as model inputs, where we
+consider local patches Pℓ with small number of points and
+global shape Pg with the whole 3D points. Hence, the fi-
+nal feature representation can be enriched by information
+from different scales. Low-level rotation-invariant descrip-
+tors are thus built on reference frames and encoded to gener-
+ate latent shape codes Fℓ and Fg following recent PCR work
+(Pan, Cai, and Liu 2022). Secondly, we (b) introduce a vari-
+ant of transformer (Vaswani et al. 2017), Aligned Integration
+Transformer (AIT), to implicitly integrate information from
+both self- and cross-attention branches for effective feature
+integration. In this way, information encoded from different
+point scales is aggregated to represent the same 3D object.
+Moreover, we consider Fℓ and Fg as unaligned since they
+are encoded from unaligned reference frames. To address
+the problem, we follow the evaluation technique proposed
+in PCR (Pan, Cai, and Liu 2022), where we use relative ro-
+tation information (T) with learnable layers to align Fℓ and
+Fg for feature consistency. Finally, to ensure RI of the inte-
+grated feature U, we follow PCR to (c) examine the corre-
+spondence map of (Fg, U) and (Fℓ, U), such that the mu-
+tual information between a local patch of a 3D object and
+the whole 3D object is maximized, and RI is further ensured
+in the final geometric feature.
+The contributions of our work are summarized as follow-
+ing three folds: (1) To our knowledge, we are the first in de-
+veloping a PCR-cored representation learning framework to-
+wards effective RI studies on 3D point clouds. (2) We intro-
+duce Aligned Integration Transformer (AIT), a transformer-
+based architecture to conduct aligned feature integration for
+a comprehensive geometry study from both local and global
+scales. (3) We propose a registration loss to maintain rota-
+tion invariance and discover semantic knowledge shared in
+different parts of the input object. Moreover, the feasibility
+of our proposed framework is successfully demonstrated on
+various 3D tasks.
+2
+Related Work
+Rotation Robust Feature Learning.
+Networks that are
+robust to rotations can be equivariant to rotations. Esteves
+et al. (2018) and Cohen et al. (2018) project 3D data
+into a spherical space for rotation equivariance and per-
+form convolutions in terms of spherical harmonic bases.
+Some (Spezialetti et al. 2020; Sun et al. 2021) learn canon-
+ical spaces to unify the pose of point clouds. Recent
+works (Luo et al. 2022; Deng et al. 2021; Jing et al. 2020)
+vectorize the scalar activations and mapping SO(3) actions
+to a latent space for easy manipulations. Although these
+works present competitive results, they cannot be strictly
+rotation-invariant. Another way for rotation robustness is to
+learn rotation-invariant features. Handcraft point-pair fea-
+tures are rotation-invariant (Zhang et al. 2019; Chen et al.
+2019; Xu et al. 2021), but they focus on local domains
+and ignore the global overview of 3D objects. Others use
+rotation-equivariant local reference frames (LRFs) (Zhang
+et al. 2020; Thomas 2020; Kim, Park, and Han 2020) or
+global reference frames (GRFs) (Li et al. 2021a) as model
+inputs based on principal component analysis (PCA). How-
+ever, they may produce inconsistent features across differ-
+ent reference frames, which would limit the representational
+power. In contrast to abovementioned methods with rotation
+robust model inputs or modules, we examine the relation be-
+tween RI and PCR and propose an effective framework.
+3D Robust Point Cloud Registration.
+Given a pair of Li-
+DAR scans, 3D PCR requires an optimal rigid transforma-
+tion to best align the two scans. Despite the recent emerging
+of ICP-based methods (Besl and McKay 1992; Wang and
+Solomon 2019b), we follow robust correspondence-based
+approaches in our work (Deng, Birdal, and Ilic 2018; Yuan
+et al. 2020; Qin et al. 2022; Pan, Cai, and Liu 2022), where
+RI is widely used to mitigate the impact of geometric trans-
+formations during feature learning. Specifically, both Pan,
+Cai, and Liu (2022) and Qin et al. (2022) analyze the en-
+coding of transformation-robust information and introduce
+a rotation-invariant module with contextual information into
+their registration pipeline. All these methods showing im-
+pressive results are closely related to rotation invariance. We
+hypothesize that the learning framework of RI can be sim-
+ilar to PCR, and we further prove in experiments that our
+network is feasible and able to achieve competitive perfor-
+mance on rotated point clouds.
+Transformers in 3D Point Clouds.
+Transformers (Doso-
+vitskiy et al. 2021; Liu et al. 2021) applied to 2D vision have
+shown great success, and they are gaining prominence in 3D
+point clouds. For example, Zhao et al. (2021) uses vector-
+ized self-attention (Vaswani et al. 2017) and positional em-
+bedding for 3D modeling. Guo et al. (2021) proposes offset
+attention for noise-robust geometric representation learning.
+Cross-attention is widely employed for semantic informa-
+tion exchange (Qin et al. 2022; Yu et al. 2021a), where fea-
+ture relations between the source and target domains are ex-
+plored. Taking advantage of both, we design a simple yet
+effective feature integration module with self and cross re-
+lations. In addition, transformation-related embeddings are
+introduced for consistent feature learning.
+Contrastive Learning with 3D Visual Correspondence.
+Based on visual correspondence, contrastive learning aims
+to train an embedding space where positive samples are
+pushed together whereas negative samples are separated
+away (He et al. 2020). The definition of positivity and neg-
+ativity follows the visual correspondence maps, where pairs
+with high confidence scores are positive otherwise negative.
+Visual correspondence is important in 3D tasks, where se-
+mantic information extracted from matched point pairs im-
+proves the network’s understanding on 3D geometric struc-
+
+tures. For example, PointContrast (Xie et al. 2020) explores
+feature correspondence across multiple views of one 3D
+point cloud with InfoNCE loss (Van den Oord, Li, and
+Vinyals 2018), increasing the model performance for down-
+stream tasks. Info3D (Sanghi 2020) and CrossPoint (Afham
+et al. 2022) minimize the semantic difference of point fea-
+tures under different poses. We follow the same idea by reg-
+istering the deep features to rotation-invariant features at in-
+termediate levels, increasing feature similarities in the em-
+bedding space to ensure rotation invariance.
+3
+Method
+Given a 3D point cloud including Nin points with xyz co-
+ordinates P = {pi ∈ R3}Nin
+i=1, we aim to learn a shape en-
+coder f that is invariant to 3D rotations: f(P) = f(RP),
+where R ∈ SO(3) and SO(3) is the rotation group. RI can
+be investigated and achieved through three stages, namely
+rotation-invariant shape encoding (Section 3.1), aligned fea-
+ture integration (Section 3.2), and deep feature registration
+(Section 3.3).
+3.1
+Rotation-Invariant Shape Encoding
+In this section, we first construct the input point pairs from
+local and global scales based on reference frames, follow-
+ing the idea of Pan, Cai, and Liu (2022) to obtain low-level
+rotation-invariant shape descriptors from LRFs and GRF di-
+rectly. Then we obtain latent shape codes via two set abstrac-
+tion layers as in PointNet++ (Qi et al. 2017b).
+Rotation Invariance for Local Patches.
+To construct
+rotation-invariant features on LRFs, we hope to construct
+an orthonormal basis for each LRF as p ∈ R3×3. Given a
+point pi and its neighbor pj ∈ N(pi), we choose #»
+xiℓ =
+#       »
+pmpi/∥#       »
+pmpi∥2, where pm is the barycenter of the local ge-
+ometry and ∥ · ∥2 is L2-norm. We then define #»
+ziℓ follow-
+ing Tombari, Salti, and Stefano (2010) to have the same di-
+rection as an eigenvector, which corresponds to the smallest
+eigenvalue via eigenvalue decomposition (EVD):
+Σℓ
+i =
+|N (pi)|
+�
+j=1
+αj (#     »
+pipj) (#     »
+pipj)⊤ , αj =
+d − ∥#     »
+pipj∥2
+�|N (pi)|
+j=1
+d − ∥#     »
+pipj∥2
+,
+(1)
+where αj is a weight parameter, allowing nearby pj to have
+large contribution to the covariance matrix, and d is the max-
+imum distance between pi and pj. Finally, we define #»
+yiℓ as
+#»
+ziℓ × #»
+xiℓ. RI is introduced to pi with respect to its neigh-
+bor pj as pℓ
+ij = #     »
+pipj⊤Mℓ
+i. Proofs of the equivariance of Mℓ
+i
+and invariance of pℓ
+ij are shown in the supplementary ma-
+terial. The latent shape code Fℓ ∈ RN×C is obtained via
+PointNet++ and max-pooling.
+Rotation Invariance for Global Shape.
+We apply PCA
+as a practical tool to obtain RI in a global scale. Similar to
+Eq. 1, PCA is performed by
+1
+N0
+�N0
+i=1(#       »
+pmpi)(#       »
+pmpi)⊤ =
+UgΛgUg⊤, where pm is the barycenter of P, Ug
+=
+[# »
+u1g, # »
+u2g, # »
+u3g] and Λg = diag(λg
+1, λg
+2, λg
+3) are eigenvec-
+tor and eigenvalue matrices. We take Ug as the orthonor-
+mal basis Mg = [#»x g, #»y g, #»z g] for GRF. By transform-
+ing point pi with Ug, the shape pose is canonicalized as
+pg
+i = piMg. Proof of the RI of pg
+i is omitted for its sim-
+plicity, and Fg ∈ RN×C is obtained following PointNet++.
+Sign Ambiguity.
+EVD introduces sign ambiguity for
+eigenvectors, which negatively impacts the model perfor-
+mance (Bro, Acar, and Kolda 2008). The description of sign
+ambiguity states that for a random eigenvector #»u, #»u and
+#»u ′, with #»u ′ having an opposite direction to #»u, are both
+acceptable solutions to EVD. To tackle this issue, we sim-
+ply force #»
+ziℓ of LRF to follow the direction of #  »
+opi, with o
+being the origin of the world coordinate. We disambiguate
+basis vectors in Mg by computing an inner product with
+#       »
+pmpi, ∀i ∈ N0. Taking #»x g for example, its direction is con-
+ditioned on the following term:
+#»x g =
+�#»x g,
+if Sx ≥ N0
+2
+#»x ′g,
+otherwise
+,
+Sx =
+N0
+�
+i=1
+1[⟨#»x g, #       »
+pmpi⟩],
+(2)
+where ⟨·, ·⟩ is the inner product, 1[·] is a binary indicator that
+returns 1 if the input argument is positive, otherwise 0. Sx
+denotes the number of points where #»x g and #       »
+pmpi point to
+the same direction. The same rule is applied to disambiguate
+#»y g and #»z g by Sy and Sz. Besides, as mentioned in Li et al.
+(2021a), Mg might be non-rotational (e.g., reflection). To
+ensure Mg a valid rotation, we simply reverse the direction
+of the basis vector whose S value is the smallest. More anal-
+yses on sign ambiguity are in the supplementary material.
+3.2
+Aligned Feature Integration
+Transformer has been widely used in 3D domain to cap-
+ture long-range dependencies (Yu et al. 2021b). In this sec-
+tion, we introduce Aligned Integration Transformer (AIT),
+an effective transformer to align latent shape codes with rel-
+ative rotation angles and integrate information via attention-
+based integration (Cheng et al. 2021). Within each AIT mod-
+ule, we first apply Intra-frame Aligned Self-attention on Fℓ
+and we do not encode Fg, which is treated as supplemen-
+tary information to assist local geometry learning with the
+global shape overview. We discuss that encoding Fg via self-
+attention can increase model overfitting, thus lowering the
+model performance. We will validate our discussion in Sec-
+tion 4.4. Inter-frame Aligned Cross-attention is applied on
+both Fℓ and Fg, and we use Attention-based Feature Inte-
+gration module for information Aggregation.
+Preliminary: Offset Attention.
+AIT utilizes offset atten-
+tion (Guo et al. 2021) for noise robustness. In the follow-
+ing, we use subscripts sa and ca to denote implementations
+related to self- and cross-attention, respectively. We first re-
+view offset attention as follows:
+F = φ(Foa) + Fin, Foa = Fin − ∥softmax(A)∥1v, A = qk⊤,
+(3)
+where q = FinWq, k = FinWk ∈ RN×d, and v =
+FinWv ∈ RN×C are query, key, and value embeddings,
+and Wq, Wk ∈ RC×d, Wv ∈ RC×C are the correspond-
+ing projection matrices. ∥ · ∥1 is L1-norm and φ denotes a
+multi-layer perceptron (MLP). Foa is offset attention-related
+feature and A ∈ RN×N is the attention logits.
+
+𝑾𝒄𝒂
+𝒗
+𝑾𝒄𝒂
+𝒌
+𝑾𝒔𝒂
+𝒒
+𝐞𝒄𝒂
+𝜶 𝑾𝒄𝒂
+𝜶
+𝑭()
+ℓ/,
+𝐹ℓ/,
+𝑨𝒄𝒂
+𝒂𝒕𝒕𝒏: 𝑁×𝑁
+𝑨𝒄𝒂
+𝒓𝒐𝒕: 𝑁×𝑁
+𝑨𝒄𝒂: 𝑁×𝑁
+𝐹,/ℓ
+𝑾𝒔𝒂
+𝒗
+𝑾𝒔𝒂
+𝒌
+𝑾𝒔𝒂
+𝒒
+𝐤𝒔𝒂: 𝑁×𝑑
+𝐪𝒔𝒂: 𝑁×𝑑
+𝐯𝒔𝒂: 𝑁×𝐶
+𝐞𝒔𝒂
+𝜶 𝑾𝒔𝒂
+𝜶
+𝑭()
+ℓ/,
+𝐹ℓ/,
+𝑨𝒔𝒂
+𝒂𝒕𝒕𝒏: 𝑁×𝑁
+𝑨𝒔𝒂
+𝒓𝒐𝒕: 	𝑁×𝑁
+𝑨1): 𝑁×𝑁
+𝑁×𝑁×𝑑
+𝑁×𝑑
+𝐯𝒄𝒂: 𝑁×𝐶
+𝐤𝒔𝒂: 𝑁×𝑑
+𝐪𝒄𝒂: 𝑁×𝑑
+(a) Intra-frame Aligned Self-attention
+(b) Inter-frame Aligned Cross-attention
+𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧
+𝐦𝐚𝐭𝐫𝐢𝐱	𝐩𝐫𝐨𝐝𝐮𝐜𝐭
+𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧
+𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧
+𝐦𝐚𝐭𝐫𝐢𝐱	𝐩𝐫𝐨𝐝𝐮𝐜𝐭
+𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧
+Figure 2: Illustrations of (a) Intra-frame Aligned Self-attention and (b) Inter-frame Aligned Cross-attention modules. Note that
+we only present processes for computing Foa in both modules.
+Intra-frame Aligned Self-attention.
+Point-wise features
+of Fℓ are encoded from unaligned LRFs, so direct imple-
+mentation of self-attention on Fℓ can cause feature inconsis-
+tency during integration. To solve this problem, rigid trans-
+formations between distinct LRFs are considered, which
+are explicitly encoded and injected into point-wise relation
+learning process. We begin by understanding the transfor-
+mation between two LRFs. For any pair of local orthonormal
+bases Mℓ
+i and Mℓ
+j, a rotation can be easily derived ∆Rji =
+Mℓ
+iMℓ
+j
+⊤ and translation is defined as ∆tji = oℓ
+i −oℓ
+j, where
+oℓ
+i/j indicates the origin. In our work, the translation part is
+intentionally ignored, where we show in the supplementary
+material that by keeping both rotation and translation infor-
+mation, the model performance decreases.
+Although ∆Rji is invariant to rotations, we do not di-
+rectly project it into the embedding space, as it is sensitive
+to the order of matrix product: ∆Rji ̸= ∆Rij, giving in-
+consistent rotation information when the product order is not
+maintained. To address this issue, we construct our embed-
+ding via the relative rotation angle ∆αji between Mℓ
+i and
+Mℓ
+j, which is normally used in most PCR works (Yew and
+Lee 2020; Pan, Cai, and Liu 2022) for evaluations. The rel-
+ative rotation angle ∆αji is computed as:
+∆αji = arccos
+�Trace (∆Rji) − 1
+2
+� 180
+π
+∈ [0, π],
+(4)
+where it is easy to see that ∆αji = ∆αij. We further apply
+sinusoidal functions on ∆αji to generate N 2 pairs of angu-
+lar embeddings eα ∈ RN×N×d for all N points as:
+eα
+i,j,2k = sin
+� ∆αji/tα
+100002k/d
+�
+, eα
+i,j,2k+1 = cos
+� ∆αji/tα
+100002k/d
+�
+,
+(5)
+where tα controls the sensitivity to angle variations.
+Finally, we inject eα into offset attention and learn intra-
+frame aligned feature Fℓ
+IAS via self-attention as follows:
+Fℓ
+IAS = φ
+�
+Fℓ
+oa
+�
++ Fℓ, Fℓ
+oa = Fℓ − ∥ softmax(Asa)∥1vsa,
+Asa = Aattn
+sa
++ Arot
+sa , Aattn
+sa
+= qsak⊤
+sa, Arot
+sa = qsa(eα
+saWα
+sa)⊤,
+(6)
+where qsa/ksa/vsa = FℓWq
+sa/FlWk
+sa/FlWv
+sa, Wα
+sa ∈
+Rd×d is a linear projection to refine the learning of eα
+sa, and
+Asa is the attention logits. The same process can be per-
+formed for Fg by swapping the index ℓ and g. Detailed il-
+lustrations are shown in Fig. 2 (a).
+Inter-frame Aligned Cross-attention.
+Semantic infor-
+mation exchange between Fℓ and Fg in the feature space
+is implemented efficiently by cross-attention (Chen, Fan,
+and Panda 2021). Since Fℓ and Fg are learned from differ-
+ent coordinate systems, inter-frame transformations should
+be considered for cross-consistency between Fℓ and Fg.
+An illustration of the cross-attention module is shown in
+Fig. 2 (b). Computation of inter-frame aligned feature Fℓ
+IAC
+via cross-attention follows a similar way as Eq. 6:
+Fℓ
+IAC = φ
+�
+Fℓ
+oa
+�
++ Fℓ, Fℓ
+oa = Fℓ − ∥ softmax(Aca)∥1vca,
+Aca = Aattn
+ca
++ Arot
+ca , Aattn
+ca
+= qcak⊤
+ca, Arot
+ca = qca(eα
+caWα
+ca)⊤,
+(7)
+where qca/kca/vca = FℓWq
+ca/FgWk
+ca/FgWv
+ca. Aca is
+cross-attention logits containing point-wise cross-relations
+over point features defined across local and global scales.
+eα
+ca ∈ RN×d is computed via Eq. 4 and Eq. 5 in terms of the
+transformation between Mℓ
+i and Mg. To this end, the geo-
+metric features learned between local and global reference
+frames can be aligned given eα
+ca, leading to a consistent fea-
+ture representation.
+
+Attention-based Feature Integration.
+Instead of simply
+adding the information from both Fℓ and Fg, we integrate
+information by incrementing attention logits. Specifically,
+we apply self-attention on Fℓ with attention logits Asa
+and cross-attention between Fℓ and Fg with attention log-
+its Aca. We combine Asa and Aca via addition, so that en-
+coded information of all point pairs from a local domain can
+be enriched by the global context of the whole shape. Illus-
+tration is shown in the supplementary material. The whole
+process is formulated as follows:
+U = φ (Foa) + Fℓ,
+Foa = Fℓ − ∥softmax(Asa + Aca)∥1(vsa + vca).
+(8)
+Hence, intra-frame point relations can be compensated by
+inter-frame information communication in a local-to-global
+manner, which enriches the geometric representations.
+3.3
+Deep Feature Registration
+Correspondence mapping (Wang and Solomon 2019a; Pan,
+Cai, and Liu 2022) plays an important role in PCR, and we
+discuss that it is also critical for achieving RI in our design.
+Specifically, although Fℓ and Fg are both rotation-invariant
+by theory, different point sampling methods and the sign
+ambiguity will cause the final feature not strictly rotation-
+invariant. To solve this issue, we first examine the correspon-
+dence map:
+m (X, Y) =
+exp
+�
+Φ1(Y)Φ2(X)⊤/t
+�
+�N
+j=1 exp (Φ1(Y)Φ2(xj)⊤/t)
+,
+(9)
+where Φ1 and Φ2 are MLPs that project latent embeddings
+X and Y to a shared space, and t controls the variation sen-
+sitivity. It can be seen from Eq. 9 that the mapping function
+m reveals feature similarities in the latent space, and it is
+also an essential part for 3D point-level contrastive learning
+in PointContrast (Xie et al. 2020) for the design of InfoNCE
+losses (Van den Oord, Li, and Vinyals 2018), which have
+been proven to be equivalent to maximize the mutual infor-
+mation. Based on this observation, we propose a registration
+loss function Lr = Lℓ
+r + Lg
+r, where Lℓ
+r and Lg
+r represent the
+registration loss of (Fℓ,U) and (Fg,U). Mathematically, Lℓ
+r
+is defined as follows:
+Lℓ
+r = −
+�
+(i,j)∈M
+log
+exp
+�
+Φ1(Uj)Φ2(f ℓ
+i )⊤/t
+�
+�
+(·,k)∈M exp
+�
+Φ1(Uk)Φ2(f ℓ
+i )⊤/t
+�. (10)
+The same rule is followed to compute Lg
+r. Although we fol-
+low the core idea of PointContrast, we differ from it in that
+PointContrast defines positive samples based on feature cor-
+respondences computed at the same layer level, while our
+positive samples are defined across layers.
+The intuition for the loss design is that the 3D shape is
+forced to learn about its local region as it has to distinguish
+it from other parts of different objects. Moreover, we would
+like to maximize the mutual information between different
+poses of the 3D shape, as features encoded from different
+poses should represent the same object, which is very use-
+ful in achieving RI in SO(3). Moreover, the mutual infor-
+mation between Fℓ and Fg is implicitly maximized, such
+Rotation Sensitive
+z/z
+z/SO(3)
+SO(3)/SO(3)
+∆
+PointNet (Qi et al. 2017a)
+89.2
+16.2
+75.5
+59.3
+PoinNet++ (Qi et al. 2017b)
+89.3
+28.6
+85.0
+56.4
+PCT (Guo et al. 2021)
+90.3
+37.2
+88.5
+51.3
+Rotation Robust
+z/z
+z/SO(3)
+SO(3)/SO(3)
+∆
+Spherical CNN* (Esteves et al. 2018)
+88.9
+76.9
+86.9
+10
+SFCNN (Rao, Lu, and Zhou 2019)
+91.4
+84.8
+90.1
+5.3
+RIConv (Zhang et al. 2019)
+86.5
+86.4
+86.4
+0.1
+ClusterNet (Chen et al. 2019)
+87.1
+87.1
+87.1
+0.0
+PR-InvNet (Yu et al. 2020)
+89.2
+89.2
+89.2
+0.0
+RI-GCN (Kim, Park, and Han 2020)
+89.5
+89.5
+89.5
+0.0
+GCAConv (Zhang et al. 2020)
+89.0
+89.1
+89.2
+0.1
+RI-Framework (Li et al. 2021b)
+89.4
+89.4
+89.3
+0.1
+VN-DGCNN (Deng et al. 2021)
+89.5
+89.5
+90.2
+0.7
+SGMNet (Xu et al. 2021)
+90.0
+90.0
+90.0
+0.0
+Li et al. (2021a)
+90.2
+90.2
+90.2
+0.0
+OrientedMP (Luo et al. 2022)
+88.4
+88.4
+88.9
+0.5
+ELGANet (Gu et al. 2022)
+90.3
+90.3
+90.3
+0.0
+Ours
+91.0
+91.0
+91.0
+0.0
+Table 1: Classification results on ModelNet40 under rota-
+tions. * denotes the input type as projected voxels of 2×642,
+while the rest take raw points of 1024×3 as inputs. ∆ is the
+absolute difference between z/SO(3) and SO(3)/SO(3).
+that shared semantic information about geometric structures
+can be learned, leading to a more geometrically accurate and
+discriminative representation. More details about Lℓ
+r can be
+found in the supplementary material.
+4
+Experiments
+We evaluate our model on 3D shape classification, part seg-
+mentation, and retrieval tasks under rotations, and exten-
+sive experiments are conducted to analyze the network de-
+sign. Detailed model architectures for the three tasks are
+shown in the supplementary material. Our evaluating proto-
+cols are the same as (Esteves et al. 2018): training and testing
+the network under azimuthal rotations (z/z); training under
+azimuthal rotations while testing under arbitrary rotations
+(z/SO(3)); and training and testing under arbitrary rotations
+(SO(3)/SO(3)).
+4.1
+3D Object Classification
+Synthetic Dataset.
+We first examine the model perfor-
+mance on the synthetic ModelNet40 (Wu et al. 2015)
+dataset. We sample 1024 points from each data with only
+xyz coordinates as input features. Hyper-parameters for
+training follow the same as (Guo et al. 2021), except
+that points are downsampled in the order of (1024, 512,
+128) with feature dimensions of (3, 128, 256). We report
+and compare our model performance with state-of-the-art
+(SoTA) methods in Table 1. Both rotation sensitive and ro-
+bust methods achieve great performance under z/z. How-
+ever, the former could not generalize well to unseen rota-
+tions. Rotation robust methods like Spherical CNN (Esteves
+et al. 2018) and SFCNN (Rao, Lu, and Zhou 2019) achieve
+competitive results under z/z, but their performance is not
+consistent on z/SO(3) and SO(3)/SO(3) due to the imperfect
+projection from points to voxels when using spherical so-
+lutions. We outperform the recent proposed methods (Luo
+et al. 2022; Xu et al. 2021; Deng et al. 2021) and achieve an
+accuracy of 91.0%, proving the superiority of our framework
+on classification.
+
+Method
+z/SO(3)
+SO(3)/SO(3)
+∆
+PointNet (Qi et al. 2017a)
+16.7
+54.7
+38.0
+PointNet++ (Qi et al. 2017b)
+15.0
+47.4
+32.4
+PCT (Guo et al. 2021)
+28.5
+45.8
+17.3
+RIConv (Zhang et al. 2019)
+78.4
+78.1
+0.3
+RI-GCN (Kim, Park, and Han 2020)
+80.5
+80.6
+0.1
+GCAConv (Zhang et al. 2020)
+80.1
+80.3
+0.2
+RI-Framework (Li et al. 2021b)
+79.8
+79.9
+0.1
+LGR-Net (Zhao et al. 2022)
+81.2
+81.4
+0.2
+VN-DGCNN (Deng et al. 2021)
+79.8
+80.3
+0.5
+OrientedMP (Luo et al. 2022)
+76.7
+77.2
+0.5
+Ours
+86.6
+86.3
+0.3
+Table 2: Classification results on ScanObjectNN OBJ BG
+under z/SO(3) and SO(3)/SO(3).
+GT
+Ours
+RI-GCN
+RIConv
+VN-DGCNN
+Figure 3: Segmentation comparisons on ShapeNetPart,
+where ground truth (GT) samples are shown for refer-
+ence. Red dotted circles indicate obvious failures on certain
+classes, and purple circles denote the slight difference be-
+tween our design and VN-DGCNN.
+Real Dataset.
+Experiments are also conducted on a real-
+scanned dataset. ScanObjectNN (Uy et al. 2019) is a com-
+monly used benchmark to explore the robustness to noisy
+and deformed 3D objects with non-uniform surface density,
+which includes 2,902 incomplete point clouds in 15 classes.
+We use OBJ BG subset with the background noise and sam-
+ple 1,024 points under z/SO(3) and SO(3)/SO(3). Table 2
+shows that our model achieves the highest results with ex-
+cellent consistency with random rotations.
+4.2
+3D Part Segmentation
+Shape part segmentation is a more challenging task than ob-
+ject classification. We use ShapeNetPart (Yi et al. 2016) for
+evaluation, where we sample 2048 points with xyz coordi-
+nates as model inputs. The training strategy is the same as
+the classification task except that the training epoch num-
+ber is 300. Part-averaged IoU (mIoU) is reported in Table
+3, and detailed per-class mIoU values are shown in the sup-
+plementary material. Representative methods such as Point-
+Method
+z/SO(3)
+SO(3)/SO(3)
+∆
+PointNet (Qi et al. 2017a)
+38.0
+62.3
+24.3
+PointNet++ (Qi et al. 2017b)
+48.3
+76.7
+28.4
+PCT (Guo et al. 2021)
+38.5
+75.2
+36.7
+RIConv (Zhang et al. 2019)
+75.3
+75.5
+0.2
+RI-GCN (Kim, Park, and Han 2020)
+77.2
+77.3
+0.1
+RI-Framework (Li et al. 2021b)
+79.2
+79.4
+0.2
+LGR-Net (Zhao et al. 2022)
+80.0
+80.1
+0.1
+VN-DGCNN (Deng et al. 2021)
+81.4
+81.4
+0.0
+OrientedMP (Luo et al. 2022)
+80.1
+80.9
+0.8
+Ours
+80.3
+80.4
+0.1
+Table 3: Segmentation results on ShapeNetPart under
+z/SO(3) and SO(3)/SO(3), where the second best results are
+underlined.
+Method
+micro mAP
+macro mAP
+Score
+Spherical CNN (Esteves et al. 2018)
+0.685
+0.444
+0.565
+SFCNN (Rao, Lu, and Zhou 2019)
+0.705
+0.483
+0.594
+GCAConv (Zhang et al. 2020)
+0.708
+0.490
+0.599
+RI-Framework (Li et al. 2021b)
+0.707
+0.510
+0.609
+Ours
+0.715
+0.510
+0.613
+Table 4: Comparisons of SoTA methods on the 3D shape
+retrieval task.
+Net++ and PCT are vulnerable to rotations. Rotation robust
+methods present competitive results under z/SO(3), where
+we achieve the second best result of 80.3%. We give more
+details of comparison between VN-DGCNN (Deng et al.
+2021) and our work in the supplementary material, where
+our method performs better than VN-DGCNN for several
+classes. Moreover, qualitative results shown in Fig. 3 present
+that we can achieve visually better results than VN-DGCNN
+in certain classes such as the airplane and car. More qualita-
+tive results are shown in the supplementary material.
+4.3
+3D Shape Retrieval
+We further conduct 3D shape retrieval experiments on
+ShapeNetCore55 (Chang et al. 2015), which contains two
+categories of datasets: normal and perturbed. We only use
+the perturbed part to validate our model performance under
+rotations. We combine the training and validation sets and
+validate our method on the testing set following the training
+policy of (Esteves et al. 2018). Experimental results are re-
+ported in Table 4, where the final score is the average value
+of micro and macro mean average of precision (mAP) as
+in (Savva et al. 2017). Similar to the classification task, our
+method achieves SoTA performance.
+4.4
+Ablation Study
+Effectiveness of Transformer Designs.
+We examine the
+effectiveness of our transformer design by conducting clas-
+sification experiments under z/SO(3). We first ablate one or
+both of the angular embeddings and report the results in Ta-
+ble 5 (models A, B, and C). Model B performs better than
+model C by 0.4%, which validates our design of feature in-
+tegration where Mℓ
+i is used as the main source of informa-
+tion. When both angular embeddings are applied, the best
+result is achieved (i.e., 91.0%). Moreover, we validate our
+discussion in Section 3.2 by comparing models D and E. We
+
+.:Model
+eα
+sa
+eα
+ca
+Fg∗
+Asa + Aca
+Lℓ
+r
+Lg
+r
+Acc.
+A
+✓
+✓
+✓
+90.0
+B
+✓
+✓
+✓
+✓
+90.6
+C
+✓
+✓
+✓
+✓
+90.2
+D
+✓
+✓
+✓
+✓
+✓
+✓
+90.2
+E
+✓
+✓
+✓
+✓
+90.4
+F
+✓
+✓
+✓
+90.0
+G
+✓
+✓
+✓
+✓
+90.2
+H
+✓
+✓
+✓
+✓
+90.6
+Ours
+✓
+✓
+✓
+✓
+✓
+91.0
+Table 5: Module analysis of AIT and loss functions. Fg∗
+means encoding Fg via Intra-frame Aligned Self-attention.
+demonstrate in model D that when encoding Fg in the same
+way as Fℓ, the model performance decreases, which indi-
+cates that encoding Fg via self-attention will increase the
+model overfitting. More analyses can be found in the supple-
+mentary material. Finally, we examine the effectiveness of
+our attention logits-based integration scheme by comparing
+our model with the conventional method (model E), which
+applies self- and cross-attention sequentially and repeatedly.
+We observe that our result is better than model E by 0.6%,
+indicating that our design is more effective.
+Registration Loss.
+We sequentially ablate Lg
+r and Lℓ
+r
+(models F, G, and H) to check the effectiveness of our reg-
+istration loss deign. Results in Table 5 demonstrate that we
+can still achieve a satisfactory result of 90.0% without fea-
+ture registration. Individual application of Lg
+r and Lℓ
+r shows
+the improvement when forcing the final representation to be
+close to rotation-invariant features. Moreover, it can be seen
+that model H performs better than model G, which indicates
+that intermediate features learned from the global scale are
+important for shape classification. The best model perfor-
+mance is hence achieved by applying both losses.
+Noise Robustness.
+In real-world applications, raw point
+clouds contain noisy signals. We conduct experiments to
+present the model robustness to noise under z/SO(3). Two
+experiments are conducted: (1) We sample and add Gaus-
+sian noise of zero mean and varying standard deviations
+N(0, σ2) to the input data; (2) We add outliers sampled
+from a unit sphere to each object. As shown in Fig. 4 (left),
+we achieve on par results to RI-Framework when std is low,
+while we perform better while std increases, indicating that
+our model is robust against high levels of noise. Besides,
+as the number of noisy points increases, most methods are
+heavily affected while we can still achieve good results.
+Visualization of Rotation Invariance.
+We further ex-
+amine RI of learned features. Specifically, we use Grad-
+CAM (Selvaraju et al. 2017) to check how the model pays
+attention to different parts of data samples under different
+rotations. Results are reported in Fig. 5 with correspondence
+between gradients and colors shown on the right. RI-GCN
+presents a good result, but its behavior is not consistent over
+some classes (e.g., vase and plant) and it does not pay atten-
+tion to regions that are critical for classification (see toilet),
+showing inferior performance to ours. PointNet++ shows no
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+std of noise
+20
+30
+40
+50
+60
+70
+80
+90
+Accuracy (%)
+RIConv
+RI-GCN
+RI-framework
+SRINet
+ours
+0
+10
+20
+30
+40
+50
+# of noisy points
+30
+40
+50
+60
+70
+80
+90
+Figure 4: Left: Results on Gaussian noise of zero mean and
+variant standard deviation values. Right: Results on differ-
+ent numbers of noisy points.
+high
+low
+high
+low
+airplane
+guitar
+vase
+plant
+toilet
+PointNet++
+RI-GCN
+high
+low
+Ours
+Figure 5: Network attention on PointNet++ (top), RI-GCN
+(mid) and our model (bot).
+resistance to rotations, while our method exhibits a consis-
+tent gradient distribution over different parts with random
+rotations, indicating our network is not affected by rotations.
+5
+Conclusion
+In this work, we rethink and investigate the close relation be-
+tween rotation invariance and point cloud registration, based
+on which we propose a PCR-cored learning framework with
+three stages. With a pair of rotation-invariant shape descrip-
+tors constructed from local and global scales, a comprehen-
+sive learning and feature integration module is proposed,
+Aligned Integration Transformer, to simultaneously effec-
+tively align and integrate shape codes via self- and cross-
+attentions. To further preserve rotation invariance in the fi-
+nal feature representation, a registration loss is proposed to
+align it with intermediate features, where shared semantic
+knowledge of geometric parts is also extracted. Extensive
+experiments demonstrated the superiority and robustness of
+our designs. In future work, we will examine efficient meth-
+ods for invariance learning on large-scale point clouds.
+
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+
+Rethinking Rotation Invariance with Point Cloud Registration
+**Supplementary Material**
+Jianhui Yu, Chaoyi Zhang, Weidong Cai
+School of Computer Science, University of Sydney, Australia
+{jianhui.yu, chaoyi.zhang, tom.cai}@sydney.edu.au
+Mathematical Proofs
+Proof of Rotation Equivariance of Orthonormal Basis
+Mℓ
+i of LRFs.
+We prove the rotation equivariance of Mℓ
+i
+designed for LRFs as mentioned in Section 3.1 of the main
+work. Given a random rotation matrix R ∈ R3×3, it is easy
+to derive that #»
+xiℓ is equivariant to rotations given the rotated
+version #»
+xiℓ
+,rot:
+#»
+xi
+ℓ
+,rot =
+Rpi − Rpm
+∥Rpi − Rpm∥2
+=
+R(pi − pm)
+�
+(R(pi − pm))⊤ R(pi − pm)
+=
+R#       »
+pmpi
+�#       »
+pmpi⊤R⊤R#       »
+pmpi
+= R
+#       »
+pmpi
+∥#       »
+pmpi∥2
+= R#»
+xi
+ℓ,
+(1)
+where the subscript rot represents the axis after rotations.
+Moreover, Σℓ
+i from Eq. (1) of the main work after rotations
+can be represented as follows:
+Σℓ
+i,rot =
+|N (pi)|
+�
+j=1
+αjR#     »
+pipj #     »
+pipj
+⊤R⊤
+= R
+�
+�
+|N (pi)|
+�
+j=1
+αj #     »
+pipj #     »
+pipj
+⊤
+�
+� R⊤ = RΣℓ
+iR⊤.
+(2)
+As mentioned in the main work, eigenvalue decomposition
+can be directly applied to Σℓ
+i, resulting in the following ex-
+pressions:
+RΣℓ
+iR⊤ = RUℓ
+iΛℓ
+iUℓ
+i
+⊤R⊤ =
+�
+RUℓ
+i
+�
+Λℓ
+i
+�
+RUℓ
+i
+�⊤ . (3)
+Since #»
+ziℓ is defined to have the same direction as the eigen-
+vector with the smallest eigenvalue, after rotation, #»
+ziℓ
+,rot =
+R#»
+ziℓ. Thus, the rotated y-axis is:
+#»
+yi
+ℓ
+,rot = #»
+zi
+ℓ
+,rot × #»
+xi
+ℓ
+,rot = R#»
+zi
+ℓ × R#»
+xi
+ℓ
+= det (R)
+�
+R−1�⊤ �#»
+zi
+ℓ × #»
+xi
+ℓ�
+= R
+�#»
+zi
+ℓ × #»
+xi
+ℓ�
+= R#»
+yi
+ℓ.
+(4)
+Since all basis vectors are rotation-equivariant, the local or-
+thonormal basis Mℓ
+i = [#»
+xiℓ, #»
+yiℓ, #»
+ziℓ] is rotation-equivariant.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Proof of Rotation Invariance of Local Shape Descriptors
+pℓ
+ij.
+Here, we show the rotation invariance of local shape
+descriptors pℓ
+ij introduced in Section 3.1 of the main work.
+Based on the proof shown above, it is easy to show the rota-
+tion invariance of pℓ
+ij after rotation R:
+pℓ
+ij,rot = #     »
+pipj
+⊤
+,rotMℓ
+i,rot = (R#     »
+pipj)⊤ �
+RMℓ
+i
+�
+= #     »
+pipj
+⊤Mℓ
+i = pℓ
+ij.
+(5)
+Model Details
+Architectures.
+The model overview for the 3D classifica-
+tion, segmentation, and retrieval tasks is shown in Fig. 1,
+where details of each module design are explained in Sec-
+tion 3 of the main work. The architecture of attention-based
+feature integration module is illustrated in Fig. 2. Please re-
+fer to Section 3.2 of the main work for more details.
+Fg in AIT.
+As mentioned in Section 3.2, to alleviate the
+model overfitting we do not apply self-attention on Fg. We
+argue that since we use global shape information as the sup-
+plementary material to assist local shape learning, this im-
+plementation allow lower-level information to flow effec-
+tively across layers to help the learning of higher-level lo-
+cal shape features, which could reduce the model overfit-
+ting. We encode Fℓ in AIT blocks, as abstracting informa-
+tion from local structures can increase the model’s ability on
+fine-grained pattern recognition and generalizability to com-
+plex scenes (Qi et al. 2017b). Hence, we find that without
+applying learnable modules on Fg is beneficial to our model
+performance.
+Registration Loss.
+In this this part, we first explain the
+benefits of applying the registration loss to preserve the ro-
+tation invariance. We then give more details about Eq. (10)
+in the main work.
+Suppose Uℓ and Ug are local and global part information
+of the final integrated feature U. By maximizing the mutual
+information between (U, Fℓ) and (U, Fg), (Uℓ, Fℓ) and
+(Ug, Fg) are implicitly maximized. In this case, the shared
+geometric information between the local Fℓ/global Fg and
+the integrated domain U are refined, increasing the repre-
+sentation power of U. Besides, the maximized similarities of
+(Uℓ, Fℓ) and (Ug, Fg) also tend to learn rotation invariance
+in an unsupervised manner. Specifically, although Fℓ/Uℓ en-
+codes local patches with different poses (since LRFs are
+arXiv:2301.00149v1  [cs.CV]  31 Dec 2022
+
+airplane
+car
+chair
+…
+toilet
+Rotational 
+Embeddin
+g
+…
+Shape Retrieval
+Classification
+GRF
+LRF
+SA x 2
+SA x 2
+AIT x 4
+U
+𝑳𝒓
+"
+𝑳𝒓ℓ
+MLP
+max
+Fℓ
+F$
+MLP
+Rotation-Invariant 
+Shape Encoding
+Aligned Feature 
+Integration
+Deep Feature
+Registration
+(a) classification/retrieval
+GRF
+LRF
+SA x 2
+SA x 2
+AIT x 4
+U
+𝑳𝒓
+"
+𝑳𝒓ℓ
+MLP
+Fℓ
+F$
+MLP
+FP x 2
+Rotational 
+Embeddin
+g
+Rotation-Invariant 
+Shape Encoding
+Aligned Feature 
+Integration
+Deep Feature
+Registration
+Part Segmentation
+(b) segmentation
+Figure 1: Model overviews for (a) classification / retrieval
+and (b) segmentation. GRF: global reference frame; LRF:
+local reference frame; SA: set abstraction; AIT: Aligned In-
+tegration Transformer; and FP: forward passing.
+𝑾!"
+#
+𝑾!"
+$
+𝑾!"
+%
+𝐤!": 𝑁×𝑑
+𝐪!": 𝑁×𝑑
+𝐯!": 𝑁×𝐶
+𝐤#": 𝑁×𝑑
+𝐯#": 𝑁×𝐶
+𝑾&"
+$
+𝑾&"
+#
+𝑭ℓ
+𝑭(
+𝑨𝒔𝒆𝒍𝒇
+𝒂𝒕𝒕𝒏: 𝑁×𝑁
+𝑨𝒔𝒆𝒍𝒇
+𝑨𝒄𝒓𝒐𝒔𝒔
+𝒓𝒐𝒕
+: 𝑁×𝑁
+𝑨𝒄𝒓𝒐𝒔𝒔
+𝑨𝒔𝒉𝒂𝒓𝒆𝒅: 𝑁×𝑁
+𝑨𝒔𝒆𝒍𝒇
+𝒓𝒐𝒕 : 𝑁×𝑁 𝑨𝒄𝒓𝒐𝒔𝒔
+𝒂𝒕𝒕𝒏 : 𝑁×𝑁
+𝑭0"
+𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧
+𝐦𝐚𝐭𝐫𝐢𝐱	𝐩𝐫𝐨𝐝𝐮𝐜𝐭
+𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧
+Figure 2: Illustrations of attention-based feature integra-
+tion, where blue and green boxes indicate self- and cross-
+attention. Brown, gold and purple colored components cor-
+respond to v, k and q implementations.
+unaligned), and Fg/Ug encodes the whole 3D object in a
+canonical pose, their feature similarity should be enforced
+to be similar as they represent the same 3D object, no matter
+what their poses are. Moreover, the mutual information be-
+tween a local scale of a 3D object and a global scale of the
+same object is maximized, which embeds U with more ac-
+curate geometric information to distinguish it from objects
+in different classes.
+We then explain the symbols in Eq. (10). M stands for
+the set of all B × N pairs of positive samples across mini-
+batches, i.e., M = {(0, 0), ..., (B ×N, B ×N)}, where B is
+the number of batch size and N is the number of points. The
+point feature Uk is the set of negative keys where (·, k) ∈ M
+and k ̸= j. Note that since the registration loss function is
+applied to both Fℓ and Fg, the point features of the same
+point encoded from the local and global scales are pushed
+close to each other, where the mutual information is in-
+creased such that shared geometric information can be dis-
+covered across the local and global scales.
+Training Details.
+For all three tasks, we set the batch
+size to 32 for training and 16 for testing. We use farthest
+point sampling to re-sample the points from the initial 10k
+points to 1024 points for classification and retrieval and 2048
+points for segmentation. Random point translation within
+[−0.2, 0.2] and rescaling within [0.67, 1.5] were adopted for
+augmentation. We trained the model for 250 epochs with
+tα = 15 and t = 0.017. SGD is adopted as the optimizer,
+where the learning rate was set to 1e-2 with momentum of
+0.9 and weight decay of 1e-4. Cosine annealing was applied
+to reschedule the learning rate for each epoch. For classi-
+fication and retrieval, we used one RTX2080Ti GPU with
+PyTorch for model implementation, and we used two GPUs
+for the segmentation task. The normal vector information is
+ignored for all experiments.
+More Analysis Experiments
+Influence of Randomness.
+We report the variance and
+mean values of each model in Table 5 of the main work to
+derive a more accurate and reliable estimate of our model
+performance. We hence report the variance and mean values
+of performance of each model in Table 5 on ModelNet40
+with 5 training rounds. As shown in Table 1, we can see
+that even for our model weights with the lowest performance
+90.6% (among our five repeated runs), it still surpasses the
+highest performance among models from A to H.
+Model
+A
+B
+C
+Acc. (%)
+89.8±0.2
+90.4±0.2
+90.1±0.1
+Model
+D
+E
+F
+Acc. (%)
+89.8±0.4
+90.1± 0.3
+89.6±0.4
+Model
+G
+H
+Best
+Acc. (%)
+90.0±0.2
+90.3±0.3
+90.8±0.2
+Table 1: Variance and Mean values of different model per-
+formances on ModelNet40 with z/SO(3).
+Point Re-sampling and Down-sampling.
+We examine
+different point re-sampling strategies from the initial 10k
+
+input points down to 1024 input points. Experimental re-
+sults of applying different sampling techniques on Model-
+Net40 are shown in Table 2, where we use random sam-
+pling (RS), farthest point sampling (FPS), uniform sampling
+(US), and inverse density importance sampling (IDIS) from
+(Groh, Wieschollek, and Lensch 2018) to examine the im-
+pact of different sampling methods on rotation invariance.
+Note that point sampling affects both LRF and GRF con-
+structions in our design, therefore we can only give analysis
+when considering both reference frames. We can see that
+random sampling gives the lowest model performance with
+89.7%, with 1.3% absolute performance drop compared to
+our method using FPS. Inverse density importance sampling
+can achieve a comparable result as our method, while it is
+not strictly invariant to rotations. We argue that due to the
+information compensation between features encoded from
+LRFs and GRF, different sampling strategies will not affect
+our model performance quite much.
+Sampling Method
+RS
+FPS (ours)
+US
+IDIS
+z/z
+89.7
+91.0
+90.2
+90.6
+z/SO(3)
+89.7
+91.0
+90.2
+90.6
+SO(3)/SO(3)
+89.7
+91.0
+90.2
+90.6
+Table 2: Classification results (%) on ModelNet40 with dif-
+ferent re-sampling techniques.
+Visualization of U.
+To better present the discriminabil-
+ity of the learned features, we summarize the shape fea-
+ture representation U by maxpooling and visualize it via t-
+SNE (Van der Maaten and Hinton 2008). Experiments are
+conducted on object classification under z/z and z/SO(3).
+Only the first 16 classes are selected for a clear represen-
+tation purpose as shown in Fig. 3. Although it is difficult to
+correctly separate all categories, we can see that some shape
+classes can be perfectly predicted, and the overall represen-
+tation ability of U under different testing protocols is satis-
+factory and consistent.
+Figure 3: t-SNE of the aggregated U with z/SO(3) (Left) and
+SO(3)/SO(3) (Right). Clusters indicate good predictions in
+object classification.
+Constructions of pℓ
+ij.
+We examine the model performance
+when using different methods to construct the local rotation-
+invariant feature pℓ
+ij. Specifically, in addition to the proposed
+Method
+PPFs
+LRFs (Ours)
+Acc. (%)
+89.3
+91.0
+Table 3: Classification results (%) on ModelNet40 with
+z/SO(3).
+method that builds pℓ
+ij based on LRFs, we examine point-
+pair features (PPFs) to build pℓ
+ij following (Deng, Birdal,
+and Ilic 2018). As reported in Table 3, we find that the model
+performance of using PPFs is lower than our LRF-based
+method. The reason is that point positions, which provide
+information of exact shape of the 3D objects, are important
+for shape learning. However, point-pair features give infor-
+mation about the topology of a 3D shape, and different 3D
+shapes can have the same topology, which introduces diffi-
+culties for exact 3D shape learning.
+Model Complexity.
+Inference model sizes of different
+methods along with the corresponding construction time
+for LRFs and inference speed are reported in Table 4.
+The construction time measured in seconds (s) shows time
+cost for different models generating their low-level rotation-
+invariant shape features, where we record the total time
+for local and global representation constructions of RI-
+Framework and our work. VN-DGCNN does not compute
+the rotation-invariant shape features, therefore no result can
+be reported. The inference speed with the unit of number of
+instances evaluated within one second (ins./s) is measured
+for each method with a batch size of 1. When computing the
+inference speed, the amount of time for low-level rotation-
+invariant feature construction of methods (Li et al. 2021b;
+Kim, Park, and Han 2020; Zhang et al. 2019; Li et al. 2021a)
+is also considered. Table 4 shows that our method only needs
+a relatively short construction time for both LRFs and GRF.
+Meanwhile, the trade-off between the accuracy and infer-
+ence speed is hard to balance. We will investigate the model
+design for a much high accuracy and faster speeds in the
+future work.
+Sign Ambiguity.
+As mentioned in the main work, we pro-
+pose simple techniques to address the sign ambiguity issue
+introduced by eigenvalue decomposition when computing
+the LRFs and GRF. We thus examine the model performance
+with no sign disambiguation techniques applied, of which
+the results are reported in Table 5. It can be seen that sign
+ambiguity negatively affects the model performance, where
+Method
+Params (M)
+Times (s)
+Speed (ins./s)
+Acc (%)
+RIConv
+0.68
+0.041
+396.4
+86.4
+RI-GCN
+4.19
+0.057
+139.1
+89.5
+RI-Framework
+2.36
+0.134
+43.1
+89.4
+VN-DGCNN
+2.77
+-
+77.3
+89.5
+Li et al. (2021a)
+2.76
+0.047
+35.8
+90.2
+Ours
+3.11
+0.043
+205.3
+91.0
+Table 4: Model complexity construction time for LRFs, and
+inference speed on ModelNet40 with z/SO(3), where Li
+et al. (2021a) is considered without test time augmentation.
+
+performances drop by 0.7% and 0.9% when uncertainty of
+vector directions is introduced to the model training. With
+our proposed solutions, the model behavior can be stabilized
+hence the classification accuracy increases.
+Method
+no@Mℓ
+no@Mg
+no@Mℓ and Mg
+Acc. (%)
+90.3
+90.1
+89.8
+Table 5: Classification results (%) on ModelNet40 with
+z/SO(3), where “no@” denotes no sign disambiguation tech-
+nique applied.
+Rotational Effect of Mg.
+As mentioned in (Li et al.
+2021a), different ways to ensure Mg is a valid rotation ma-
+trix would result in different model performances. In this
+part, we examine four different methods to ensure Mg is a
+valid rotation as follows: (a) we randomly permute two basis
+vectors regardless of the S value; (b) we randomly negate the
+value of a basis vector regardless of the S value; (c) we per-
+mute two basis vectors of S values being the smallest two;
+and (d) we simply reverse the direction of the basis vector
+whose S value is the smallest, which is the proposed method
+in our implementation. We can see from Table 6 that our
+simple design achieves the highest value, while all the others
+decrease the model performance, which shows the effective-
+ness of our proposed method.
+3D Semantic Segmentation.
+To check our model’s effec-
+tiveness on real-world large scenes, additional experiments
+are conducted on S3DIS dataset (Armeni et al. 2016), which
+includes six indoor areas of three different buildings. Each
+point is labeled by one of the 13 categories (e.g., ceiling,
+chair or clutter). Following the same pre-processing steps as
+(Qi et al. 2017b; Wang et al. 2019), each room is divided into
+1m×1m blocks and for each block 4096 points are sampled
+during training process. We use area-5 for testing and all the
+other areas for training. The quantitative results are shown
+in Table 7 following (Zhao et al. 2022), where it shows that
+under random rotations, our model outperforms LGR-Net
+by 7.8%, showing a more effective way to process large in-
+door scenes. For a more intuitive understanding of our model
+performance, qualitative results are shown in Fig. 4 for ref-
+erence.
+3D Part Segmentation.
+For visualization purposes (see
+Fig. 4 in the main work) as well as a detailed analysis of
+model behavior for each category, we report the per-class
+mIoU accuracies under z/SO(3) and SO(3)/SO(3) in Ta-
+bles 8 and 9, where bold numbers indicate the best results
+for each category. As per-class mIoU scores are not reported
+in VN-DGCNN (Deng et al. 2021), we follow the official
+implementation1 and report per-class mIoU results in both
+tables. However, the reproduced results of VN-DGCNN
+are much lower than the ones reported in their work, and
+our model achieves better segmentation results than VN-
+DGCNN (Deng et al. 2021) for most categories. Our model
+also outperforms the state-of-the-art methods (Zhao et al.
+1https://github.com/FlyingGiraffe/vnn-pc
+Method
+a
+b
+c
+d (ours)
+Acc. (%)
+90.1
+90.1
+90.5
+91.0
+Table 6: Classification results (%) on ModelNet40 with
+z/SO(3).
+Method
+z/z
+z/SO(3)
+SO(3)/SO(3)
+PointNet (Qi et al. 2017a)
+41.1
+4.1
+29.3
+DGCNN (Wang et al. 2019)
+48.4
+3.6
+34.3
+RIConv (Zhang et al. 2019)
+22.0
+22.0
+22.0
+LRG-Net (Zhao et al. 2022)
+43.4
+43.4
+43.4
+Ours
+51.2
+51.2
+51.2
+Table 7: Semantic segmentation results (mIoU) on S3DIS
+area-5.
+2022; Luo et al. 2022) in several classes (e.g., airplane, chair,
+and table) under different testing conditions. Furthermore,
+it is also obvious that our model performance is consistent
+across all categories when tested under different rotations.
+In addition, we present more qualitative examples in Fig. 5,
+which includes all 16 classes. For each class, we show two
+pairs of ground truth and our predicted samples. We can see
+that although errors occur when the boundary between the
+different parts have marginal difference, our model achieves
+great performance for most classes.
+References
+Armeni, I.; Sener, O.; Zamir, A. R.; Jiang, H.; Brilakis, I.;
+Fischer, M.; and Savarese, S. 2016. 3D semantic parsing of
+large-scale indoor spaces. In CVPR.
+Deng, C.; Litany, O.; Duan, Y.; Poulenard, A.; Tagliasac-
+chi, A.; and Guibas, L. J. 2021. Vector neurons: A general
+framework for SO (3)-equivariant networks. In ICCV.
+Deng, H.; Birdal, T.; and Ilic, S. 2018. PPFNet: Global con-
+text aware local features for robust 3d point matching. In
+CVPR.
+Groh, F.; Wieschollek, P.; and Lensch, H. 2018.
+Flex-
+convolution. In ACCV.
+Guo, M.-H.; Cai, J.-X.; Liu, Z.-N.; Mu, T.-J.; Martin, R. R.;
+and Hu, S.-M. 2021. PCT: Point cloud transformer. In CVM.
+Kim, S.; Park, J.; and Han, B. 2020.
+Rotation-Invariant
+Local-to-Global Representation Learning for 3D Point
+Cloud. In NeurIPS.
+Li, F.; Fujiwara, K.; Okura, F.; and Matsushita, Y. 2021a. A
+Closer Look at Rotation-Invariant Deep Point Cloud Analy-
+sis. In ICCV.
+Li, X.; Li, R.; Chen, G.; Fu, C.-W.; Cohen-Or, D.; and Heng,
+P.-A. 2021b. A rotation-invariant framework for deep point
+cloud analysis. In TVCG.
+Luo, S.; Li, J.; Guan, J.; Su, Y.; Cheng, C.; Peng, J.; and
+Ma, J. 2022. Equivariant Point Cloud Analysis via Learning
+Orientations for Message Passing. In CVPR.
+Qi, C. R.; Su, H.; Mo, K.; and Guibas, L. J. 2017a. Point-
+Net: Deep learning on point sets for 3D classification and
+segmentation. In CVPR.
+
+Method
+mIoU
+air
+bag
+cap
+car chair
+ear
+guitar knife lamp laptop motor mug pistol rocket skate table
+plane
+phone
+bike
+board
+PointNet (Qi et al. 2017a)
+37.8
+40.4 48.1 46.3 24.5 45.1
+39.4
+29.2
+42.6 52.7
+36.7
+21.2
+55.0 29.7
+26.6
+32.1
+35.8
+PointNet++ (Qi et al. 2017b)
+48.3
+51.3 66.0 50.8 25.2 66.7
+27.7
+29.7
+65.6 59.7
+70.1
+17.2
+67.3 49.9
+23.4
+43.8
+57.6
+PCT (Guo et al. 2021)
+38.5
+32.2 44.8 36.3 26.1 36.2
+40.2
+48.1
+42.1 54.0
+40.9
+18.7
+50.5 25.6
+27.7
+44.7
+47.6
+RIConv (Zhang et al. 2019)
+75.3
+80.6 80.0 70.8 68.8 86.8
+70.3
+87.3
+84.7 77.8
+80.6
+57.4
+91.2 71.5
+52.3
+66.5
+78.4
+GCAConv (Zhang et al. 2020)
+77.2
+80.9 82.6 81.0 70.2 88.4
+70.6
+87.1
+87.2 81.8
+78.9
+58.7
+91.0 77.9
+52.3
+66.8
+80.3
+RI-Framework (Li et al. 2021b)
+79.2
+81.4 82.3 86.3 75.3 88.5
+72.8
+90.3
+82.1 81.3
+81.9
+67.5
+92.6 75.5
+54.8
+75.1
+78.9
+LGR-Net (Zhao et al. 2022)
+80.0
+81.5 80.5 81.4 75.5 87.4
+72.6
+88.7
+83.4 83.1
+86.8
+66.2
+92.9 76.8
+62.9
+80.0
+80.0
+VN-DGCNN⋆ (Deng et al. 2021)
+75.3
+81.1 74.8 72.9 73.8 87.8
+55.9
+91.4
+83.8 80.2
+84.4
+44.5
+92.8 74.6
+57.2
+70.2
+78.9
+OrientedMP (Luo et al. 2022)
+80.1
+81.7 79.0 85.0 78.1 89.7
+76.5
+91.6
+85.9 81.6
+82.1
+67.6
+95.0 79.6
+64.4
+76.9
+80.7
+Ours
+80.3
+84.5 82.7 83.9 76.6 90.2
+76.1
+91.6
+86.6 83.5
+84.6
+50.1
+94.4 81.9
+60.3
+75.3
+81.8
+Table 8: Segmentation results of class-wise and averaged mIoU on ShapeNetPart under z/SO(3), where ⋆ means our reproduced
+results of VN-DGCNN using the official code.
+Method
+mIoU
+air
+bag
+cap
+car chair
+ear
+guitar knife lamp laptop motor mug pistol rocket skate table
+plane
+phone
+bike
+board
+PointNet (Qi et al. 2017a)
+74.4
+81.6 68.7 74.0 70.3 87.6
+68.5
+88.9
+80.0 74.9
+83.6
+56.5
+77.6 75.2
+53.9
+69.4
+79.9
+PointNet++ (Qi et al. 2017b)
+76.7
+79.5 71.6 87.7 70.7 88.8
+64.9
+88.8
+78.1 79.2
+94.9
+54.3
+92.0 76.4
+50.3
+68.4
+81.0
+PCT (Wang et al. 2019)
+75.2
+80.1 69.0 82.5 66.8 88.4
+69.4
+90.4
+85.3 81.8
+79.6
+39.9
+89.2 76.5
+51.8
+72.6
+80.0
+RIConv (Zhang et al. 2019)
+75.5
+80.6 80.2 70.7 68.8 86.8
+70.4
+87.2
+84.3 78.0
+80.1
+57.3
+91.2 71.3
+52.1
+66.6
+78.5
+GCAConv (Zhang et al. 2020)
+77.3
+81.2 82.6 81.6 70.2 88.6
+70.6
+86.2
+86.6 81.6
+79.6
+58.9
+90.8 76.8
+53.2
+67.2
+81.6
+RI-Framework (Li et al. 2021b)
+79.4
+81.4 84.5 85.1 75.0 88.2
+72.4
+90.7
+84.4 80.3
+84.0
+68.8
+92.6 76.1
+52.1
+74.1
+80.0
+LGR-Net (Zhao et al. 2022)
+80.1
+81.7 78.1 82.5 75.1 87.6
+74.5
+89.4
+86.1 83.0
+86.4
+65.3
+92.6 75.2
+64.1
+79.8
+80.5
+VN-DGCNN⋆ (Deng et al. 2021)
+74.7
+80.0 79.4 79.1 71.5 89.2
+66.1
+89.0
+83.5 80.6
+82.0
+29.3
+91.4 73.4
+51.5
+67.8
+81.0
+OrientedMP (Luo et al. 2022)
+80.9
+81.8 78.8 85.4 78.0 89.6
+76.7
+91.6
+85.7 81.7
+82.1
+67.6
+95.0 79.1
+63.5
+76.5
+81.0
+Ours
+80.4
+84.3 82.2 84.6 77.9 89.9
+76.6
+91.3
+86.7 84.1
+84.3
+50.1
+93.4 79.0
+63.7
+75.3
+82.3
+Table 9: Segmentation results of class-wise and averaged mIoU on ShapeNetPart under SO(3)/SO(3).
+Qi, C. R.; Yi, L.; Su, H.; and Guibas, L. J. 2017b. Point-
+Net++: Deep hierarchical feature learning on point sets in a
+metric space. In NeurIPS.
+Van der Maaten, L.; and Hinton, G. 2008. Visualizing data
+using t-SNE. In JMLR.
+Wang, Y.; Sun, Y.; Liu, Z.; Sarma, S. E.; Bronstein, M. M.;
+and Solomon, J. M. 2019. Dynamic graph CNN for learning
+on point clouds. In ACM ToG.
+Zhang, Z.; Hua, B.-S.; Chen, W.; Tian, Y.; and Yeung, S.-K.
+2020. Global context aware convolutions for 3D point cloud
+understanding. In 3DV.
+Zhang, Z.; Hua, B.-S.; Rosen, D. W.; and Yeung, S.-K. 2019.
+Rotation invariant convolutions for 3D point clouds deep
+learning. In 3DV.
+Zhao, C.; Yang, J.; Xiong, X.; Zhu, A.; Cao, Z.; and Li, X.
+2022. Rotation invariant point cloud analysis: Where local
+geometry meets global topology. In Pattern Recognition.
+
+Input
+GT
+Ours
+ceiling
+floor
+wall
+beam
+column
+window
+door
+table
+chair
+sofa
+bookcase
+board
+clutter
+Figure 4: Visualization of semantic segmentation results on S3DIS area-5. The first row is the original inputs, the second row
+is the ground truth (GT) samples and the last row is our predicted results.
+
+GT
+Ours
+GT
+Ours
+GT
+Ours
+GT
+Ours
+Figure 5: Segmentation comparisons between the ground truth (GT) and our model on ShapeNetPart dataset under z/SO(3).
+
+1111

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+arXiv:2301.05381v1  [math.AT]  13 Jan 2023
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH
+Z2 COEFFICIENTS
+KATE POIRIER AND THOMAS TRADLER
+Abstract. Luc Menichi showed that the BV algebras on H●(LS2;Z2)[−2] coming from
+string topology and the one on HH●(H●(S2;Z2),H●(S2;Z2)) using Poincar´e duality on
+H●(S2;Z2) are not isomorphic.
+In this note we show how one can obtain the string
+topology BV algebra on Hochschild cohomology using a Poincar´e duality structure with
+higher homotopies.
+This Poincar´e duality (with higher homotopies) on cohomology is
+induced by a local Poincar´e duality (with higher homotopies) on the cochain level.
+1. Main Statements
+1.1. BV algebras for the 2-sphere with Z2 coefficients. In [M, Theorem 24] Luc
+Menichi calculated the string topology BV algebra (defined by Moira Chas and Dennis
+Sullivan in [CS]) for the 2-sphere M = S2 with Z2 = Z/2Z coefficients to be
+H●(LS2;Z2) ≅ Λa ⊗ Z2[u] ≅ ⊕
+k≥0
+Z2.αk ⊕ ⊕
+k≥0
+Z2.βk,
+(1.2)
+where αk ⋅ αℓ = αk+ℓ,βk ⋅ βℓ = 0, and αk ⋅ βℓ = βℓ ⋅ αk = βk+ℓ,
+and ∆ST(αk) = 0 and ∆ST(βk) = k ⋅ αk−1 + k ⋅ βk+1.
+Here, the degrees are given by ∣a∣ = −2, ∣u∣ = 1, and thus, setting αk = 1⊗uk and βk = a⊗uk,
+we have ∣αk∣ = k, ∣βk∣ = k − 2.
+Moreover, in [M, Proposition 20], Menichi calculated the BV algebra for Hochschild
+cohomology of the cohomology of S2 with Z2 coefficients to be
+HH●(H●(S2;Z2);H●(S2;Z2)) ≅ Λg ⊗ Z2[f] ≅ ⊕
+k≥0
+Z2.φk ⊕ ⊕
+k≥0
+Z2.ψk,
+(1.3)
+where φk ⋅ φℓ = φk+ℓ,ψk ⋅ ψℓ = 0, and φk ⋅ ψℓ = ψℓ ⋅ φk = ψk+ℓ,
+and ∆(φk) = 0 and ∆(ψk) = k ⋅ φk−1.
+There degrees are similarly given by ∣g∣ = −2, ∣f∣ = 1, and so, setting φk = 1 ⊗ f k and
+ψk = g ⊗ f k, we have ∣φk∣ = k, ∣ψk∣ = k − 2.
+Note, that these BV algebras differ in their ∆ operators, and, in fact, Menichi obtained
+the following result.
+Theorem 1.4. [M, Corollary 30] There is no isomorphism of BV algebras between
+(H●(LS2;Z2),⋅,∆ST) and (HH●(H●(S2;Z2);H●(S2;Z2)),⋅,∆).
+It is worth noting though, that the induced Gerstenhaber algebras on H●(LS2;Z2) and
+HH●(H●(S2;Z2);H●(S2;Z2)) are isomorphic; cf. [M, Corollary 23].
+2010 Mathematics Subject Classification. 55P50, 16E40 (primary), 57P10, 08A65 (secondary).
+Key words and phrases. String topology, Hochschild cohomology, BV algebra, Poincar´e duality.
+1
+
+2
+K. POIRIER AND T. TRADLER
+1.5. The BV algebra coming from local Poincar´e duality. A crucial ingredient for de-
+termining the ∆ operator on Hochschild cohomology (1.3) comes from a choice of Poincar´e
+duality structure given as a bimodule isomorphism F ∶ H●(S2;Z2)
+≅
+�→ H●(S2;Z2). For the
+(Z2-)cohomology H●(S2;Z2) and homology H●(S2;Z2), an obvious choice is to define the
+degree +2 isomorphism F as follows:
+degree −2
+degree −1
+degree 0
+degree 1
+degree 2
+H●(S2;Z2)
+= ...
+Z2
+0
+Z2
+0
+0
+...
+H●(S2;Z2)
+= ...
+0
+0
+Z2
+0
+Z2
+...
+F
+F
+F
+Indeed, this choice of F, which is the map given by capping with the Z2-fundamental class
+of S2, leads to the BV algebra in (1.3) as we will confirm in theorem 1.6 (together with
+5.3) below.
+Note that on the cochain level, capping with a fundamental cycle is not a bimodule
+map; see observation 4.4. Using a generalization of bimodule maps, which, in addition
+to F, allows for higher homotopies, we show in this paper that one can obtain the string
+topology BV algebra (1.2) on Hochschild cohomology. The precise definition of bimodule
+maps with higher homotopies was studied in [T1] and will be reviewed in section 2 below.
+Applying the concept of bimodule maps with higher homotopies, we will construct a spe-
+cific example of such a map for the case of a local cochain model of the 2-sphere C●(S2;Z2)
+in example 4.20. Then we pull this map back to obtain a bimodule map with homotopies
+̃F for cohomology H●(S2;Z2) (see proposition 4.23). The resulting map ̃F, recorded in
+example 4.2, has precisely one higher homotopy, when compared to F (see (4.3)). Now,
+using this ̃F, we obtain the following main theorem.
+Theorem 1.6. The BV algebra on HH●(H●(S2;Z2);H●(S2;Z2)) induced by ̃F (coming
+from a local bimodule map with higher homotopies transferred to cohomology) is isomorphic
+to the string topology BV algebra (1.2) on H●(LS2;Z2).
+Proof. We compute both BV algebras coming from F and ̃F, respectively.
+Denote by
+H● ∶= H●(S2;Z2) and H● ∶= H●(S2;Z2). Then the bimodule map F and the bimodule map
+̃F (with higher homotopies) induce respective (graded module) isomorphisms F and ̃
+F of
+Hochschild cohomologies (see 5.3 and 5.4)
+(1.7)
+F, ̃
+F ∶ HH●(H●,H●)
+≅
+�→ HH●(H●,H●).
+Next, we use the explicit description (as reviewed in 5.1 and 5.2) stating that
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.φk ⊕ ⊕
+k≥0
+Z2.ψk,
+where ∣φk∣ = k, ∣ψk∣ = k − 2,
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.θk ⊕ ⊕
+k��0
+Z2.χk,
+where ∣θk∣ = k + 2, ∣χk∣ = k.
+With this notation, F induces in (1.7) the map F(φk) = θk,F(ψk) = χk (see 5.3), while ̃F
+induces the map ̃
+F(φk) = θk + χk+2, ̃
+F(ψk) = χk (computed in 5.4).
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+3
+The BV ∆ operator on HH●(H●,H●) is given by transferring Connes’ B-operator from
+HH●(H●,H●) to HH●(H●,H●) via the given Poincar´e duality bimodule isomorphism,
+where, Connes’ B-operator is the map (computed in 5.2):
+B ∶ HH●(H●,H●) → HH●(H●,H●),
+B(θk) = 0,B(χk) = k ⋅ θk−1.
+Thus the induced operators ∆ = F −1○B○F and ̃∆ = ̃
+F −1○B○ ̃
+F from F and ̃F, respectively,
+are:
+∆(φk) = 0,
+∆(ψk) = F −1(k ⋅ θk−1) = k ⋅ φk−1,
+(1.8)
+̃∆(φk−2) = ̃∆(ψk) = ̃
+F −1(k ⋅ θk−1) = k ⋅ (φk−1 + ψk+1),
+(1.9)
+where we used that ̃
+F −1 is given by ̃
+F −1(θk) = φk + ψk+2 and ̃
+F −1(χk) = ψk.
+Note that the ∆ operator in (1.8) coincides with ∆ from (1.3). To see that the BV algebra
+with ̃∆ is isomorphic to the string topology BV algebra (1.2), we use an isomorphism which
+was essentially described in [M, last paragraph of Lemma 21]: the map Θ ∶ HH●(H●,H●) →
+H●(LS2;Z2) with Θ(φk) = αk +k ⋅βk+2 and Θ(ψk) = βk is an algebra isomorphism such that
+∆ST ○ Θ = Θ ○ ̃∆.
+□
+1.10. Organization of the paper. In section 2, we review basics about Hochschild
+cochain complexes and morphisms between them coming from bimodule maps up to higher
+homotopies. In section 3 we review the induced BV algebra coming from a chosen Poincar´e
+duality structure. In section 4 we give various computations of bimodule maps (with and
+without higher homotopies). In particular, the construction of a bimodule map (up to
+certain controlled structures) for the 2-simplex coming from locality (see example 4.9) con-
+stitutes the main computational aspect of this paper, with its proof being spelled out in
+appendix A. In section 5 we compute the BV algebras coming from the Poincar´e duality
+structures on cohomology of the 2-sphere from section 4, and with this complete the proof
+of theorem 1.6.
+Acknowledgments. We would like to thank Mahmoud Zeinalian for discussions about
+this topic. The second author was partially supported by a PSC-CUNY research award.
+2. Bimodule maps with higher homotopies
+In this section we review bimodule maps, bimodule maps up to higher homotopies,
+and homotopy inner products; see also [T1]. Moreover we review the induced maps on
+Hochschild cohomology.
+2.1. Basic setup. Denote by R a commutative ring with unit; our main example of interest
+is R = Z2. Let A be a unital dg-algebra over R, where we note that all the differentials in
+this paper will always go down, i.e., d ∶ Aj → Aj−1. We denote by A the space A shifted up
+by one, i.e., Aj ∶= Aj−1.
+Let M be a dg-bimodule over A. If we want to emphasize the corresponding algebra A,
+then we will also write M/A instead of M. From now on, all modules as well as module
+maps will be written in bold.
+Note that if M is a dg-bimodule over A, then so is its dual space M∗ with M∗
+j ∶= (M−j)∗ =
+HomR(M−j,R) with differential d ∶ M∗
+j → M∗
+j−1,d(n)(m) ∶= (−1)∣n∣+1n(d(m)), and module
+maps (n.a)(m) ∶= n(a.m) and (a.n)(m) ∶= (−1)∣a∣⋅(∣n∣+∣m∣)n(m.a) for n ∈ M∗,m ∈ M,a ∈ A,
+
+4
+K. POIRIER AND T. TRADLER
+where ∣.∣ denotes the degree. Moreover, the dg-algebra A is itself a dg-module A ∶= A over
+A (with module structure given by the algebra multiplication), and thus A∗ is a dg-module
+over A as well.
+Define the Hochschild cochain complex of A with values in M to be given by CH●(A,M) ∶=
+∏r≥0 Hom(A⊗r,M), where the differential D = D0 + D1, with D2 = 0, is defined for ϕ ∈
+Hom(A⊗r,M) by setting
+D0(ϕ)(a1,... ,ar) ∶= d(ϕ(a1,... ,ar)) +
+r
+∑
+j=1
+(−1)∣ϕ∣+∣a1∣+⋅⋅⋅+∣aj−1∣ ⋅ ϕ(a1,... ,daj,... ,ar),
+D1(ϕ)(a1,... ,ar+1) ∶= (−1)∣ϕ∣⋅∣a1∣ ⋅ a1.ϕ(a2,... ,ar+1)
++
+r
+∑
+j=1
+(−1)∣ϕ∣+∣a1∣+⋅⋅⋅+∣aj∣ ⋅ ϕ(a1,... ,aj ⋅ aj+1,... ,ar+1) + (−1)∣ϕ∣+1+∣a1∣+⋅⋅⋅+∣ar∣ ⋅ ϕ(a1,... ,ar).ar+1.
+We will mainly use the normalized Hochschild cochain complex CH
+●(A,M), which is the
+subcomplex of CH●(A,M) consisting of those ϕ ∈ CH●(A,M) which vanish when any of the
+inputs is the unit 1 ∈ A. The inclusion CH
+●(A,M) ↪ CH●(A,M) is a quasi-isomorphism;
+see [L, 1.5.7].
+2.2. Inner products and homotopy inner products. Let M and N be dg-bimodules
+over A, and let F ∶ M → N be a dg-bimodule map. Then, there is an induced cochain map
+CH(F) ∶ CH
+●(A,M) → CH
+●(A,N), ϕ ↦ F ○ ϕ. An inner product on M is a dg-bimodule
+map F ∶ M → M∗.
+We will need a more general notion of inner product F ∶ M → M∗ which also allows
+for higher homotopies. To this end, consider a sequence of maps F = {Fp,q ∶ A⊗p ⊗ M ⊗
+A⊗q → M∗}p,q≥0, and define its differential by setting DF = {(DF)p,q}p,q≥0 to be (DF)p,q =
+(D0F)p,q + (D1F)p,q, where ∀a1,... ,ap,b1,... ,bq,∈ A and m,n ∈ M:
+(D0F)p,q(a1,... ,ap;m;b1,... ,bq)(n)
+(2.3)
+∶=
+p
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣aj−1∣ ⋅ Fp,q(a1,... ,daj,... ,ap;m;b1,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+1 ⋅ Fp,q(a1,... ,ap;dm;b1,... ,bq)(n)
++
+q
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bj−1∣ ⋅ Fp,q(a1,... ,ap;m;b1,... ,dbj,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq∣+1 ⋅ Fp,q(a1,... ,ap;m;b1,... ,bq)(dn)
+and, using the notation F−1,q = Fp,−1 = 0:
+(D1F)p,q(a1,... ,ap;m;b1,... ,bq)(n)
+(2.4)
+∶= (−1)∣F∣+∣a1∣⋅(∣a2∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq∣+∣n∣) ⋅ Fp−1,q(a2,... ,ap;m;b1,... ,bq)(n.a1)
++
+p−1
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣aj∣ ⋅ Fp−1,q(a1,... ,aj ⋅ aj+1,... ,ap;m;b1,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap−1∣+1 ⋅ Fp−1,q(a1,... ,ap−1;ap.m;b1,... ,bq)(n)
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+5
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣ ⋅ Fp,q−1(a1,... ,ap;m.b1;b2,... ,bq)(n)
++
+q−1
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bj∣ ⋅ Fp,q−1(a1,... ,ap;m;b1,... ,bj ⋅ bj+1,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq−1∣+1 ⋅ Fp,q−1(a1,... ,ap;m;b1,... ,bq−1)(bq.n)
+Then we call F a homotopy inner product for M if DF = 0. By slight abuse of notation,
+we will still use the notation F ∶ M → M∗ for F = {Fp,q}p,q with all its homotopies.
+We will sometimes depict evaluations of F as follows:
+(2.5)
+Fp,q(a1,... ,ap;m;b1,... ,bq)(n)
+a1
+a2
+...
+ap
+m
+b1
+b2
+...
+bq
+n
+Note, that a homotopy inner product F = {Fp,q}p,q with Fp,q = 0 for p + q > 0 is precisely
+an inner product, i.e., a dg-bimodule map F0,0 ∶ M → M∗.
+Let F be a homotopy inner product for M. We will always assume that F vanishes
+when any of the algebra inputs is the unit 1 ∈ A. Then, there is an induced map F ∶=
+CH(F) ∶ CH
+●(A,M) → CH
+●(A,M∗) by setting F = CH(F) = ∑p,q≥0 CH(F)p,q, where
+CH(F)p,q ∶ Hom(A⊗r,M) → Hom(A⊗p+r+q,M∗) is
+(2.6)
+(CH(F)p,q(ϕ))(a1,... ,ap+r+q)
+∶= (−1)∣ϕ∣⋅(∣a1∣+⋅⋅⋅+∣ap∣) ⋅ Fp,q(a1,... ,ap;ϕ(ap+1,... ,ap+r);ap+r+1,... ,ap+r+q).
+a1
+a2
+...
+ap
+ap+r+1
+...
+ap+r+q
+ϕ
+ap+1
+ap+r
+...
+A direct but lengthy calculation shows that
+(2.7)
+CH(DF)(ϕ) = D ○ CH(F)(ϕ) − (−1)∣F∣ ⋅ CH(F) ○ D(ϕ),
+∀ϕ ∈ CH●(A,M).
+Corollary 2.8. If F is a homotopy inner product, i.e., DF = 0, then F = CH(F) ∶
+CH
+●(A,M) → CH
+●(A,M∗) is a cochain map. By abuse of notation, we often write F
+for the induced map on Hochschild cohomology. If F = DF′ for some F′, then the induced
+map on Hochschild cohomology vanishes, i.e., HH(F) = 0.
+2.9. Pullback under a dg-morphism. Let A and B be two unital dg-algebras, and
+let f ∶ B → A be a dg-algebra map (which is necessarily of degree 0).
+Then any dg-
+bimodule M = M/A over A induces a dg-bimodule M/B ∶= M over B via the module structure
+b.m ∶= f(b).m and m.b ∶= m.f(b) for any m ∈ M,b ∈ B. Moreover, any dg-bimodule map
+F/A ∶ M/A → N/A also induces a dg-bimodule map F/B ∶ M/B → N/B. Moreover, a homotopy
+inner product F/A for M/A also induces a homotopy inner product for M/B via
+(F/B)p,q(b1,... ,bp;m;c1,... ,cq)(n) ∶= (F/A)p,q(f(b1),... ,f(bp);m;f(c1),... ,f(cq))(n)
+
+6
+K. POIRIER AND T. TRADLER
+for all b1,... ,bp,c1,... ,cq,∈ B and m,n ∈ M/B.
+Moreover, for a dg-algebra map f ∶ B → A, we have the dg-modules B/B and B∗
+/B, and,
+from A/A and A∗
+/A, we also get A/B and A∗
+/B. The dg-algebra map f then induces a dg-
+bimodule map f/B ∶ B/B → A/B and, by dualizing, the dg-bimodule map f∗
+/B ∶ A∗
+/B → B∗
+/B.
+Combining the last two paragraphs, assume that f ∶ B → A is a dg-algebra map, and that
+F/A ∶ A/A → A∗
+/A is a homotopy inner product for A. Then, we get a transferred homotopy
+inner product f(F)/B ∶ B/B → B∗
+/B for B, given by f(F)/B ∶= f∗
+/B ○ F/B ○ f/B,
+B/B
+f/B
+�
+f(F)/B
+�
+A/B
+F/B
+�
+B∗
+/B
+A∗
+/B
+f∗
+/B
+�
+(2.10)
+(f(F)/B)p,q(b1,... ,bp;m;c1,... ,cq)(n)
+= (F/A)p,q(f(b1),... ,f(bp);f(m);f(c1),... ,f(cq))(f(n)),
+where b1,... ,bp,m,c1,... ,cq,n ∈ B.
+Finally, for a dg-algebra map f ∶ B → A, and a dg-bimodule M/A, there is an induced
+cochain map on Hochschild cochains, CH(f,M) ∶ CH
+●(A,M/A) → CH
+●(B,M/B),
+(2.11)
+CH(f;M)(ϕ)(b1,... ,br) ∶= ϕ(f(b1),... ,f(br)),
+∀ϕ ∈ CH
+●(A,M/A),b1,... ,br ∈ B.
+With this we get the following lemma.
+Lemma 2.12. If f ∶ B → A is a dg-algebra map, and F/A ∶ A/A → A∗
+/A is a homotopy inner
+product for A/A, then the following diagram commutes.
+(2.13)
+CH
+●(B,B/B)
+CH(f/B)
+�
+CH(f(F)/B)
+�
+CH
+●(B,A/B)
+CH(F/B)
+�
+CH
+●(A,A/A)
+CH(f,A)
+�
+CH(F/A)
+�
+CH
+●(B,B∗
+/B)
+CH
+●(B,A∗
+/B)
+CH(f∗
+/B)
+�
+CH
+●(A,A∗
+/A)
+CH(f,A∗)
+�
+2.14. Isomorphism on Hochschild cohomology. A particular case of interest is when
+a homotopy inner product F ∶ A → A∗ induces an isomorphism on Hochschild cohomology
+F ∶= HH(F) ∶ HH●(A,A) → HH●(A,A∗). In this case, we can transfer any structure
+between these Hochschild cohomologies, in particular we can transfer the B operator as it
+was done in theorem 1.6.
+Note that in equation (2.13), if the four horizontal maps are quasi-isomorphisms, then,
+obviously, the right vertical map CH(F/A) is a quasi-isomorphism iff the left vertical map
+CH(f(F)/B) is a quasi-isomorphism.
+3. BV algebra on Hochschild cohomology
+We now review how homotopy inner products induce a BV algebra on Hochschild co-
+homology; see theorem 3.7. Our main reference for this is [T2]. We start by defining two
+operators B and ∆F.
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+7
+Definition 3.1. Consider a unital dg-algebra A.
+(1) We define Connes’ B-operator (or more precisely the dual of Connes’ B-operator)
+to be B ∶ CH
+●(A,A∗) → CH
+●(A,A∗) given for ϕ ∈ Hom(A⊗r,A∗) by B(ϕ) ∈
+Hom(A⊗r−1,A∗) with
+(3.2)
+(B(ϕ))(a1,... ,ar−1)(ar)
+∶=
+r
+∑
+j=1
+(−1)(∣aj∣+⋅⋅⋅+∣ar∣)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+∣ar∣ ⋅ ϕ(aj,... ,ar,a1,... ,aj−1)(1),
+A direct but lengthy computation shows that D○B(ϕ) = −B○D(ϕ). By abuse of no-
+tation, we denote the induced map on Hochschild cohomology by B ∶ HH●(A,A∗) →
+HH●(A,A∗) as well.
+(2) Next, assume that we also have a homotopy inner product F = {Fp,q ∶ A⊗p ⊗ A ⊗
+A⊗q → A∗}p,q≥0.
+Then, we define the operator ZF ∶=
+∑
+p,q≥0ZF
+p,q ∶ CH
+●(A,A) →
+CH
+●(A,A∗), where, for ϕ ∈ Hom(A⊗r,A), we set ZF
+p,q(ϕ) ∈ Hom(A⊗p+r+q−1,A∗) to
+be given by
+(ZF
+p,q(ϕ))(a1,... ,ap+r+q−1)(ap+r+q)
+(3.3)
+∶=
+p+q
+∑
+j=p+1
+(−1)∣F∣+(∣ϕ∣+1)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+1
+⋅ Fp,q(a1,... ,ap;1;ap+1,... ,aj−1,ϕ(aj,... ,aj+r−1),aj+r,... ,ap+r+q−1)(ap+r+q)
++
+r
+∑
+j=1
+(−1)(∣aj∣+⋅⋅⋅+∣ap+r+q∣)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+∣ap+r+q∣+∣ϕ∣⋅(∣aj∣+⋅⋅⋅+∣aj+p+q−1∣)
+⋅ Fp,q(aj,... ,aj+p−1;1;aj+p,... ,aj+p+q−1)(ϕ(ap+j+q,... ,ap+r+q,a1,... ,aj−1))
++
+p
+∑
+j=1
+(−1)∣F∣+(∣ϕ∣+1)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+1
+⋅ Fp,q(a1,... ,aj−1,ϕ(aj,... ,aj+r−1),aj+r,... ,ap+r−1;1;ap+r,... ,ap+r+q−1)(ap+r+q)
+One can check again that D ○ ZF(ϕ) = −(−1)∣F∣ ⋅ ZF ○ D(ϕ). By abuse of notation,
+we denote the induced map on Hochschild cohomology by ZF ∶ HH●(A,A) →
+HH●(A,A∗) as well.
+We remark that ZF has appeared in [T2, lemma 17] as the operation associated
+to the symbol
+(−1)µ ⋅ 1
+1
+1
++1
++1
+1
+(3) Let F be a homotopy inner product for A. Assume, moreover, that F ∶ A → A∗
+induces an isomorphism on Hochschild cohomology, F ∶= HH(F) ∶ HH●(A,A) →
+
+8
+K. POIRIER AND T. TRADLER
+HH●(A,A∗). Then denote by ∆F ∶ HH●(A,A) → HH●(A,A) the composition
+∆F ∶= F −1 ○ ZF,
+HH●(A,A)
+ZF
+�→ HH●(A,A∗)
+F−1
+�→ HH●(A,A)
+Lemma 3.4. Let A be a unital dg-algebra.
+(1) On Hochschild cochains HH●(A,A∗), we have that B2 = 0.
+(2) Assume that F ∶ A → A∗ is a homotopy inner product which induces an isomorphism
+on Hochschild cohomology F ∶ HH●(A,A) → HH●(A,A∗). Then the deviation of
+∆F from being a derivation of the cup product is the usual Gerstenhaber bracket.
+Here the cup product on Hochschild cohomology is given for Hochschild cochains
+ϕ ∈ Hom(A⊗r,A) and ρ ∈ Hom(A⊗s,A) to be ϕ ⌣ ρ ∈ Hom(A⊗r+s,A) with
+(3.5)
+(ϕ ⌣ ρ)(a1,... ,ar+s) ∶= (−1)∣ρ∣⋅(∣a1∣+⋅⋅⋅+∣ar∣) ⋅ ϕ(a1,... ,ar) ⋅ ρ(ar+1,... ,ar+s).
+Proof. For (1), note that B2 = 0 follows since normalized Hochschild cochains vanish when
+any input is the unit 1. Part (2) was proved in [T2, section 3.3].
+□
+Since lemma 3.4 (1) and (2) are conditions needed for a BV algebra, i.e., a square zero
+operator (here: B) whose deviation from being a derivation is a Gerstenhaber bracket (here:
+∆F), we make the following definition.
+Definition 3.6. Let A be a dg-algebra, and let F ∶ A → A∗ be a homotopy inner product
+for A. We call F a Poincar´e duality structure for A, if it satisfies if it satisfies the following
+two conditions:
+(1) F induces an isomorphism of graded modules on Hochschild cohomology F =
+HH(F) ∶ HH●(A,A)
+≅
+�→ HH●(A,A∗).
+(2) Transferring B from HH●(A,A∗) to HH●(A,A) via F equals ∆F, i.e.,
+HH●(A,A)
+F
+�
+HH●(A,A∗)
+F−1
+�
+B
+�
+F −1 ○ B ○ F = ∆F = F −1 ○ ZF
+HH●(A,A)
+∆F
+�
+=
+HH●(A,A)
+ZF
+�
+HH●(A,A∗)
+F−1
+�
+With this, we obtain the following theorem, which is [T2, theorem 2].
+Theorem 3.7. Let A be a dg-algebra, and let F ∶ A → A∗ be a Poincar´e duality structure
+for A. Then, HH●(A,A) together with the cup product and ∆F is a BV algebra.
+Examples 3.8. We now give a few examples for how one can check whether a homotopy
+inner product is a Poincar´e duality structure.
+(1) Let F be an inner product for a dg-algebra A (i.e., the only non-zero component is
+F0,0), and assume that F induces an isomorphism on Hochschild cohomology. Then
+F is a Poincar´e duality structure.
+(To see this, we note that this is a special case of the next example (2) below,
+since the condition (D1F)0,1 = 0 gives (using (2.4)) that F(mb)(n) = F(m)(bn),
+and (D1F)1,0 = 0 gives F(am)(n) = (−1)∣a∣⋅(∣m∣+∣n∣)F(m)(na) for all m,n,a,b ∈
+A, and, thus F(m)(n) = F(1 ⋅ m)(n) = F(1)(m ⋅ n) = (−1)∣m∣⋅∣n∣F(n ⋅ 1)(m) =
+(−1)∣m∣⋅∣n∣F(n)(m).)
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+9
+(2) Let F be a homotopy inner product for a dg-algebra A, and assume that F in-
+duces an isomorphism on Hochschild cohomology. Assume further that F is in-
+variant under cyclic rotation of the first p + 1 and last q + 1 inputs, i.e., ∀p,q ≥ 0,
+∀a1,... ,ap,m,b1,... ,bq,n ∈ A:
+Fp,q(a1,... ,ap;m;b1,... ,bq)(n) = (−1)ε ⋅ Fq,p(b1,... ,bq;n;a1,... ,ap)(m)
+a1
+a2
+...
+ap
+m
+b1
+b2
+...
+bq
+n = (−1)ε ⋅
+b1
+b2
+...
+bq
+n
+a1
+a2
+...
+ap
+m
+where (−1)ε = (−1)(∣a1∣+⋅⋅⋅+∣ap∣+∣m∣)⋅(∣b1∣+⋅⋅⋅+∣bq∣+∣n∣). Then F is a Poincar´e duality struc-
+ture. This was proved in [T2, lemma 17].
+(3) In the examples in section 5 of this paper we will check if a given homotopy inner
+product F is a Poincar´e duality structure or not by explicitly computing F, B and
+ZF; cf. the examples given in 5.3, 5.4 and 5.5.
+In particular, it is worth noting that there exist homotopy inner products F for
+which F is an isomorphism, but for which F −1 ○ B ○ F ≠ ∆F. We provide an explicit
+example for such a homotopy inner product in 5.5.
+4. Computations of bimodule maps with higher homotopies
+In this section, we will give explicit bimodule maps and bimodule maps with higher
+homotopies for specific dg-algebras. In particular we will compute the homotopy inner
+product on H●(S2;Z2) coming from a pullback of a local homotopy inner product. In the
+remainder of this paper –with the exception of observation 4.4– the ground ring will always
+be R = Z2.
+Example 4.1 (H●(S2;Z2) inner product without homotopies). Consider the dg-algebra
+A ∶= H●(S2;Z2) ≅ Z2.e ⊕ Z2.s with zero differential and degrees ∣e∣ = 0 and ∣s∣ = −2, and
+where e is the unit and s ⋅ s = 0. For the dg module and dual dg module of A we use
+the notation A ≅ Z2.e ⊕ Z2.s and A∗ ≅ Z2.e∗ ⊕ Z2.s∗, respectively, where e∗ and s∗ are
+the duals of e and s with ∣e∗∣ = 0 and ∣s∗∣ = 2 and the module structure is given by
+e.e∗ = e∗.e = e∗,e.s∗ = s∗.e = s∗,s.e∗ = e∗.s = 0,s.s∗ = s∗.s = e∗ (see 2.1).
+Define the dg bimodule map F ∶ A → A∗ by F(s) = e∗ and F(e) = s∗,
+A
+= ...
+⟨s⟩
+0
+⟨e⟩
+0
+0
+...
+A∗
+= ...
+0
+0
+⟨e∗⟩
+0
+⟨s∗⟩
+...
+F
+F
+F
+Thus, F ∶ A → A∗ is the map F(s)(e) = 1,F(s)(s) = 0,F(e)(e) = 0,F(e)(s) = 1 (in other
+words, F is given by capping with s∗). One can check directly that F is a dg bimodule
+map. Note, that when interpreting F as an inner product <,>= F ∶ A ⊗ A → Z2, the only
+non-vanishing inner products are < s,e >= 1 and < e,s >= 1.
+
+10
+K. POIRIER AND T. TRADLER
+Example 4.2 (H●(S2;Z2) inner product with homotopies). We next consider the same
+algebra A and module structures A and A∗ as in example 4.1, but we define an inner
+product ̃F for A with higher homotopies (in the sense of 2.2). In fact, ̃F has its only
+non-zero components given by ̃F0,0 ∶ A → A∗ and ̃F2,0 ∶ A ⊗ A ⊗ A → A∗ (see 2.2) via
+̃F0,0(s)(e) = 1,
+̃F0,0(e)(s) = 1,
+̃F2,0(s,s;e)(e) = 1.
+(4.3)
+s
+e
+e
+s
+s
+s
+e
+e
+It is again a direct check that ̃F is a homotopy inner product, i.e., it satisfies all equation
+required by 0 = DF = D1F given by (2.4) (since d and thus D0 vanishes). Explicitly, these
+equations are:
+̃F0,0(a ⋅ a1)(̃a) =̃F0,0(a)(a1 ⋅ ̃a),
+̃F0,0(a1 ⋅ a)(̃a) =̃F0,0(a)(̃a ⋅ a1),
+̃F2,0(a1,a2;a ⋅ a3)(̃a) =̃F2,0(a1,a2;a)(a3 ⋅ ̃a),
+0 =̃F2,0(a1,a2;a3 ⋅ a)(̃a) + ̃F2,0(a1,a2 ⋅ a3;a)(̃a)
++ ̃F2,0(a1 ⋅ a2,a3;a)(̃a) + ̃F2,0(a2,a3;a)(̃a ⋅ a1)
+for all a,̃a ∈ A, and a1,a2,a3 ∈ A.
+We next give formulas for calculating homotopy inner products (over Z2) for (triangu-
+lated) 2-dimensional spaces on the cochain level.
+The following observation notes that
+higher homotopies naturally appear for inner products on the cochain level.
+Observation 4.4. Let A be a unital dg-algebra over any commutative ring R, so that
+both A ∶= A and A∗ are dg-modules over A, (see 2.1). Now, let x ∈ A∗ be a fixed closed
+element, d(x) = 0. Define F ∶ A → A∗ for a ∈ A by setting F(a) ∈ A∗ evaluated on some
+̃a ∈ A to be F(a)(̃a) ∶= (x ⌢ a)(̃a) = x(a ⋅ ̃a).
+Claim: F is chain map, and a graded right module map. F is in general not a graded
+left module map. A sufficient condition for F being a graded left module map is that A is
+graded commutative.
+Proof. First, F is chain map, since for a,̃a ∈ A and dx = 0, we have F(da)(̃a) = x(da ⋅ ̃a) =
+x(d(a ⋅ ̃a) − (−1)∣a∣a ⋅ d̃a) = (−1)∣x∣+1dx(a ⋅ ̃a) − (−1)∣a∣ ⋅ F(a)(d̃a) = −(−1)∣a∣ ⋅ (−1)∣F(a)∣+1 ⋅
+d(F(a))(̃a) = (−1)∣F∣⋅d(F(a))(̃a). Next, F is a graded right module map, since for a,̃a ∈ A,
+a1 ∈ A, we have (F(a).a1)(̃a) = F(a)(a1 ⋅ ̃a) = x(a ⋅ a1 ⋅ ̃a) = F(a.a1)(̃a). To check when
+F is a graded left module map, compute F(a1.a)(̃a) = x(a1 ⋅ a ⋅ ̃a) and (a1.F(a))(̃a) =
+(−1)∣a1∣⋅(∣F(a)∣+∣̃a∣) ⋅ F(a)(̃a ⋅ a1) = (−1)∣a1∣⋅(∣F∣+∣a∣+∣̃a∣) ⋅ x(a ⋅ ̃a ⋅ a1). Thus, F is in general not a
+graded left module map. Moreover, if a1 ⋅ a ⋅ ̃a = (−1)∣a1∣⋅(∣a∣+∣̃a∣)a ⋅ ̃a ⋅ a1 for all a,̃a,a1, then
+(a1.F(a))(̃a) = (−1)∣a1∣⋅∣F∣ ⋅ F(a1.a)(̃a), and thus F will be a graded left module map.
+□
+In particular, assume that X is a closed, oriented manifold, and R is a commutative
+ring. In order to calculate the BV algebra on HH●(C●(X;R);(C●(X;R))∗) one might try
+to use capping a cochain with a fundamental cycle x of X as the appropriate dg-bimodule
+map F = x ⌢ − ∶ C●(X;R) → C●(X;R)
+incl
+↪ (C●(X;R))∗. However, the above observation
+shows that capping on the (co-)chain level is in general not a dg-bimodule map, and thus
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+11
+does not induce a chain map on the Hochschild cochains. One way to resolve this issue
+is to provide higher homotopies for the left module structure, i.e., to provide a homotopy
+inner product, which then does give a corresponding chain map on Hochschild cochains.
+This is what is done in this paper for S2 with Z2 coefficients.
+Example 4.5 (The 0-simplex). Let A[0] ∶= Z2.e0, where e0 is the unit. A homotopy inner
+product F[0] ∶ A[0] → (A[0])∗ is given by F[0](e0)(e0) = 1.
+We will also vary the superscript in an obvious way, i.e., the dg-algebra A[1] ∶= Z2.e1 has
+the homotopy inner product F[1] ∶ A[1] → (A[1])∗,F[1](e1)(e1) = 1, etc.
+Example 4.6 (The 1-simplex). Let A[01] ∶= Z2.e0 ⊕ Z2.e1 ⊕ Z2.b01, where ∣e0∣ = ∣e1∣ = 0 and
+∣b01∣ = −1 with differential d(e0) = d(e1) = b01. The product is the usual cup product of
+simplicial cochains, i.e., e0 ⋅ e0 = e0, e1 ⋅ e1 = e1, and e0 ⋅ b01 = b01 ⋅ e1 = b01. Note that the unit
+of A[01] is e0 + e1.
+We want to define an inner product F with lowest component non-vanishing only for
+F(e0)(b01) = F(b01)(e1) = 1. Note that F is not a chain map, but F(d(a))(̃a)+F(a)(d(̃a)) =
+F[0](a)(̃a) + F[1](a)(̃a) for all a,̃a ∈ A[01]. Moreover, F is a graded right module map, i.e.,
+F(a ⋅ a1)(̃a) = F(a)(a1 ⋅ ̃a) for all a,̃a,a1 ∈ A[01], but F is not a graded left module map,
+since F(e0 ⋅ e0)(b01) = 1 ≠ 0 = F(e0)(b01 ⋅ e0).
+There is an inductive procedure (involving choices at each stage) for obtaining higher
+homotopies that provide F with left modules homotopies, making it into a homotopy inner
+product up to F[0] and F[1] from example 4.5 (interpreted as maps A[01] → (A[01])∗) . This
+procedure was described in [TZS, Proposition 3.1.2]. Performing the induction leads to a
+sequence of maps, which was first stated in [TZS, Proposition B.2]:
+∀k ≥ 0 ∶
+F[01]
+k,0 (b01,... ,b01;e0)(b01) = 1,
+F[01]
+k,0 (b01,... ,b01;b01)(e1) = 1,
+(4.7)
+b01
+b01
+...
+b01
+e0
+b01
+b01
+b01
+...
+b01
+b01
+e1
+All other inner products are zero. Note, this resolves the above problem of F[01]
+0,0 not being a
+graded left module map, since, now, F[01]
+0,0 is a graded left module map up to the homotopy
+F[01]
+1,0 ; for example:
+(DF[01])1,0(e0;e0)(b01) = F[01]
+1,0 (d(e0);e0)(b01) + F[01]
+1,0 (e0;d(e0))(b01)
++ F[01]
+0,0 (e0 ⋅ e0)(b01) + F[01]
+0,0 (e0)(b01 ⋅ e0) = 1 + 0 + 1 + 0 = 0.
+From [TZS, Proposition B.2], which we will also prove in appendix A, we have that:
+(4.8)
+DF[01] = F[0] + F[1]
+Again, we will need to vary the superscript in an obvious way, i.e., for the dg-algebra
+A[12] there are maps F[12] which are given by replacing 0 and 1 in the above with 1 and 2,
+respectively, etc.
+Example 4.9 (The 2-simplex). We now describe inner product maps for the 2-simplex,
+which extends the previous two examples of the 0- and 1-simplex. Let A[012] ∶= Z2.e0 ⊕
+Z2.e1 ⊕ Z2.e2 ⊕ Z2.b01 ⊕ Z2.b02 ⊕ Z2.b12 ⊕ Z2.c012 with ∣ej∣ = 0, ∣bij∣ = −1 and ∣c012∣ = −2, and
+
+12
+K. POIRIER AND T. TRADLER
+differential d(e0) = b01 + b02, d(e1) = b01 + b12, d(e2) = b02 + b12 and d(bij) = c012 for all
+0 ≤ i < j ≤ 2. The multiplication is non-zero only for
+∀j ∶
+ej ⋅ ej = ej,
+∀i < j ∶
+ei ⋅ bij = bij,
+bij ⋅ ej = bij,
+(4.10)
+e0 ⋅ c012 = c012,
+c012 ⋅ e2 = c012,
+b01 ⋅ b12 = c012.
+Now, following the procedure (which uses locality) from [TZS, Proposition 3.1.2], we
+define maps F[012]
+k,0
+whose only non-zero maps are given by the following equations (4.11)-
+(4.18):
+∀k ≥ 0 ∶
+F[012]
+k,0 (b02,... ,b02;c012)(e2) = 1,
+(4.11)
+F[012]
+k,0 (b02,... ,b02;e0)(c012) = 1,
+(4.12)
+F[012]
+k,0 (b02,... ,b02;b01)(b12) = 1,
+(4.13)
+b02
+b02
+...
+b02
+e2
+c012
+b02
+b02
+...
+b02
+c012
+e0
+b02
+b02
+...
+b02
+b12
+b01
+∀k ≥ 0 ∶ ∀1 ≤ ℓ ≤ k + 1 ∶
+F[012]
+k+1,0(b01,... ,b01, c012
+�
+ℓth
+,b02,... ,b02;e0)(b01) = 1,
+(4.14)
+F[012]
+k+1,0(b01,... ,b01, c012
+�
+ℓth
+,b02,... ,b02;b01)(e1) = 1,
+(4.15)
+F[012]
+k+1,0(b02,... ,b02, c012
+�
+ℓth
+,b12,... ,b12;e1)(b12) = 1,
+(4.16)
+F[012]
+k+1,0(b02,... ,b02, c012
+�
+ℓth
+,b12,... ,b12;b12)(e2) = 1,
+(4.17)
+b01
+...
+b01
+c012
+b02...b02
+b01
+e0
+b01
+...
+b01
+c012
+b02...b02
+e1
+b01
+b02
+...
+b02
+c012
+b12...b12
+b12
+e1
+b02
+...
+b02
+c012
+b12...b12
+e2
+b12
+(4.18)
+∀k ≥ 0 ∶ ∀1 ≤ ℓ1 < ℓ2 ≤ k + 2 ∶
+F[012]
+k+2,0(b01,... ,b01, c012
+�
+ℓ1th
+,b02,... ,b02, c012
+�
+ℓ2th
+,b12,... ,b12;e1)(e1) = 1.
+b01...
+b01
+c012
+b02
+...
+b02
+c012
+b12...b12
+e1
+e1
+We claim that the following equation holds, which will be proved in appendix A:
+(4.19)
+DF[012] = F[01] + F[02] + F[12]
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+13
+As before, the above will also be applied to obvious variations of the superscript, such
+as, e.g, A[123] with maps F[123] given by replacing 0, 1 and 2 in the above with 1, 2 and 3,
+respectively, etc.
+Example 4.20 (C●(S2;Z2) inner product with homotopies). We now use a tetrahedral
+triangulation of the 2-sphere. More precisely, we set
+A = ⊕
+0≤j≤3
+Z2.ej ⊕
+⊕
+0≤i<j≤3
+Z2.bij ⊕
+⊕
+0≤i<j<l≤3
+Z2.cijl
+e0
+e1
+e2
+e3
+with ∣ej∣ = 0, ∣bij∣ = −1 and ∣cijl∣ = −2 and differential d(ej) = ∑i≠j b{ij}, d(bij) = ∑l≠i,j c{ijl},
+and d(cijl) = 0, where the bracket “{}” denotes indices in ascending order, i.e., b{ij} =
+{bij, for i < j
+bji, for j < i and similarly for c{ijl}. The multiplication is similar to (4.10), where the
+indices “012” are replaced by any 0 ≤ i < j < l ≤ 3:
+∀j ∶
+ej ⋅ ej = ej,
+∀i < j ∶
+ei ⋅ bij = bij,
+bij ⋅ ej = bij,
+(4.21)
+∀i < j < l ∶
+ei ⋅ cijl = cijl,
+cijl ⋅ el = cijl,
+bij ⋅ bjl = cijl,
+and all other multiplications vanish. Note in particular that A has the unit e0 +e1 +e2 +e3.
+We define the homotopy inner product F for A ∶= A to be given by (4.11)-(4.18) with
+“012” in (4.11)-(4.18) replaced by any 0 ≤ i < j < l ≤ 3, i.e.,
+(4.22)
+F ∶= F[012] + F[013] + F[023] + F[123].
+Claim: F is a homotopy inner product, i.e., DF = 0.
+Proof. We compute DF = DF[012] + DF[013] + DF[023] + DF[123]. We claim that we can use
+equation (4.19) to evaluate this expression, which is not completely obvious since D in
+(4.19) is for the dg-algebra A[012], whereas we need to apply D for A.
+To see that (4.19) holds for the dg-algebra A, note that, for example, DF[012](a1,... )(ar)
+applies a differential or a multiplication to the inputs aj ∈ A (as stated in (2.3) and (2.4)).
+Now, if any of the inputs aj are generators of A which has a 3 in the index (such as, e.g.,
+b13 ∈ A), then taking the differential or a multiplication will consist of sums of generators
+with 3 in their indices, and thus DF[012] applied to it would vanish. On the other hand, if
+all inputs aj applied to DF[012] are only given by generators indexed by 0, 1, and 2, then
+DF[012] applied to it coincides with what we would get for A[012], because, the multiplication
+in A equals the one in A[012], and the differential in A equals the differential in A[012] up
+to generators with indices of 3. (For example, applying d(b12) = c012 + c123 in DF[012] will
+vanish on c123 and will coincide with (4.19) on c012 = dA[012](b12) =(differential of b12 in
+A[012]).)
+Thus, DF = DF[012] + DF[013] + DF[023] + DF[123] = (F[01] + F[02] + F[12]) + (F[01] + F[03] +
+F[13]) + (F[02] + F[03] + F[23]) + (F[12] + F[13] + F[23]) = 0.
+□
+
+14
+K. POIRIER AND T. TRADLER
+Proposition 4.23. Let A be the dg-algebra from example 4.20, and let B = H●(S2;Z2) ≅
+Z2.e ⊕ Z2.s be the dg-algebra from examples 4.1 and 4.2.
+Then, the map f ∶ B → A,
+f(e) ∶= e0 + e1 + e2 + e3 and f(s) ∶= c012, is a dg-algebra quasi-isomorphism.
+Transferring the homotopy inner product F/A defined in (4.22) via the map f gives
+f(F)/B = ̃F/B, where ̃F/B is the homotopy inner product defined in equation (4.3) from
+example 4.2.
+Proof. The transfered map of F/A from (4.22) is given by applying f to inputs from B and
+then applying F/A; see (2.10). Since the image of f is spanned by c012 and e0 + e1 + e2 + e3,
+it follows from (4.11)-(4.18) that the only non-zero components (after applying f) are:
+F[012]
+0,0 (c012)(e2) = 1,
+F[012]
+0,0 (e0)(c012) = 1,
+F[012]
+2,0 (c012,c012;e1)(e1) = 1.
+Thus, we get precisely the non-zero components of ̃F from (4.3) for f(F)/B.
+□
+5. Computations of the BV algebras
+We now compute the BV algebras on Hochschild cohomology induced by the homotopy
+inner products considered in examples 4.1 and 4.2 in section 4. In particular, we com-
+plete the computations from theorem 1.6 for the 2-sphere by computing HH●(H●,H●) and
+HH●(H●,H●) for H● = H● = H●(S2;Z2) and H● = H●(S2;Z2), and we will check the for-
+mulas for the maps F and ̃
+F in (1.7) stated in theorem 1.6. In the section the ground ring
+is R = Z2.
+5.1. Hochschild cohomology HH●(H●,H●) and the cup product. Denote by H● =
+H●(S2;Z2) ≅ Z2.e ⊕ Z2.s from example 4.1 with ∣e∣ = 0 and ∣s∣ = −2. The generators of
+normalized Hochschild cochains CH
+●(H●,H●) are φk, ψk for k ≥ 0 given by
+φk(s,... ,s
+��������������������
+k many
+) ∶= e,
+ψk(s,... ,s
+��������������������
+k many
+) ∶= s.
+Note that ∣φk∣ = k, ∣ψk∣ = k − 2, since the shifted generator s is of degree ∣s∣ = −1. Applying
+the Hochschild differential (see 2.1, and using d = 0 and s ⋅ s = 0, s ⋅ e = e ⋅ s = s), we see that
+all φk and ψk are closed (over Z2). Thus,
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.φk ⊕ ⊕
+k≥0
+Z2.ψk.
+The cup product (3.5) is readily checked to be φk ⌣ φℓ = φk+ℓ, ψk ⌣ ψℓ = 0, and φk ⌣ ψℓ =
+ψℓ ⌣ φk = ψk+ℓ.
+5.2. Hochschild cohomology HH●(H●,H●) and Connes’ B-operator. Denote by H● =
+H●(S2;Z2) ≅ Z2.e∗ ⊕ Z2.s∗ from example 4.1, with ∣e∗∣ = 0 and ∣s∗∣ = 2 and s.e∗ = e∗.s =
+0,s.s∗ = s∗.s = e∗. The generators of normalized Hochschild cochains CH
+●(H●,H●) are θk,
+χk for k ≥ 0 given by
+θk(s,... ,s
+��������������������
+k many
+) ∶= s∗,
+χk(s,... ,s
+��������������������
+k many
+) ∶= e∗.
+Note that ∣θk∣ = k + 2, ∣χk∣ = k. All θk and χk are closed under the Hochschild cochain
+differential (over Z2), so that
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.θk ⊕ ⊕
+k≥0
+Z2.χk.
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+15
+Whereas HH●(H●,H●) has the cup product (3.5), HH●(H●,H●) has Connes’ B-operator
+(3.2). We compute B from (3.2) to be (B(ϕ))( s,... ,s
+��������������������
+(r−1) many
+)(a) ∶= r ⋅ ϕ(s,... ,s,a,s,... ,s)(e),
+which applied to θk and χk yields:
+B(θr) = 0,
+B(χr) = r ⋅ θr−1.
+This confirms the expressions for B stated in theorem 1.6.
+5.3. F induced by example 4.1 and the BV delta ∆F. The dg-bimodule map F ∶ H● →
+H●, F(s) = e∗ and F(e) = s∗, from example 4.1 induced a map on Hochschild cohomologies
+F ∶ HH●(H●,H●) → HH●(H●,H●) via (2.6). Since F only has a lowest component F0,0
+this is simply
+F(φk) = F ○ φk = θk,
+F(ψk) = F ○ ψk = χk.
+F is clearly an isomorphism, and confirms F from theorem 1.6.
+We also compute ∆F and check it is equal to B transferred to HH●(H●,H●), i.e., con-
+dition (2) from definition 3.6 for a Poincar´e duality structure.
+From (3.3), we get for
+ϕ ∈ Hom(H●⊗r,H●) that ZF(ϕ)(s,... ,s)(a) = r ⋅ F(e)(ϕ(s,... ,a,... ,s)), which gives
+ZF(φk) = 0 = B(θk) = B ○ F(φk),
+ZF(ψk) = k ⋅ θk−1 = B(χk) = B ○ F(ψk).
+Therefore, F −1 ○ B ○ F = F −1 ○ ZF = ∆F.
+5.4. ̃
+F induced by example 4.2 and the BV delta ∆̃F. Consider the dg-bimodule
+map ̃F ∶ H● → H● from example 4.2. Since ̃F = ̃F0,0 + ̃F2,0 is ̃F0,0(s) = e∗, ̃F0,0(e) = s∗ with
+one extra homotopy ̃F2,0 which is non-zero only for ̃F2,0(s,s;e) ∶= e∗ (see (4.3)), we get
+̃
+F(φk) = ̃F0,0 ○ φk + ̃F2,0(−,−;φk(−)) = θk + χk+2,
+̃
+F(ψk) = ̃F0,0 ○ ψk = χk.
+This confirms ̃
+F from theorem 1.6. As was noted in theorem 1.6, ̃
+F is an isomorphism with
+inverse ̃
+F −1 given by ̃
+F −1(θk) = φk + ψk+2 and ̃
+F −1(χk) = ψk.
+We also check that in this case ̃
+F −1○B○ ̃
+F = ∆̃F. From (3.3), we get for ϕ ∈ Hom(H●⊗r,H●),
+that
+Z
+̃F(ϕ)(s,... ,s)(a) =r ⋅ ̃F0,0(e)(ϕ(s,... ,a,... ,s)) + r ⋅ ̃F2,0(s,s;e)(ϕ(s,... ,a,... ,s))
++ ̃F2,0(ϕ(s,... ,s),s;e)(a) + ̃F2,0(s,ϕ(s,... ,s);e)(a)
+which gives
+Z
+̃F(φk) = 0 + k ⋅ θk+1 + 0 + 0 = k ⋅ θk+1,
+Z
+̃F(ψk) = k ⋅ θk−1 + 0 + χk+1 + χk+1 = k ⋅ θk−1.
+Since B ○ ̃
+F(φk) = B(θk + χk+2) = k ⋅ θk+1 and B ○ ̃
+F(ψk) = B(χk) = k ⋅ θk−1, this shows that
+Z ̃F = B ○ ̃
+F and, thus, ̃
+F −1 ○ B ○ ̃
+F = ̃
+F −1 ○ Z ̃F = ∆̃F. Explicitly, ∆̃F is given by:
+∆
+̃F(φk) = ̃
+F −1 ○ Z
+̃F(φk) = ̃
+F −1(k ⋅ θk+1) = k ⋅ (φk+1 + ψk+3)
+∆
+̃F(ψk) = ̃
+F −1 ○ Z
+̃F(ψk) = ̃
+F −1(k ⋅ θk−1) = k ⋅ (φk−1 + ψk+1)
+This confirms ̃∆ from (1.9) and thus theorem 1.6.
+
+16
+K. POIRIER AND T. TRADLER
+5.5. Example of a homotopy inner product that is not a Poincar´e duality struc-
+ture. We end this section with an example of a homotopy inner product which is not
+a Poincar´e duality structure due to its failure of condition (2) from definition 3.6 (while
+condition (1) holds). We therefore do not get a BV algebra on Hochschild cohomology
+HH●(A,A) with underlying algebra A, since there is no ∆ operator, but only an operator,
+F −1 ○B ○F, that squares to zero, and a different operator, ∆F, whose deviation from being
+a derivation is a Gerstenhaber bracket; cf. lemma 3.4.
+Let A = Z2.e ⊕ Z2.b ⊕ Z2.c with zero differential, ∣e∣ = 0, ∣b∣ = −1, ∣c∣ = −2, and where
+e is the unit, b ⋅ b = c and all other products vanish.
+We denote A = A and its dual
+A∗ = Z2.e∗ ⊕ Z2.b∗ ⊕ Z2.c∗ where e∗,b∗,c∗ are the duals of e,b,c, respectively, so that
+∣e∗∣ = 0, ∣b∗∣ = 1, ∣c∗∣ = 2. Define the homotopy inner product F whose only non-vanishing
+components are F0,0 and F3,0 given by
+F0,0(c)(e) = F0,0(b)(b) = F0,0(e)(c) = 1,
+F3,0(c,b,c;e)(e) = F3,0(b,b,b;c)(e) = F3,0(b,b,b;e)(c) = F3,0(b,b,b;b)(b) = 1.
+One can check explicitly that the conditions for a homotopy inner product 0 = DF = D1F,
+where D1 is given by (2.4), are satisfied.
+Note, that F induces an isomorphism on Hochschild cochains F ∶ CH
+●(A,A)
+≅
+�→
+CH
+●(A,A∗), since F0,0 ∶ A → A∗ is an isomorphism, and thus F satisfies 3.6 condition (1).
+We next show that 3.6 condition (2) fails, i.e., that B○F ≠ ZF. Consider ϕ ∈ Hom(A⊗0,A),
+ϕ(1) = e. Note that ϕ is a closed element in the Hochschild cochains ϕ ∈ CH
+●(A,A). Plug-
+ging ϕ into (3.3) gives ZF(ϕ) = 0, since each individual summand of (3.3) vanishes. Next,
+to compute F(ϕ), we see the only non-zero evaluation of F(ϕ) are (the ones with “e” in
+the input spot for the module of F):
+F(ϕ)(1) = F0,0(ϕ(1)) = c∗
+F(ϕ)(c,b,c) = F3,0(c,b,c;ϕ(1)) = e∗
+F(ϕ)(b,b,b) = F3,0(b,b,b;ϕ(1)) = c∗
+Applying B from (3.2) (note that the unit in A is in this example written as e) to this is
+only non-zero on the middle term, for which we get the only non-zero evaluations:
+B(F(ϕ))(c,b)(c) = 1,
+B(F(ϕ))(b,c)(c) = 1,
+B(F(ϕ))(c,c)(b) = 1.
+Then, B(F(ϕ)) is non-vanishing, and closed in CH
+●(A,A∗), which follows either because
+ϕ is closed and B ○ F is a chain map or by direct inspection.
+Claim: B(F(ϕ)) is not exact in CH
+●(A,A∗).
+The claim implies that B ○F(ϕ) ≠ 0 = ZF(ϕ) in HH●(A,A∗), and so F is not a Poincar´e
+duality structure as it fails condition (2) from definition 3.6.
+Proof of the claim: To prove the claim, note that d = 0 in A implies that the Hochschild
+differential D = D0 + D1 = D1 from 2.1 must increase the number of tensor factors of A for
+the inputs. Thus any ̃ϕ which makes B(F(ϕ)) exact, i.e., D(̃ϕ) = B(F(ϕ)), must have
+non-vanishing components with exaclty one tensor input. Since the degree ∣B(F(ϕ))∣ = 3,
+it must be that ∣̃ϕ∣ = 4. However, all Hochschild cochains in CH
+●(A,A∗) with one or no
+input are of degree ≤ 3. Thus, B(F(ϕ)) cannot be exact.
+□
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+17
+Appendix A. Proof of equations (4.8) and (4.19)
+In this appendix we prove the identity (4.8), i.e., DF[01] = F[0] + F[1], for the maps F[01]
+given by (4.7) from example 4.6, as well as the identity (4.19), i.e., DF[012] = F[01] + F[02] +
+F[12], for the maps F[012] given by (4.11)-(4.18) from example 4.9. The ground ring in this
+appendix is R = Z2.
+A.1. Preliminaries. Any Fp,q is a map Fp,q ∶ A⊗p ⊗ A ⊗ A⊗q → A∗, which is an element
+of Fp,q ∈ (A∗)⊗p ⊗ A∗ ⊗ (A∗)⊗q ⊗ A∗. Consider A[01] with generators e0,e1,b01, respectively
+A[012] with generators e0,e1,e2,b01,b12,b02,c012). For a multi-index I ∈ {0,1,2,01,02,12,012},
+define a∗
+I to be a∗
+0 ∶= e∗
+0, a∗
+1 ∶= e∗
+1, a∗
+2 ∶= e∗
+2, a∗
+01 ∶= b∗
+01, a∗
+02 ∶= b∗
+02, a∗
+12 ∶= b∗
+12, a∗
+012 ∶= c∗
+012. With
+this we use the following notation, for a given sequence of multi-indices I1,... ,Ip+q+2:
+(A.2)
+[ I1,... ,Ip ∣ Ip+1 ∣ Ip+2,... ,Ip+q+1 ∣ Ip+q+2 ]
+∶= a∗
+I1 ⊗ ... ⊗ a∗
+Ip ⊗ a∗
+Ip+1 ⊗ a∗
+Ip+2 ⊗ ... ⊗ a∗
+Ip+q+1 ⊗ a∗
+Ip+q+2 ∈ (A∗)⊗p ⊗ A∗ ⊗ (A∗)⊗q ⊗ A∗
+Moreover, placing a tilde under the multi-index sums over any number of these indices, i.e.,
+for tildes at positions 1 ≤ k1 < k2 < ⋅⋅⋅ < kr ≤ p, we define
+(A.3)
+[ I1,... ,Ik1
+̃
+,... ,Ik2
+̃
+,... ,Ip ∣ Ip+1 ∣ Ip+2,... ,Ip+q+1 ∣ Ip+q+2 ]
+∶=
+∑
+n1,n2,...,nr≥0
+[ I1,... ,Ik1,... ,Ik1
+���������������������������������������������
+n1 many
+,... ,Ik2,... ,Ik2
+���������������������������������������������
+n2 many
+,... ,Ip ∣ Ip+1 ∣ Ip+2,... ,Ip+q+1 ∣ Ip+q+2 ]
+Since our examples are chosen to be “strict” on the right (see observation 4.4), we only
+need higher homotopies (and thus tildes) on the first p tensor factors, but not on the other
+q tensor factors of A∗.
+We apply D = D0 +D1 from (2.3) and (2.4) to the above. Here, D0 applies the differential
+of A∗, which, in the multi-indiex notation, removes one index, (for example, 01 ↦ 0 + 1,
+012 ↦ 01 + 02 + 12, etc.). D1 applies a multiplication, which on A∗ applies the Alexander-
+Whitney coproduct (for example, 01 ↦ 0 ⊗ 01 + 01 ⊗ 1, 012 ↦ 0 ⊗ 012 + 01 ⊗ 12 + 012 ⊗ 2,
+etc.); see also the calculation below.
+A.4. Proof of (4.8). F[01] from (4.7) is given by F[01] = [ 01
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃ ∣ 01 ∣ ∣ 1 ].
+We calculate D = D0 + D1 applied to F[01].
+D0([ 01
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0, 01
+̃ ∣ 0 ∣ ∣ 01 ] 01
++[ 01
+̃,1, 01
+̃ ∣ 0 ∣ ∣ 01 ] 02
++[ 01
+̃ ∣ 0 ∣ ∣ 0 ] 03
++[ 01
+̃ ∣ 0 ∣ ∣ 1 ] 04
+D1([ 01
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0,01, 01
+̃ ∣ 0 ∣ ∣ 01 ] 01
++[ 01
+̃,01,1, 01
+̃ ∣ 0 ∣ ∣ 01 ] 02
++[ 01
+̃,0 ∣ 0 ∣ ∣ 01 ] 01
++[ 01
+̃ ∣ 0 ∣ 0 ∣ 01 ] 05
++[ 01
+̃ ∣ 0 ∣ 0 ∣ 01 ] 05
++[ 01
+̃ ∣ 0 ∣ 01 ∣ 1 ] 06
++[ 01, 01
+̃ ∣ 0 ∣ ∣ 0 ] 03
++[ 1, 01
+̃ ∣ 0 ∣ ∣ 01 ] 02
+D0([ 01
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0, 01
+̃ ∣ 01 ∣ ∣ 1 ] 07
++[ 01
+̃,1, 01
+̃ ∣ 01 ∣ ∣ 1 ] 08
++[ 01
+̃ ∣ 0 ∣ ∣ 1 ] 04
++[ 01
+̃ ∣ 1 ∣ ∣ 1 ] 09
+
+18
+K. POIRIER AND T. TRADLER
+D1([ 01
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0,01, 01
+̃ ∣ 01 ∣ ∣ 1 ] 07
++[ 01
+̃,01,1, 01
+̃ ∣ 01 ∣ ∣ 1 ] 08
++[ 01
+̃,0 ∣ 01 ∣ ∣ 1 ] 07
++[ 01
+̃,01 ∣ 1 ∣ ∣ 1 ] 09
++[ 01
+̃ ∣ 0 ∣ 01 ∣ 1 ] 06
++[ 01
+̃ ∣ 01 ∣ 1 ∣ 1 ] 10
++[ 01
+̃ ∣ 01 ∣ 1 ∣ 1 ] 10
++[ 1, 01
+̃ ∣ 01 ∣ ∣ 1 ] 08
+The gray box on the right of each expression indicates which terms can be combined. Note
+that for
+01 ,
+02 ,
+07 , and
+08 , there are always 3 terms that can be combined and cancel. In
+fact, all terms cancel, except for terms labeled
+03 , which has a left over term of [ ∣ 0 ∣ ∣ 0 ],
+and terms labeled
+09 , which has a left over term of [ ∣ 1 ∣ ∣ 1 ]. These two left-over terms
+are F[0] and F[1], which shows that DF[01] = F[0] + F[1], i.e., equation (4.8).
+A.5. Proof of (4.19). F[012] from (4.11)-(4.18) is given by
+F[012] =[ 02
+̃ ∣ 012 ∣ ∣ 2 ] + [ 02
+̃ ∣ 0 ∣ ∣ 012 ] + [ 02
+̃ ∣ 01 ∣ ∣ 12 ]
++ [ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ]
++ [ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] + [ 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ]
++ [ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ]
+We calculate D = D0 + D1 applied to F[012]. Again, the gray box indicates which terms
+combine and cancel. Left-over terms will be collected below.
+D0([ 02
+̃ ∣ 012 ∣ ∣ 2 ]) =
+[ 02
+̃,0, 02
+̃ ∣ 012 ∣ ∣ 2 ] 01
++[ 02
+̃,2, 02
+̃ ∣ 012 ∣ ∣ 2 ] 02
++[ 02
+̃ ∣ 01 ∣ ∣ 2 ] 03
++[ 02
+̃ ∣ 02 ∣ ∣ 2 ] 04
++[ 02
+̃ ∣ 12 ∣ ∣ 2 ] 05
+D1([ 02
+̃ ∣ 012 ∣ ∣ 2 ]) =
+[ 02
+̃,0,02, 02
+̃ ∣ 012 ∣ ∣ 2 ] 01
++[ 02
+̃,02,2, 02
+̃ ∣ 012 ∣ ∣ 2 ] 02
++[ 02
+̃,0 ∣ 012 ∣ ∣ 2 ] 01
++[ 02
+̃,01 ∣ 12 ∣ ∣ 2 ] 06
++[ 02
+̃,012 ∣ 2 ∣ ∣ 2 ] 07
++[ 02
+̃ ∣ 012 ∣ 2 ∣ 2 ] 08
++[ 02
+̃ ∣ 01 ∣ 12 ∣ 2 ] 09
++[ 02
+̃ ∣ 0 ∣ 012 ∣ 2 ] 10
++[ 02
+̃ ∣ 012 ∣ 2 ∣ 2 ] 08
++[ 2, 02
+̃ ∣ 012 ∣ ∣ 2 ] 02
+D0([ 02
+̃ ∣ 0 ∣ ∣ 012 ]) =
+[ 02
+̃,0, 02
+̃ ∣ 0 ∣ ∣ 012 ] 11
++[ 02
+̃,2, 02
+̃ ∣ 0 ∣ ∣ 012 ] 12
++[ 02
+̃ ∣ 0 ∣ ∣ 01 ] 13
++[ 02
+̃ ∣ 0 ∣ ∣ 02 ] 14
++[ 02
+̃ ∣ 0 ∣ ∣ 12 ] 15
+D1([ 02
+̃ ∣ 0 ∣ ∣ 012 ]) =
+[ 02
+̃,0,02, 02
+̃ ∣ 0 ∣ ∣ 012 ] 11
++[ 02
+̃,02,2, 02
+̃ ∣ 0 ∣ ∣ 012 ] 12
+[ 02
+̃,0 ∣ 0 ∣ ∣ 012 ] 11
++[ 02
+̃ ∣ 0 ∣ 0 ∣ 012 ] 16
+[ 02
+̃ ∣ 0 ∣ 0 ∣ 012 ] 16
++[ 02
+̃ ∣ 0 ∣ 01 ∣ 12 ] 17
+[ 02
+̃ ∣ 0 ∣ 012 ∣ 2 ] 10
++[ 2, 02
+̃ ∣ 0 ∣ ∣ 012 ] 12
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+19
+[ 12, 02
+̃ ∣ 0 ∣ ∣ 01 ] 18
++[ 012, 02
+̃ ∣ 0 ∣ ∣ 0 ] 19
+D0([ 02
+̃ ∣ 01 ∣ ∣ 12 ]) =
+[ 02
+̃,0, 02
+̃ ∣ 01 ∣ ∣ 12 ] 20
++[ 02
+̃,2, 02
+̃ ∣ 01 ∣ ∣ 12 ] 21
++[ 02
+̃ ∣ 0 ∣ ∣ 12 ] 15
++[ 02
+̃ ∣ 1 ∣ ∣ 12 ] 22
++[ 02
+̃ ∣ 01 ∣ ∣ 1 ] 23
++[ 02
+̃ ∣ 01 ∣ ∣ 2 ] 03
+D1([ 02
+̃ ∣ 01 ∣ ∣ 12 ]) =
+[ 02
+̃,0,02, 02
+̃ ∣ 01 ∣ ∣ 12 ] 20
++[ 02
+̃,02,2, 02
+̃ ∣ 01 ∣ ∣ 12 ] 21
++[ 02
+̃,0 ∣ 01 ∣ ∣ 12 ] 20
++[ 02
+̃,01 ∣ 1 ∣ ∣ 12 ] 24
++[ 02
+̃ ∣ 01 ∣ 1 ∣ 12 ] 25
++[ 02
+̃ ∣ 0 ∣ 01 ∣ 12 ] 17
++[ 02
+̃ ∣ 01 ∣ 1 ∣ 12 ] 25
++[ 02
+̃ ∣ 01 ∣ 12 ∣ 2 ] 09
++[ 2, 02
+̃ ∣ 01 ∣ ∣ 12 ] 21
++[ 12, 02
+̃ ∣ 01 ∣ ∣ 1 ] 26
+D0([ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 27
++[ 01
+̃,1, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 28
++[ 01
+̃,01, 02
+̃ ∣ 0 ∣ ∣ 01 ] 13
++[ 01
+̃,02, 02
+̃ ∣ 0 ∣ ∣ 01 ] 13
++[ 01
+̃,12, 02
+̃ ∣ 0 ∣ ∣ 01 ] 18
++[ 01
+̃,012, 02
+̃,0, 02
+̃ ∣ 0 ∣ ∣ 01 ] 29
++[ 01
+̃,012, 02
+̃,2, 02
+̃ ∣ 0 ∣ ∣ 01 ] 30
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 0 ] 19
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 1 ] 31
+D1([ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0,01, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 27
++[ 01
+̃,01,1, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 28
++[ 01
+̃,0,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 27
++[ 01
+̃,01,12, 02
+̃ ∣ 0 ∣ ∣ 01 ] 18
++[ 01
+̃,012,2, 02
+̃ ∣ 0 ∣ ∣ 01 ] 30
++[ 01
+̃,012, 02
+̃,0,02, 02
+̃ ∣ 0 ∣ ∣ 01 ] 29
++[ 01
+̃,012, 02
+̃,02,2, 02
+̃ ∣ 0 ∣ ∣ 01 ] 30
++[ 01
+̃,012, 02
+̃,0 ∣ 0 ∣ ∣ 01 ] 29
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 0 ∣ 01 ] 32
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 0 ∣ 01 ] 32
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 01 ∣ 1 ] 33
++[ 1, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 28
++[ 01, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 0 ] 19
+D0([ 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 34
++[ 01
+̃,1, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 35
++[ 01
+̃,01, 02
+̃ ∣ 01 ∣ ∣ 1 ] 23
++[ 01
+̃,02, 02
+̃ ∣ 01 ∣ ∣ 1 ] 23
++[ 01
+̃,12, 02
+̃ ∣ 01 ∣ ∣ 1 ] 26
++[ 01
+̃,012, 02
+̃,0, 02
+̃ ∣ 01 ∣ ∣ 1 ] 36
++[ 01
+̃,012, 02
+̃,2, 02
+̃ ∣ 01 ∣ ∣ 1 ] 37
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 1 ] 31
++[ 01
+̃,012, 02
+̃ ∣ 1 ∣ ∣ 1 ] 38
+
+20
+K. POIRIER AND T. TRADLER
+D1([ 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0,01, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 34
++[ 01
+̃,01,1, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 35
++[ 01
+̃,0,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 34
++[ 01
+̃,01,12, 02
+̃ ∣ 01 ∣ ∣ 1 ] 26
++[ 01
+̃,012,2, 02
+̃ ∣ 01 ∣ ∣ 1 ] 37
++[ 01
+̃,012, 02
+̃,0,02, 02
+̃ ∣ 01 ∣ ∣ 1 ] 36
++[ 01
+̃,012, 02
+̃,02,2, 02
+̃ ∣ 01 ∣ ∣ 1 ] 37
++[ 01
+̃,012, 02
+̃,0 ∣ 01 ∣ ∣ 1 ] 36
++[ 01
+̃,012, 02
+̃,01 ∣ 1 ∣ ∣ 1 ] 39
++[ 01
+̃,012, 02
+̃ ∣ 01 ∣ 1 ∣ 1 ] 40
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 01 ∣ 1 ] 33
++[ 01
+̃,012, 02
+̃ ∣ 01 ∣ 1 ∣ 1 ] 40
++[ 1, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 35
+D0([ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ]) =
+[ 02
+̃,0, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 41
++[ 02
+̃,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 42
++[ 02
+̃,01, 12
+̃ ∣ 1 ∣ ∣ 12 ] 24
++[ 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 12 ] 22
++[ 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] 22
++[ 02
+̃,012, 12
+̃,1, 12
+̃ ∣ 1 ∣ ∣ 12 ] 43
++[ 02
+̃,012, 12
+̃,2, 12
+̃ ∣ 1 ∣ ∣ 12 ] 44
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 2 ] 45
+D1([ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ]) =
+[ 02
+̃,0,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 41
++[ 02
+̃,02,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 42
++[ 02
+̃,0,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 41
++[ 02
+̃,01,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] 24
++[ 02
+̃,012,2, 12
+̃ ∣ 1 ∣ ∣ 12 ] 44
++[ 02
+̃,012, 12
+̃,1,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] 43
++[ 02
+̃,012, 12
+̃,12,2, 12
+̃ ∣ 1 ∣ ∣ 12 ] 44
++[ 02
+̃,012, 12
+̃,1 ∣ 1 ∣ ∣ 12 ] 43
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 12 ] 46
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 12 ] 46
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 12 ∣ 2 ] 47
++[ 2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 42
++[ 12, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 48
+D0([ 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ]) =
+[ 02
+̃,0, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 49
++[ 02
+̃,2, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 50
++[ 02
+̃,01, 12
+̃ ∣ 12 ∣ ∣ 2 ] 06
++[ 02
+̃,02, 12
+̃ ∣ 12 ∣ ∣ 2 ] 05
++[ 02
+̃,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] 05
++[ 02
+̃,012, 12
+̃,1, 12
+̃ ∣ 12 ∣ ∣ 2 ] 51
++[ 02
+̃,012, 12
+̃,2, 12
+̃ ∣ 12 ∣ ∣ 2 ] 52
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 2 ] 45
++[ 02
+̃,012, 12
+̃ ∣ 2 ∣ ∣ 2 ] 07
+D1([ 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ]) =
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+21
+[ 02
+̃,0,02, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 49
++[ 02
+̃,02,2, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 50
++[ 02
+̃,0,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 49
++[ 02
+̃,01,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] 06
++[ 02
+̃,012,2, 12
+̃ ∣ 12 ∣ ∣ 2 ] 52
++[ 02
+̃,012, 12
+̃,1,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] 51
++[ 02
+̃,012, 12
+̃,12,2, 12
+̃ ∣ 12 ∣ ∣ 2 ] 52
++[ 02
+̃,012, 12
+̃,1 ∣ 12 ∣ ∣ 2 ] 51
++[ 02
+̃,012, 12
+̃,12 ∣ 2 ∣ ∣ 2 ] 07
++[ 02
+̃,012, 12
+̃ ∣ 12 ∣ 2 ∣ 2 ] 53
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 12 ∣ 2 ] 47
++[ 02
+̃,012, 12
+̃ ∣ 12 ∣ 2 ∣ 2 ] 53
++[ 2, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 50
+D0([ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ]) =
+[ 01
+̃,0, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 54
++[ 01
+̃,1, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 55
++[ 01
+̃,01, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,12, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 48
++[ 01
+̃,012, 02
+̃,0, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 56
++[ 01
+̃,012, 02
+̃,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 57
++[ 01
+̃,012, 02
+̃,01, 12
+̃ ∣ 1 ∣ ∣ 1 ] 39
++[ 01
+̃,012, 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,012, 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,012, 02
+̃,012, 12
+̃,1, 12
+̃ ∣ 1 ∣ ∣ 1 ] 58
++[ 01
+̃,012, 02
+̃,012, 12
+̃,2, 12
+̃ ∣ 1 ∣ ∣ 1 ] 59
+D1([ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ]) =
+[ 01
+̃,0,01, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 54
++[ 01
+̃,01,1, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 55
++[ 01
+̃,0,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 54
++[ 01
+̃,01,12, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 48
++[ 01
+̃,012,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 57
++[ 01
+̃,012, 02
+̃,0,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 56
++[ 01
+̃,012, 02
+̃,02,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 57
++[ 01
+̃,012, 02
+̃,0,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 56
++[ 01
+̃,012, 02
+̃,01,12, 12
+̃ ∣ 1 ∣ ∣ 1 ] 39
++[ 01
+̃,012, 02
+̃,012,2, 12
+̃ ∣ 1 ∣ ∣ 1 ] 59
++[ 01
+̃,012, 02
+̃,012, 12
+̃,1,12, 12
+̃ ∣ 1 ∣ ∣ 1 ] 58
++[ 01
+̃,012, 02
+̃,012, 12
+̃,12,2, 12
+̃ ∣ 1 ∣ ∣ 1 ] 59
++[ 01
+̃,012, 02
+̃,012, 12
+̃,1 ∣ 1 ∣ ∣ 1 ] 58
++[ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 1 ] 60
++[ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 1 ] 60
++[ 1, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 55
+In the above sum, there are several terms which appear twice and which therefore cancel
+(over Z2). The terms that are labeled with the following numbers appear twice and cancel:
+03 , 08 , 09 , 10 , 15 , 16 , 17 , 25 , 31 , 32 , 33 , 40 , 45 , 46 , 47 , 53 , 60 .
+
+22
+K. POIRIER AND T. TRADLER
+Moreover, in some instances the sum of three terms cancel. The following labels appear
+three times and also cancel:
+01 , 02 , 06 , 07 , 11 , 12 , 18 , 19 , 20 , 21 , 24 , 26 , 27 , 28 , 29 , 30 , 34 , 35 ,
+36 , 37 , 39 , 41 , 42 , 43 , 44 , 48 , 49 , 50 , 51 , 52 , 54 , 55 , 56 , 57 , 58 , 59 .
+There are six terms labeled with
+38 , whose sum vanishes:
+38 =([ 01
+̃,012, 02
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,012, 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,012, 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 1 ])
++ ([ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,01, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ])
+=[ 01
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] = 0
+The left-over terms that do not cancel are:
+04 = [ 02
+̃ ∣ 02 ∣ ∣ 2 ]
+05 = [ 02
+̃ ∣ 12 ∣ ∣ 2 ] + [ 02
+̃,02, 12
+̃ ∣ 12 ∣ ∣ 2 ] + [ 02
+̃,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] = [ 12
+̃ ∣ 12 ∣ ∣ 2 ]
+13 = [ 02
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃,01, 02
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃,02, 02
+̃ ∣ 0 ∣ ∣ 01 ] = [ 01
+̃ ∣ 0 ∣ ∣ 01 ]
+14 = [ 02
+̃ ∣ 0 ∣ ∣ 02 ]
+22 = [ 02
+̃ ∣ 1 ∣ ∣ 12 ] + [ 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 12 ] + [ 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] = [ 12
+̃ ∣ 1 ∣ ∣ 12 ]
+23 = [ 02
+̃ ∣ 01 ∣ ∣ 1 ] + [ 01
+̃,01, 02
+̃ ∣ 01 ∣ ∣ 1 ] + [ 01
+̃,02, 02
+̃ ∣ 01 ∣ ∣ 1 ] = [ 01
+̃ ∣ 01 ∣ ∣ 1 ]
+These terms are precisely F[02], F[12], and F[01], so that DF[012] = D0F[012] + D1F[012] =
+F[01] + F[02] + F[12], which is (4.19).
+References
+[CS]
+Moira Chas, Dennis Sullivan. String Topology. preprint arXiv:math/9911159
+[L]
+Jean-Louis Loday. Cyclic Homology, Grundlehren der mathematischen WIssenschaften 301 (1998),
+Springer.
+[M]
+Luc Menichi. String Topology for Spheres, Comment. Math. Helv. 84 (2009), 135–157.
+[T1]
+Thomas Tradler. Infinity Inner Products on A-Infinity Algebras. J. Homotopy Relat. Struct. 3
+(2008), no. 1, p. 245–271.
+[T2]
+Thomas Tradler. The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products.
+Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, p. 2351–2379.
+[TZS]
+Thomas Tradler, Mahmoud Zeinalian, Dennis Sullivan Infinity structure of Poincar´e duality spaces.
+Algebr. Geom. Topol. 7 (2007), p. 233–260.
+Kate Poirier, Department of Mathematics, New York City College of Technology,
+City University of New York, 300 Jay Street, Brooklyn, NY 11201
+Email address: [email protected]
+Thomas Tradler, Department of Mathematics, New York City College of Technology,
+City University of New York, 300 Jay Street, Brooklyn, NY 11201
+Email address: [email protected]
+

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+MATHEMATICA MONTISNIGRI 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+  
+ 
+2010 Mathematics Subject Classification:  30B10, 11M05, 11M41, 33B15.  
+Key words and Phrases: Riemann’s zeta function, Lerch transcendent, polylogarithm, digamma function, 
+Euler’s constant. 
+SEVERAL CLASSICAL IDENTITIES VIA MELLIN’S TRANSFORM 
+KHRISTO N. BOYADZHIEV 
+Department of mathematics, Ohio Northern University 
+Ada, Ohio, 45810, USA 
+ E-mail: [email protected] 
+DOI: 
+Summary. We present a summation rule using Mellin’s transform to give short proofs of 
+some important classical relations between special functions and Bernoulli and Euler 
+polynomials. For example, the values of the Hurwitz zeta function at the negative integers are 
+expressed in terms of Bernoulli polynomials. We also show identities involving exponential 
+and Hermite polynomials. 
+1. INTRODUCTION 
+Throughout we use the notation ( )
+,
+a
+a
+it t
+=
++
+∈  for the vertical line with abscissa
+0
+1
+a
+<
+< , oriented from minus to plus infinity. First we recall the formulas for the Mellin 
+transform 
+1
+0
+( )
+( )
+s
+G s
+x
+g x dx
+∞
+−
+= ∫
+ 
+and its inverse  
+ 
+( )
+1
+( )
+( )
+,
+0
+2
+s
+a
+g x
+x G s ds
+x
+i
+π
+−
+=
+>
+∫
+. 
+(1) 
+Definition. For 
+0
+n >
+ consider integrals of the form 
+( )
+( )
+,
+0
+s
+L n x G s ds
+x
+−
+>
+∫
+ 
+where ( )
+L n  consists of the line segment [
+,
+]
+ni ni
+−
+ together with the semicircle 
+( )
+R n  in the left 
+half plane for which the line segment is the diagonal. If the integrals 
+( )
+( )
+s
+R n x G s ds
+−
+∫
+ 
+approach zero when n → ∞, we say that the line of integration in (1) can be closed to the left. 
+In a similar manner we define integrals where the line of integration can be closed to the right. 
+Proposition. Suppose that the function 
+( )
+G s  is meromorphic on the half plane 
+( )
+Re s
+a
+ε
+<
++
+for some small 
+0
+ε >
+, and has only simple poles at  
+0, 1, 2,...
+s =
+− −
+, with residues 
+0
+1
+2
+,
+,
+,...
+c c c
+. 
+If the line of integration in (1) can be closed to the left, the residue theorem provides the 
+representation 
+
+Khristo N. Boyadzhiev 
+ 
+0
+( )
+n
+n
+n
+g x
+c x
+∞
+=
+= ∑
+ 
+(2) 
+for the function 
+( )
+g x  from (1), i.e. this function is a power series. If now 
+( )
+f s  is an 
+appropriate holomorphic function on 
+( )
+Re s
+a
+ε
+<
++
+ without poles, we can write  
+ 
+( )
+0
+1
+( ) ( )
+(
+)
+2
+s
+n
+n
+a
+n
+x
+f s G s ds
+c f
+n x
+i
+π
+∞
+−
+=
+=
+−
+∑
+∫
+, 
+(3) 
+when the power series on the right side converges.  
+Formulas of this type are used for summation of series or interpolation, and are present in 
+many publications (see [2, 4, 9, 10] and the references there).  
+In this note we focus on a special area of applications for the proposition - obtaining some 
+classical identities by using Mellin inversion and comparing coefficients. In order to keep the 
+paper short we omit details and do not discuss convergence of some integrals and series. The 
+validity of such formulas is considered in [4, 9, 10]. 
+The illustration of the method is given in the following examples.  
+2 EXAMPLES 
+Remind that the residues of the gamma function at zero and the negative integers are given 
+by 
+( 1)
+Res( ,
+)
+!
+n
+n
+n
+−
+Γ −
+=
+ for 
+0,1,2,...
+n =
+. Also ( )s
+Γ
+ has rapid decay on vertical lines. We have 
+the estimate ([15, (20)] 
+(
+)
+it
+a +
+Γ
+ ~ 
+1
+2
+2
+2
+| |
+,
+a
+t
+t
+e
+a
+π
+π
+−
+−
+∈  
+when | |t → ∞ . The estimate helps for the convergence of our integrals.  
+The above facts will be used in the following examples. We note that in these examples the 
+lines of integration can be closed to the left (proofs are standard, using the growth estimate for 
+( )s
+Γ
+). Note also that when we replace 
+( )
+G s  by ( )s
+Γ
+ in (3) we have 
+ 
+( )
+0
+1
+( 1)
+( ) ( )
+(
+)
+2
+!
+n
+s
+n
+a
+n
+x
+f s
+s ds
+f
+n x
+i
+n
+π
+∞
+−
+=
+−
+Γ
+=
+−
+∑
+∫
+. 
+(4) 
+Example 1. Ramanujan’s Master Theorem. As Hardy writes in [8], Ramanujan was very fond 
+of his integral formula [8, p.186] 
+ 
+1
+0
+0
+( )(
+)
+(
+) ( )
+!
+n
+s
+n
+f n
+x
+x
+dx
+f
+s
+s
+n
+∞
+∞
+−
+=
+
+
+−
+=
+−
+Γ
+
+
+
+
+∑
+∫
+ 
+ 
+(5) 
+and used it for many applications. Berndt rightly calls it Ramanujan’s Master Theorem [2, 
+Entry 11, p.105]. Details, comments, and applications of (5) are given in these two books and 
+also in [1, 4, 6]. Clearly, after replacing 
+( )
+f s  by 
+(
+)
+f
+s
+−
+ equation (5) turns into (4) after 
+Mellin inversion. Ramanujan did not use the residue theorem for his proof but only standard 
+calculus (see [2, p.106]).  
+
+Khristo N. Boyadzhiev 
+Example 2. The Hurwitz zeta function is defined by 
+0
+1
+( , )
+(Re( )
+0, Re( )
+1)
+(
+)s
+n
+s z
+z
+s
+n
+z
+ζ
+∞
+=
+=
+>
+>
++
+∑
+ 
+with integral representation 
+(1
+)
+1
+0
+( , ) ( )
+(Re( )
+1)
+1
+t
+z
+s
+t
+e
+s z
+s
+t
+dt
+s
+e
+ζ
+∞
+−
+−
+Γ
+=
+>
+−
+∫
+ 
+When 
+1
+z = , 
+( ,1)
+( )
+s
+s
+ζ
+ζ
+=
+ is the Riemann zeta function [5]. 
+We have also the modified integral representation (argument is the same as on pp. 61-62 in 
+[13]) 
+(1
+)
+1
+0
+1
+( , ) ( )
+(0
+Re( )
+1)
+1
+t
+z
+s
+t
+e
+s z
+s
+t
+dt
+s
+e
+t
+ζ
+∞
+−
+− 
+
+Γ
+=
+<
+<
+−
+
+
+−
+
+
+∫
+ 
+which is a Mellin transform formula. By Mellin inversion 
+(1
+)
+( )
+0
+1
+1
+( 1)
+( , ) ( )
+(
+, )
+1
+2
+!
+t
+z
+n
+s
+t
+a
+n
+e
+t
+s z
+s ds
+n z
+e
+t
+i
+n
+ζ
+ζ
+π
+−
+∞
+−
+=
+−
+−
+=
+Γ
+=
+−
+−
+∑
+∫
+ 
+At the same time, the Bernoulli polynomials 
+( )
+n
+B z  have the generating function 
+0
+( )
+1
+!
+xz
+k
+k
+x
+k
+B z
+xe
+x
+e
+k
+∞
+=
+=
+−
+∑
+ 
+and from this 
+(1
+)
+1
+1
+1
+0
+(1
+)
+(1
+)
+1
+( , )
+1
+!
+(
+1)!
+x
+z
+k
+n
+k
+n
+x
+k
+n
+B
+z
+B
+z
+e
+g x z
+x
+x
+e
+x
+k
+n
+−
+∞
+∞
+−
++
+=
+=
+−
+−
+≡
+−
+=
+=
+−
++
+∑
+∑
+. 
+Therefore, by comparing coefficients for 
+0, 1,...
+n =
+we find the classical formula 
+1
+1
+( 1)
+(1
+)
+( )
+(
+, )
+1
+1
+n
+n
+n
+B
+z
+B
+z
+n z
+n
+n
+ζ
++
++
+−
+−
+−
+=
+= −
++
++
+ 
+(using the property ( 1)
+(1
+)
+( )
+n
+n
+n
+B
+z
+B z
+−
+−
+=
+, so that 
+1
+1
+( 1)
+(1
+)
+( )
+n
+n
+n
+B
+z
+B
+z
++
++
+−
+−
+= −
+).  
+In particular,  
+1
+1
+(0, )
+( )
+2
+z
+B z
+z
+ζ
+= −
+=
+− . 
+For the Bernoulli numbers 
+(0)
+( 1)
+(1)
+n
+n
+n
+n
+B
+B
+B
+=
+= −
+ we have 
+1
+1
+( 1)
+(0)
+( 1)
+(
+)
+(
+,1)
+1
+1
+n
+n
+n
+n
+B
+B
+n
+n
+n
+n
+ζ
+ζ
++
++
+−
+−
+−
+=
+−
+=
+=
++
++
+. 
+The odd Bernoulli numbers are zeros except 
+1
+1/ 2
+B = −
+. Thus 
+(0)
+1/ 2
+ζ
+= −
+. 
+
+Khristo N. Boyadzhiev 
+Next we give a new proof of a result of Kenneth Williams and Zhang Nan-Yue [14]. 
+Example 3. Consider now the alternating Hurwitz zeta function 
+0
+( 1)
+( , )
+(Re( )
+0, Re( )
+0)
+(
+)
+n
+s
+n
+s z
+z
+s
+n
+z
+η
+∞
+=
+−
+=
+>
+>
++
+∑
+ 
+which extends to the entire complex plane as analytic in the variable s  and has the integral 
+representation 
+(1
+)
+1
+0
+( , ) ( )
+(Re( )
+0)
+1
+t
+z
+s
+t
+e
+s z
+s
+t
+dt
+s
+e
+η
+∞
+−
+−
+Γ
+=
+>
++
+∫
+. 
+By inversion 
+(1
+)
+( )
+0
+1
+( 1)
+( , ) ( )
+(
+, )
+1
+2
+!
+t
+z
+n
+s
+n
+t
+a
+n
+e
+t
+s z
+s ds
+n z x
+e
+i
+n
+η
+η
+π
+−
+∞
+−
+=
+−
+=
+Γ
+=
+−
++
+∑
+∫
+. 
+Euler’s polynomials 
+( )
+n
+E
+z  are defined by the generating function 
+0
+2
+( )
+(| |
+)
+1
+!
+t x
+n
+n
+t
+n
+e
+t
+E
+x
+t
+e
+n
+π
+∞
+=
+=
+<
++
+∑
+. 
+Comparing coefficients gives 
+( 1)
+1
+(
+, )
+(1
+)
+( )
+2
+2
+n
+n
+n
+n z
+E
+z
+E
+z
+η
+−
+−
+=
+−
+=
+ 
+by the property 
+(1
+)
+( 1)
+( )
+n
+n
+n
+E
+z
+E
+z
+−
+= −
+. 
+Example 4. Euler worked with the function 
+0
+( 1)
+( )
+(Re
+0)
+(2
+1)
+n
+s
+n
+L s
+s
+n
+∞
+=
+−
+=
+>
++
+∑
+ 
+which we call here Euler’s L -function (sometimes it is called Dirichlet’s L -function). This 
+function has the integral representation 
+ 
+1
+0
+2 ( ) ( )
+cosh
+sx
+s L s
+dx
+x
+∞
+−
+Γ
+= ∫
+. 
+It also has analytic extension on the complex plane.  
+By Mellin inversion and using equation (4) 
+( )
+0
+1
+1
+( 1)
+( ) ( )
+(
+)
+2cosh( )
+2
+!
+n
+s
+n
+a
+n
+x L s
+s ds
+L
+n x
+x
+i
+n
+π
+∞
+−
+=
+−
+=
+Γ
+=
+−
+∑
+∫
+ 
+Euler’s numbers 
+n
+E  are defined by the generating function 
+0
+1
+cosh
+!
+n
+n
+n
+E x
+x
+n
+∞
+=
+= ∑
+. 
+
+Khristo N. Boyadzhiev 
+This function is even, so the Euler numbers with odd indices are zeros. By comparing 
+coefficients we find 
+2
+1
+( 2 )
+(
+0,1, 2,...)
+2
+n
+L
+n
+E
+n
+−
+=
+=
+. 
+Example 5.  The exponential polynomials 
+n
+ϕ  are defined by the generating function 
+ 
+(
+1)
+0
+( )
+!
+x
+n
+z e
+n
+n
+x
+e
+z n
+ϕ
+∞
+−
+=
+= ∑
+ 
+(see [3, 12]). We will use the function   
+ 
+(
+1)
+0
+( , )
+( 1)
+( )
+!
+x
+n
+z e
+n
+n
+n
+x
+x z
+e
+z n
+ψ
+ϕ
+−
+∞
+−
+=
+=
+=
+−
+∑
+   
+ which has Mellin transform 
+ 
+1
+1
+0
+0
+( , )
+( , )
+s
+z
+s
+x
+ze
+s z
+x
+x z dx
+e
+x
+e
+dx
+ψ
+∞
+∞
+−
+−
+−
+−
+Ψ
+=
+=
+∫
+∫
+ 
+ 
+1
+0
+0
+0
+( )
+( ) ( , )
+!
+!
+n
+n
+z
+s
+z
+s
+n
+n
+nx
+z
+z
+e
+x
+e
+dx
+e
+s
+s f s z
+n
+n n
+∞
+∞
+∞
+−
+−
+−
+−
+=
+=
+
+
+=
+=
+Γ
+= Γ
+
+
+
+
+∑
+∑
+∫
+ 
+with  
+0
+( , )
+!
+n
+z
+s
+n
+z
+f s z
+e
+n n
+∞
+−
+=
+=
+∑
+. 
+Now we have from equation (4) 
+( )
+( )
+0
+1
+1
+( 1)
+( , )
+( , )
+( , ) ( )
+(
+, )
+2
+2
+!
+n
+s
+s
+n
+a
+a
+n
+x z
+x
+x s ds
+x
+f s z
+s ds
+f
+n z x
+i
+i
+n
+ψ
+π
+π
+∞
+−
+−
+=
+−
+=
+Ψ
+=
+Γ
+=
+−
+∑
+∫
+∫
+ 
+Thus for 
+0
+n ≥
+(with the agreent 
+00
+1
+= ) 
+ 
+0
+( )
+(
+, )
+!
+n
+z
+k
+n
+k
+k
+z
+f
+n z
+e
+z
+k
+ϕ
+∞
+−
+=
+=
+−
+=
+∑
+ 
+(6) 
+which is one of the fundamental properties of the exponential polynomials. 
+Note that identity (6) can be used for a meaningful extension of 
+( )
+n z
+ϕ
+ to 
+( )z
+λ
+ϕ
+ with non-
+integer index λ . For instance, we have (with 0
+0
+λ =
+) 
+ 
+1
+( )
+!
+z
+k
+k
+k
+z
+e
+z
+k
+λ
+λ
+ϕ
+∞
+−
+=
+=
+∑
+. 
+Example 6.  The generating function for the Hermite polynomials is 
+
+Khristo N. Boyadzhiev 
+ 
+2
+2
+0
+( , )
+( )
+!
+n
+xz
+x
+n
+n
+x
+x z
+e
+H
+z n
+ψ
+∞
+−
+=
+=
+= ∑
+ 
+with Mellin transform [11, p. 27] 
+ 
+ 
+2
+/2
+2
+( , )
+2
+( 2 ) ( )
+z
+s
+s
+s z
+e
+D
+z
+s
+−
+−
+Ψ
+=
+Γ
+. 
+where 
+p
+D  are the parabolic cylinder functions [7, pp. 1065-1067]. Let now  
+2
+/2
+2
+( , )
+2
+( 2 )
+z
+s
+s
+f s z
+e
+D
+z
+−
+−
+=
+. 
+Then 
+( )
+( 1)
+(
+, )
+n
+n
+H
+z
+f
+n z
+= −
+−
+ and since  
+( )
+( 1)
+(
+)
+n
+n
+n
+H
+z
+H
+z
+= −
+−
+ one finds the classical result 
+ 
+2
+2
+2
+( )
+2
+( 2 )
+n
+z
+n
+n
+H
+z
+e D
+z
+=
+ 
+([7, entry 9.253, p.1067]). 
+6.  CONCLUSIONS 
+In this note we presented a rule how Mellin’s transform can be used to give short proofs of 
+several classical identities connecting, in particular, the Bernoulli and Euler polynomials to 
+the values of the Hurwitz and alternating Hurwitz functions at the negative integers. We also 
+proved identities for the exponential and Hermite polynomials. 
+REFERENCES 
+[1] 
+T. Amdeberhan, I. Gonzalez, M. Harrison, V. H. Moll, A. Straub, “Ramanujan's Master 
+Theorem”, The Ramanujan Journal, 29, 103–120 (2012). 
+[2] 
+B. C. Berndt, Ramanujan’s Notebooks, Parts I and II, Springer-Verlag, New York (1985, 
+1989). 
+[3] 
+K. N. Boyadzhiev, “Exponential polynomials, Stirling numbers, and evaluation of some 
+Gamma integrals”, Abstract and Applied Analysis, Article ID 168672 (2009).  
+[4] 
+G. Dahlquist, “On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-
+Christoffel rules, I”, BIT Numer. Math., 37(2), 256-295 (1997); “On summation formulas 
+due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, II”, BIT Numer. Math., 
+37(4), 804-832 (1997); “On Summation Formulas Due to Plana, Lindelöf and Abel, and 
+Related Gauss-Christoffel Rules, III”, BIT Numer. Math. 39, 51–78 (1999).  
+[5] 
+H. M. Edwards, Riemann’s Zeta Function, Academic Press, Boston, (1974). 
+[6] 
+I. González, V. H. Moll, I. Schmidt, “A generalized Ramanujan Master Theorem applied to 
+the evaluation of Feynman diagrams”, Adv. Appl. Math., 63, 214-230 (2015). 
+[7] 
+I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic 
+Press, (1980). 
+[8] 
+G. H. Hardy, Ramanujan, Cambridge University Press, (1940). 
+[9] 
+E. Lindelöf, Le Calcul des Résidues et ses Applications à la Theorie des Fonctions, 
+Gauthier-Villars, Paris, (1905). 
+[10] D.S. Mitrinovic and J. D. Keckic, The Cauchy Method of Residues, D. Reidel Publ. Co., 
+Dordrecht/Boston, (1984). 
+[11] F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, New York,1974. 
+ 
+
+Khristo N. Boyadzhiev 
+[12] Gian-Carlo Rota, Finite Operator Calculus, Academic Press, new York, (1975). 
+[13] D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, (1971). 
+[14] K. S. Williams and Z. Nan-Yue, “Special values of the Lerch zeta function and the 
+evaluation of certain integrals”, Proc. Amer. Math. Soc., 119, 35-49 (1993). 
+[15] W. Heap, Notes on the gamma function and the Riemann zeta function (online publication). 
+https://wiki.math.ntnu.no/_media/ma3001/2014v/analytisktallteori/the_riemann_zeta_functi
+on_notes.pdf 
+ 
+ 
+ 
+

UtAzT4oBgHgl3EQf0_7l/content/tmp_files/load_file.txt

@@ -0,0 +1,194 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf,len=193
+page_content='MATHEMATICA MONTISNIGRI  2010 Mathematics Subject Classification:  30B10, 11M05, 11M41, 33B15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Key words and Phrases: Riemann’s zeta function, Lerch transcendent, polylogarithm, digamma function,  Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  SEVERAL CLASSICAL IDENTITIES VIA MELLIN’S TRANSFORM  KHRISTO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' BOYADZHIEV  Department of mathematics, Ohio Northern University  Ada, Ohio, 45810, USA  E-mail: k-boyadzhiev@onu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='edu  DOI:  Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We present a summation rule using Mellin’s transform to give short proofs of  some important classical relations between special functions and Bernoulli and Euler  polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' For example, the values of the Hurwitz zeta function at the negative integers are  expressed in terms of Bernoulli polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We also show identities involving exponential  and Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' INTRODUCTION  Throughout we use the notation ( )  ,  a  a  it t  =  +  ∈\uf0a1  for the vertical line with abscissa  0  1  a  <  < , oriented from minus to plus infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' First we recall the formulas for the Mellin  transform  1  0  ( )  ( )  s  G s  x  g x dx  ∞  −  = ∫  and its inverse  ( )  1  ( )  ( )  ,  0  2  s  a  g x  x G s ds  x  i  π  −  =  >  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  (1)  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' For  0  n >  consider integrals of the form  ( )  ( )  ,  0  s  L n x G s ds  x  −  >  ∫  where ( )  L n  consists of the line segment [  ,  ]  ni ni  −  together with the semicircle  ( )  R n  in the left  half plane for which the line segment is the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' If the integrals  ( )  ( )  s  R n x G s ds  −  ∫  approach zero when n → ∞, we say that the line of integration in (1) can be closed to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  In a similar manner we define integrals where the line of integration can be closed to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Suppose that the function  ( )  G s  is meromorphic on the half plane  ( )  Re s  a  ε  <  +  for some small  0  ε >  , and has only simple poles at  0, 1, 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  s =  − −  , with residues  0  1  2  ,  ,  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  c c c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  If the line of integration in (1) can be closed to the left, the residue theorem provides the  representation  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  0  ( )  n  n  n  g x  c x  ∞  =  = ∑  (2)  for the function  ( )  g x  from (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' this function is a power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' If now  ( )  f s  is an  appropriate holomorphic function on  ( )  Re s  a  ε  <  +  without poles, we can write  ( )  0  1  ( ) ( )  (  )  2  s  n  n  a  n  x  f s G s ds  c f  n x  i  π  ∞  −  =  =  −  ∑  ∫  ,  (3)  when the power series on the right side converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Formulas of this type are used for summation of series or interpolation, and are present in  many publications (see [2, 4, 9, 10] and the references there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  In this note we focus on a special area of applications for the proposition - obtaining some  classical identities by using Mellin inversion and comparing coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' In order to keep the  paper short we omit details and do not discuss convergence of some integrals and series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The  validity of such formulas is considered in [4, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  The illustration of the method is given in the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  2 EXAMPLES  Remind that the residues of the gamma function at zero and the negative integers are given  by  ( 1)  Res( ,  )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  n  n  −  Γ −  =  for  0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Also ( )s  Γ  has rapid decay on vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We have  the estimate ([15, (20)]  (  )  it  a +  Γ  ~  1  2  2  2  | |  ,  a  t  t  e  a  π  π  −  −  ∈\uf0a1  when | |t → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The estimate helps for the convergence of our integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  The above facts will be used in the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We note that in these examples the  lines of integration can be closed to the left (proofs are standard, using the growth estimate for  ( )s  Γ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Note also that when we replace  ( )  G s  by ( )s  Γ  in (3) we have  ( )  0  1  ( 1)  ( ) ( )  (  )  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  n  a  n  x  f s  s ds  f  n x  i  n  π  ∞  −  =  −  Γ  =  −  ∑  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  (4)  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Ramanujan’s Master Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' As Hardy writes in [8], Ramanujan was very fond  of his integral formula [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='186]  1  0  0  ( )(  )  (  ) ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  n  f n  x  x  dx  f  s  s  n  ∞  ∞  −  =  \uf8f1  \uf8fc  −  =  −  Γ  \uf8f2  \uf8fd  \uf8f3  \uf8fe  ∑  ∫  (5)  and used it for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Berndt rightly calls it Ramanujan’s Master Theorem [2,  Entry 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Details, comments, and applications of (5) are given in these two books and  also in [1, 4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Clearly, after replacing  ( )  f s  by  (  )  f  s  −  equation (5) turns into (4) after  Mellin inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Ramanujan did not use the residue theorem for his proof but only standard  calculus (see [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='106]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The Hurwitz zeta function is defined by  0  1  ( , )  (Re( )  0, Re( )  1)  (  )s  n  s z  z  s  n  z  ζ  ∞  =  =  >  >  +  ∑  with integral representation  (1  )  1  0  ( , ) ( )  (Re( )  1)  1  t  z  s  t  e  s z  s  t  dt  s  e  ζ  ∞  −  −  Γ  =  >  −  ∫  When  1  z = ,  ( ,1)  ( )  s  s  ζ  ζ  =  is the Riemann zeta function [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  We have also the modified integral representation (argument is the same as on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 61-62 in  [13])  (1  )  1  0  1  ( , ) ( )  (0  Re( )  1)  1  t  z  s  t  e  s z  s  t  dt  s  e  t  ζ  ∞  −  − \uf8eb  \uf8f6  Γ  =  <  <  −  \uf8ec  \uf8f7  −  \uf8ed  \uf8f8  ∫  which is a Mellin transform formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' By Mellin inversion  (1  )  ( )  0  1  1  ( 1)  ( , ) ( )  (  , )  1  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  t  z  n  s  t  a  n  e  t  s z  s ds  n z  e  t  i  n  ζ  ζ  π  −  ∞  −  =  −  −  =  Γ  =  −  −  ∑  ∫  At the same time, the Bernoulli polynomials  ( )  n  B z  have the generating function  0  ( )  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  xz  k  k  x  k  B z  xe  x  e  k  ∞  =  =  −  ∑  and from this  (1  )  1  1  1  0  (1  )  (1  )  1  ( , )  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  (  1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  x  z  k  n  k  n  x  k  n  B  z  B  z  e  g x z  x  x  e  x  k  n  −  ∞  ∞  −  +  =  =  −  −  ≡  −  =  =  −  +  ∑  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Therefore, by comparing coefficients for  0, 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n =  we find the classical formula  1  1  ( 1)  (1  )  ( )  (  , )  1  1  n  n  n  B  z  B  z  n z  n  n  ζ  +  +  −  −  −  =  = −  +  +  (using the property ( 1)  (1  )  ( )  n  n  n  B  z  B z  −  −  =  , so that  1  1  ( 1)  (1  )  ( )  n  n  n  B  z  B  z  +  +  −  −  = −  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  In particular,  1  1  (0, )  ( )  2  z  B z  z  ζ  = −  =  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  For the Bernoulli numbers  (0)  ( 1)  (1)  n  n  n  n  B  B  B  =  = −  we have  1  1  ( 1)  (0)  ( 1)  (  )  (  ,1)  1  1  n  n  n  n  B  B  n  n  n  n  ζ  ζ  +  +  −  −  −  =  −  =  =  +  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  The odd Bernoulli numbers are zeros except  1  1/ 2  B = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Thus  (0)  1/ 2  ζ  = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  Next we give a new proof of a result of Kenneth Williams and Zhang Nan-Yue [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Consider now the alternating Hurwitz zeta function  0  ( 1)  ( , )  (Re( )  0, Re( )  0)  (  )  n  s  n  s z  z  s  n  z  η  ∞  =  −  =  >  >  +  ∑  which extends to the entire complex plane as analytic in the variable s  and has the integral  representation  (1  )  1  0  ( , ) ( )  (Re( )  0)  1  t  z  s  t  e  s z  s  t  dt  s  e  η  ∞  −  −  Γ  =  >  +  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  By inversion  (1  )  ( )  0  1  ( 1)  ( , ) ( )  (  , )  1  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  t  z  n  s  n  t  a  n  e  t  s z  s ds  n z x  e  i  n  η  η  π  −  ∞  −  =  −  =  Γ  =  −  +  ∑  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Euler’s polynomials  ( )  n  E  z  are defined by the generating function  0  2  ( )  (| |  )  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  t x  n  n  t  n  e  t  E  x  t  e  n  π  ∞  =  =  <  +  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Comparing coefficients gives  ( 1)  1  (  , )  (1  )  ( )  2  2  n  n  n  n z  E  z  E  z  η  −  −  =  −  =  by the property  (1  )  ( 1)  ( )  n  n  n  E  z  E  z  −  = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Euler worked with the function  0  ( 1)  ( )  (Re  0)  (2  1)  n  s  n  L s  s  n  ∞  =  −  =  >  +  ∑  which we call here Euler’s L -function (sometimes it is called Dirichlet’s L -function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' This  function has the integral representation  1  0  2 ( ) ( )  cosh  sx  s L s  dx  x  ∞  −  Γ  = ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  It also has analytic extension on the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  By Mellin inversion and using equation (4)  ( )  0  1  1  ( 1)  ( ) ( )  (  )  2cosh( )  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  n  a  n  x L s  s ds  L  n x  x  i  n  π  ∞  −  =  −  =  Γ  =  −  ∑  ∫  Euler’s numbers  n  E  are defined by the generating function  0  1  cosh  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  n  n  E x  x  n  ∞  =  = ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  This function is even, so the Euler numbers with odd indices are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' By comparing  coefficients we find  2  1  ( 2 )  (  0,1, 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=')  2  n  L  n  E  n  −  =  =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The exponential polynomials  n  ϕ  are defined by the generating function  (  1)  0  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  x  n  z e  n  n  x  e  z n  ϕ  ∞  −  =  = ∑  (see [3, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We will use the function  (  1)  0  ( , )  ( 1)  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  x  n  z e  n  n  n  x  x z  e  z n  ψ  ϕ  −  ∞  −  =  =  =  −  ∑  which has Mellin transform  1  1  0  0  ( , )  ( , )  s  z  s  x  ze  s z  x  x z dx  e  x  e  dx  ψ  ∞  ∞  −  −  −  −  Ψ  =  =  ∫  ∫  1  0  0  0  ( )  ( ) ( , )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  n  z  s  z  s  n  n  nx  z  z  e  x  e  dx  e  s  s f s z  n  n n  ∞  ∞  ∞  −  −  −  −  =  =  \uf8f1  \uf8fc  =  =  Γ  = Γ  \uf8f2  \uf8fd  \uf8f3  \uf8fe  ∑  ∑  ∫  with  0  ( , )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  z  s  n  z  f s z  e  n n  ∞  −  =  =  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Now we have from equation (4)  ( )  ( )  0  1  1  ( 1)  ( , )  ( , )  ( , ) ( )  (  , )  2  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  s  n  a  a  n  x z  x  x s ds  x  f s z  s ds  f  n z x  i  i  n  ψ  π  π  ∞  −  −  =  −  =  Ψ  =  Γ  =  −  ∑  ∫  ∫  Thus for  0  n ≥  (with the agreent  00  1  = )  0  ( )  (  , )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  z  k  n  k  k  z  f  n z  e  z  k  ϕ  ∞  −  =  =  −  =  ∑  (6)  which is one of the fundamental properties of the exponential polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Note that identity (6) can be used for a meaningful extension of  ( )  n z  ϕ  to  ( )z  λ  ϕ  with non-  integer index λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' For instance, we have (with 0  0  λ =  )  1  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  z  k  k  k  z  e  z  k  λ  λ  ϕ  ∞  −  =  =  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The generating function for the Hermite polynomials is  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  2  2  0  ( , )  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  xz  x  n  n  x  x z  e  H  z n  ψ  ∞  −  =  =  = ∑  with Mellin transform [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 27]  2  /2  2  ( , )  2  ( 2 ) ( )  z  s  s  s z  e  D  z  s  −  −  Ψ  =  Γ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  where  p  D  are the parabolic cylinder functions [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 1065-1067].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Let now  2  /2  2  ( , )  2  ( 2 )  z  s  s  f s z  e  D  z  −  −  =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Then  ( )  ( 1)  (  , )  n  n  H  z  f  n z  = −  −  and since  ( )  ( 1)  (  )  n  n  n  H  z  H  z  = −  −  one finds the classical result  2  2  2  ( )  2  ( 2 )  n  z  n  n  H  z  e D  z  =  ([7, entry 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='253, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='1067]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' CONCLUSIONS  In this note we presented a rule how Mellin’s transform can be used to give short proofs of  several classical identities connecting, in particular, the Bernoulli and Euler polynomials to  the values of the Hurwitz and alternating Hurwitz functions at the negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We also  proved identities for the exponential and Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
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+A reappraisal of the principle of equivalent time based on
+physicochemical methods
+M. Rufinoa, A.L. Lixandrão-Filhoa and S. Guedesa,∗
+aDepartamento de Raios Cósmicos e Cronologia, Grupo de Cronologia, Instituto de Física “Gleb Wataghin", Universidade Estadual de
+Campinas, R. Sérgio Buarque de Holanda, 777 - Cidade Universitária, Campinas - SP, 13083-859, Brazil
+A R T I C L E I N F O
+Keywords:
+Fission-track thermochronology
+Equivalent time
+Effective rate constant
+Physicochemical techniques
+A B S T R A C T
+The main feature of the Fission-Track Thermochronology is its ability to infer the thermal histo-
+ries of mineral samples in regions of interest for geological studies. The ingredients that make
+the thermal history inference possible are the annealing models, which capture the annealing ki-
+netics of fission tracks for isothermal heating experiments, and the Principle of Equivalent Time
+(PET), which allows the application of the annealing models to variable temperatures. It turns
+out that the PET only applies to specific types of annealing models describing single activation
+energy annealing mechanisms (parallel models). However, the PET has been extensively applied
+to models related to multiple activation energy mechanisms (fanning models). This procedure is
+an approximation that has been overlooked due to the lack of a suitable alternative. To deal with
+this difficult, a formalism, based on physicochemical techniques, that allows to quantify the ef-
+fects of annealing on the fission tracks for variable temperatures, is developed. It is independent
+of the annealing mechanism and, therefore, is applicable to any annealing model. In the cases
+in which the PET is valid, parallel models, the proposed method and the PET predict the same
+degrees of annealing. However, deviations appear when the methods are applied to the fanning
+models, with the PET underestimating annealing effects. The consequences for the inference of
+thermal histories are discussed.
+1. Introduction
+The Principle of Equivalent Time (PET) is one of the basic ingredients for the inference of thermal histories in
+Fission-Track Thermochronology (FTT). The PET states that the rate of track shortening due to temperature is inde-
+pendent of its previous thermal history. Thus, any thermal history may be replaced with a constant temperature heating
+for an equivalent time resulting in the current fission-track length. Since the models that constrain the annealing ki-
+netics are only applicable to constant temperature heating events, it is the PET that allows for the inference of variable
+thermal histories from the fission-track age and from the distribution of fission-track lengths measured in the sample.
+The PET was first proposed by (Goswami et al., 1984). Later on, Duddy et al. (1988) established a practical method
+of finding thermal histories, applying the PET, that has been mostly unchanged since then. They also demonstrated
+that the PET is only valid for the single activation energy Arrhenius annealing equation. Such equation is represented
+as parallel straight isoretention (same fission-track length) curves on the pseudo-Arrhenius space (logarithm of time as
+a function of inverse temperature, Fig. 1a) and is called Parallel Arrhenius equation. However, they applied the PET to
+the Fanning Arrhenius (FA) model (Laslett et al., 1987), in which the isoretention curves diverge from a single point
+with different slopes, implying different activation energies. They recognized that this procedure is an approximation,
+since the PET only applies to parallel models, but argued that their fanning model deviated only slightly from a parallel
+one.
+Annealing models continued to evolve. Carlson (1990) presented a modified version of the parallel model (Fig. 1a).
+Crowley et al. (1991) proposed versions of the Parallel and Fanning equations that are curved in the pseudo-Arrhenius
+space (Fig. 1b), implying activation energies that vary with the temperature of annealing. The Fanning Curvilinear
+(FC) model has been shown to produce better geological extrapolations than the other models for the apatite (Ketcham
+et al., 2007) and zircon (Guedes et al., 2013) fission-track systems. It is currently the model of choice in most geological
+studies using the FTT (Ketcham, 2019) in thermal history codes relying on the PET to apply the annealing equations for
+∗Principal corresponding author
+[email protected] (M. Rufino); [email protected] (A.L. Lixandrão-Filho); [email protected] (S.
+Guedes)
+ORCID(s): 0000-0003-4871-5120 (M. Rufino); 0000-0002-8343-8942 (A.L. Lixandrão-Filho); 0000-0002-7753-8584 (S. Guedes)
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 1 of 15
+arXiv:2301.03401v1  [physics.geo-ph]  6 Jan 2023
+
+athe inference of thermal histories (Ketcham, 2005; Gallagher, 2012). The joint application of the PET and FC equation
+is an approximation that has been overlooked due to the lack of an alternative to deal with the variable temperature
+thermal histories.
+Recently, Rufino and Guedes (2022) applied a physicochemical technique to the Fission-Tack Arrhenius equations
+and were able to formulate the annealing kinetics in terms of the reaction rate constant, which is the fundamental
+quantity related to the activation energy (Arrhenius, 1889; Cohen et al., 2007). The reaction for the annealing process
+is the recombination of displaced atoms and vacant sites that form the track. Once the rate constant is determined for
+the annealing equations, they can be represented in the Arrhenius space (logarithm of the rate constant as a function
+of the inverse temperature) and their trends can be used to retrieve the general mechanisms underlying the Arrhenius
+models. The rate constant encodes the most fundamental features of annealing. Once it is determined, the shortening
+of the fission tracks may in principle be quantified not only for constant temperatures but also for varying ones, being
+an alternative to the PET, without the single activation energy restriction.
+The Arrhenius annealing equations can be derived from the rate constant for the case of constant temperature an-
+nealing (Rufino and Guedes, 2022). The next step is to apply the physicochemical approach to the variable temperature
+annealing of the fission tracks and compare the results to the ones obtained with the PET. The length shortening of
+fission tracks is calculated for cooling temperature-time (T-t) paths with different slopes using the parallel, fanning and
+Carlson models as they are the representations of different activation energy mechanisms. The temperature indexes
+Closure and Total Annealing temperatures, calculated using the PET and the rate constant techniques, are presented
+and compared to illustrate the differences between both approaches for the geological extrapolation.
+2. Method
+2.1. The physicochemical perspective of fission track annealing
+The kinetics of chemical reactions can be described by the Arrhenius equation (Arrhenius, 1889), which relates
+the temperature derivative of the reaction rate 푘, the universal gas constant 푅 and a constant 푞, related to a change in
+the standard internal energy (Laidler, 1984, p.494):
+d ln 푘(푇 )
+d푇
+=
+푞
+푅푇 2 .
+(1)
+Eq. (1) can be solved for the reaction rate as a function of temperature 푘(푇 ) using a pre-exponential factor 퐴 and
+the Arrhenius activation energy 퐸푎:
+푘(푇 ) = 퐴 exp (−퐸푎∕푅푇 ) .
+(2)
+Chemical processes that obey Eq. (2) result in straight lines with slope −퐸푎∕푅 in Arrhenius plots (ln 푘 × 1∕푇 ).
+Among the fission-track annealing models, the Parallel Arrhenius is the only one that actually fits this formulation
+of a single constant activation energy. Deviations from Eq. (2) are quite common. To enable a more complete study
+of chemical reactions, the International Union of Pure and Applied Chemistry (IUPAC) has defined the Arrhenius
+activation energy (Cohen et al., 2007):
+퐸푎 = −푅 d ln(푘)
+d(1∕푇 ).
+(3)
+The Arrhenius activation energy, as defined by Eq. (3), is an empirical quantity aimed to be a kinetic parameter that
+can vary with the temperature of the reaction medium. Its determination depends on the previous knowledge of the rate
+constant, the quantity that encodes the reaction kinetics. Thus, for application to the fission-track system, the reaction
+rate constant associated with the annealing mechanisms must be found, which can be done using the formalism of
+studies in solid state processes (Vyazovkin, 2015).
+The annealing kinetics of fission tracks is described by empirical (Laslett et al., 1987; Crowley et al., 1991; Laslett
+and Galbraith, 1996; Rana et al., 2021) or semi-empirical (Carlson, 1990; Guedes et al., 2006, 2013) equations relating
+the reduced track length, 푟 = 퐿∕퐿0 (where 퐿 is the length of the fission track after heating and 퐿0 is the unannealed
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 2 of 15
+
+fission-track length), with the duration, 푡, of the constant temperature (푇 ) heating. The general form of the annealing
+equations is:
+푔(푟) = 푓(푡, 푇 ),
+(4)
+in which 푔(푟) is a transformation of 푟 and 푓(푡, 푇 ) defines the geometrical characteristics of the isoretention curves
+in the pseudo-Arrhenius space (ln 푡 × 1∕푇 , Fig. 1). The Parallel Arrhenius (PA, Eq. (PA1)) and Fanning Arrhenius
+(FA, Eq. (FA1)) equations (Laslett et al., 1987), the Parallel Curvilinear (PC, Eq. (PC1)) and Fanning Curvilinear
+(FC, Eq. (FC1)) models (Crowley et al., 1991) as well as the Carlson Model (CM, Eq. (CM1)) that mixes the Parallel
+Arrhenius and Parallel Curvilinear models in the same equation (Carlson, 1990), are used in this analysis. The trans-
+formation function 푔(푟) = ln(1 − 푟) was chosen because it carries no fitting parameters and was shown to produce
+good fits to annealing data (Guedes et al., 2022). In addition, it arises naturally from the physicochemical formulation
+of the fission track annealing (Rufino and Guedes, 2022), as will be shown below. More comprehensive descriptions
+of the annealing models can be found elsewhere (Carlson, 1990; Ketcham, 2019; Guedes et al., 2022).
+(a)
+(b)
+Figure 1: Representation of the Arrhenius fission-track annealing models in the pseudo-Arrhenius plot.
+(a) Fanning
+Arrhenius, Parallel Arrhenius, and Carlson models. (b) Fanning Curvilinear and Parallel Curvilinear models. Laboratory
+annealing data are c-axis projected reduced fission-track lengths from Durango apatite (Carlson et al., 1999). Data from
+the geological benchmark KTB (Wauschkuhn et al., 2015) are included only for reference. The models are represented as
+isoretention curves. Points on these curves are the temperature and time of constant temperature heating resulting in the
+same reduced length.
+The annealing data set on the c-axis projected fission tracks for Durango apatite (Carlson et al., 1999) was used for
+model fitting. Durango apatite annealing data was chosen because Durango is a well-known standard sample often used
+in methodological studies (Green et al., 1986; Carlson, 1990; Ketcham et al., 1999, 2007; Rana et al., 2021; Guedes
+et al., 2022; Rufino and Guedes, 2022). The fitting parameters for PA, PC, FA and FC models are the same presented in
+Rufino and Guedes (2022). They were numerically determined using the function nlsLM of the package minpack.lm
+(Elzhov et al., 2016) written in R language, which applies the Levenberg– Marquardt algorithm to minimize the residual
+sum of square (RSS), using the squared inverse of 푟 uncertainties as weights. With the same method, fitting parameters
+were also obtained for the CM model. The fitting parameters are presented in the last column of Table 1.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 3 of 15
+
+40
+Model
+r = 0.55
+=0.9
+ParallelArrhenius
+In(time), time in hours
+Carlson Model
+20
+ Fanning Arrhenius
+88880
+0
+Length reduction exp. data
+-20
+r> 0.9
+A
+0.8 < r <0.9
+0.7 < r < 0.8
+-40
+口
+0.6
+5 < r<0.7
+0
+1
+2
+3
+4
+1000/T, K-140
+r = 0.55
+Model
+r = 0.9
+Parallel Curvilinear
+In(time), time in hours
+AM
+Fanning Curvilinear
+20
+88880
+0
+Length reduction exp. data
+-20
+r> 0.9
+A
+0.8 < r < 0.9
+0.7 < r < 0.8
+-40
+口
+0.6
+5 < r<0.7
+0
+1
+2
+3
+4
+1000/T, K-1Table 1
+Effective reaction rate constant, 푟 reduction and equivalent time equations associated with
+the fission-track annealing models
+FT models
+Equations
+Parameters (standard error)
+Parallel Arrhenius
+(PA)
+푓푃퐴(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐2
+푅푇
+(PA1)
+푘푒푓(푇 )푃퐴 = 푐1푒푐0∕푐1 exp
+(푐2∕푐1
+푅푇
+)
+(PA2)
+퐸푃퐴
+푎
+= −푐2
+푐1
+(PA3)
+푐0 = 5.631 (0.220)
+푐1 = 0.1865 (0.0066)
+푐2 = -10.46 (0.31) kcal/mol
+휒2
+휈 = 2.65
+Parallel curvilinear
+(PC)
+푓푃퐶(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐2 ln
+( 1
+푅푇
+)
+(PC1)
+푘푒푓(푇 )푃퐶 = 푐1푒푐0∕푐1 (푅푇 )−푐2∕푐1
+(PC2)
+퐸푃퐶
+푎 (푇 ) = −푐2
+푐1
+푅푇
+(PC3)
+푐0 = -4.910 (0.096)
+푐1 = 0.1944 (0.0060)
+푐2 = -9.610 (0.244)
+휒2
+휈 = 2.12
+Carlson Model
+(CM)
+푓퐶푀(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐1 ln(푅푇 ) + 푐2
+푅푇
+(CM1)
+푘푒푓(푇 )퐶푀 = 푐1푒푐0∕푐1 exp
+(푐2∕푐1
+푅푇
+)
+푅푇
+(CM2)
+퐸퐶푀
+푎
+(푇 ) = −푐2
+푐1
++ 푅푇
+(CM3)
+푐0 = 5.426 (0.2155)
+푐1 = 0.1867 (0.0066)
+푐2 = -10.25 (0.2994) kcal/mol
+휒2
+휈 = 2.63
+Fanning Arrhenius
+(FA)
+푓퐹퐴(푡, 푇 ) = 푐0 + 푐1
+ln(푡) − 푐2
+1
+푅푇 − 푐3
+(FA1)
+푘푒푓(푡, 푇 )퐹퐴 =
+푐1 exp[(1 − 푛)푓퐹퐴(푡, 푇 )]
+푡(1 − 푐3푅푇 )
+(FA2)
+퐸퐹퐴
+푎 (푡, 푇 ) =
+(푅푇 )2 [푐1(푛 − 1)(푐2 − ln 푡) − 푐3 + 1∕푅푇 ]
+(푐3푅푇 − 1)2
+(FA3)
+푐0 = -8.518 (1.072)
+푐1 = 0.1266 (0.0191) mol/kcal
+푐2 = -20.99 (5.81)
+푐3 = 0.2985 (0.1026) mol/kcal
+휒2
+휈 = 1.66
+0.5 ≤ 푛 < 1
+Fanning curvilinear
+(FC)
+푓퐹퐶(푡, 푇 ) = 푐0 + 푐1
+ln(푡) − 푐2
+ln
+(
+1
+푅푇
+)
+− 푐3
+(FC1)
+푘푒푓(푡, 푇 )퐹퐶 =
+푐1 exp[(1 − 푛)푓퐹퐶(푡, 푇 )]
+푡
+(
+ln
+(
+1
+푅푇
+)
+− 푐3
+)
+(FC2)
+퐸퐹퐶
+푎 (푡, 푇 ) =
+푅푇
+(
+푐1푐2푛 − 푐1푐2 + (푐1 − 푐1푛) ln(푡) − 푐3 + ln
+(
+1
+푅푇
+))
+(
+푐3 − ln
+(
+1
+푅푇
+))2
+(FC3)
+푐0 = -9.449 (1.480)
+푐1 = 0.1627 (0.0298)
+푐2 = -24.58 (7.75)
+푐3 = -0.8626 (0.1549)
+휒2
+휈 = 1.88
+0.5 ≤ 푛 < 1
+Notes: 1. For each fission-track annealing model (Eqs. (PA1), (PC1), (CM1) (FA1), (FC1)), the reaction rate constants,
+푘푒푓 and Arrhenius Activation energies were obtained using 푔(푟) = ln(1 − 푟) and 푓푟(푟) = (1 − 푟)푛. 2. Rate constants (Eqs.
+(PA2), (PC2), (CM2) (FA2), (FC2)) were calculated after Eq. (15). 3. Arrhenius activation energies (Eqs. (PA3), (PC3),
+(CM3) (FA3), (FC3)) were obtained by the application of Eq. (3), and are average values for constant heating
+experiments.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 4 of 15
+
+Fission tracks are formed by displaced atoms and vacant sites, in concentrations high enough to change the structure
+of the mineral in a volume of about 2-10 nm in diameter and around 20 휇m in length. The annealing process is the
+recombination of defects and vacancies, which also changes the neighbor structure and consequently the recombination
+rate. This kind of solid-state reaction is described by the conversion rate equation (Vyazovkin, 2015):
+d훼
+d푡 = 푘(푇 )푓훼(훼).
+(5)
+The Eq. (5) relates the rate of conversion of the reactant 훼 with the constant rate and with the reaction function
+푓훼(훼). For the fission tracks, 훼 is the concentration of recombined atoms, and 푓훼(훼) describes how the recombination
+process changes the surrounding structure. The track length can be used as a proxy for the concentration of displaced
+atoms (Rufino and Guedes, 2022) and:
+훼 = 퐿0 − 퐿
+퐿0
+= 1 − 푟
+(6)
+and with this change of variable:
+d푟
+d푡 = −푘푒푓(푡, 푇 )푓푟(푟).
+(7)
+The rate constant has been replaced with an effective rate constant, 푘푒푓(푡, 푇 ), which may depend on time and
+temperature and is suitable to describe more complex reactions (Vyazovkin, 2016). For the reaction function, the
+reaction-order function has already been shown to produce consistent results mainly for the single activation energy
+mechanisms of annealing (Green et al., 1988; Rufino and Guedes, 2022):
+푓푟 = (1 − 푟)푛
+(8)
+in which 푛 is the reaction order.
+Eq. (7) is a differential equation that can be solved by the separation of variables. To define the limits of the integral,
+consider that at the beginning of the thermal history (푡 = 0), the track is unannealed (푟 = 1). After a heating duration
+푡, the track length has been shortened to 푟. Then:
+∫
+1
+푟
+d푟
+푓푟(푟) = ∫
+푡
+0
+−푘푒푓(푡, 푇 )d푡.
+(9)
+Eq. (9) is the basic equation from which the annealing kinetics can be studied from a physicochemical perspective.
+Once the reaction function and the rate constant are chosen, the dependence of the reduced fission-track length can be
+calculated over any T-t path. Let’s start with the known case of constant temperature heating, from which the annealing
+equations should be obtained. For the single activation energy models, PA, PC, and CM, the rate constants are given
+by:
+푘푒푓(푇 )푃 퐴 = 퐴1 exp
+(−푄1
+푅푇
+)
+,
+(10a)
+푘푒푓(푇 )푃 퐶 = 퐴2(푅푇 )푚,
+(10b)
+푘푒푓(푇 )퐶푀 = 퐴3(푅푇 ) exp
+(
+− 푄3
+푅푇
+)
+,
+(10c)
+where 퐴푖, 푄푖, and 푚 are constants. Eq. (10a) is the original Arrhenius equation from which the PA equation is derived.
+푄1 can be directly identified with the activation energy only in this case. Eq. (10b) generates the PC equation with
+a temperature-dependent activation energy (Table 1, Eq. (PC3)). Eq. (10c) generates the Carlson Model, also with a
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 5 of 15
+
+temperature-dependent activation energy (Table 1, Eq. (CM3)). It is the product of Eqs. (10a) and (10b), with 푚 = 1,
+and has been proposed soon after the original Arrhenius equation to deal with reactions that deviate from the expected
+Arrhenius behavior (Kooij, 1893). Note that although the activation energies in the Eqs. (10b) and (10c) depend on
+temperature, they still fall into the category of single activation energy processes, meaning that all recombination events
+at a given temperature have the same activation energy.
+Annealing experiments are isothermal heating procedures. Then, substituting the effective rate constants (Eqs. (10))
+into the integral equation (Eq. 9) together with the reaction-order function defined in Eq. (8) and solving it considering
+the temperature as a constant results in:
+ln(1 − 푟) =
+ln [퐴1(1 − 푛)]
+1 − 푛
++
+1
+1 − 푛 ln(푡) −
+푄1
+1 − 푛
+1
+푅푇 ,
+(11a)
+ln(1 − 푟) = ln[퐴2(1 − 푛)]
+1 − 푛
++
+1
+1 − 푛 ln(푡) −
+푚
+1 − 푛 ln
+( 1
+푅푇
+)
+,
+(11b)
+ln(1 − 푟) =
+ln [퐴3(1 − 푛)]
+1 − 푛
++
+1
+1 − 푛 ln(푡) −
+푄3
+1 − 푛
+1
+푅푇 −
+1
+1 − 푛 ln
+( 1
+푅푇
+)
+,
+(11c)
+which are the equations for the PA (11a), PC (11b), CM (11c) models with 푔(푟) = ln(1 − 푟). For the chosen reaction
+function, the integral only has a real solution if 푛 < 1. Comparing the right sides of these equations respectively with
+Eqs. (PA1), (PC1), and (CM1), one can find out that the rate constant parameters are related to the fitting parameters
+of the annealing equations as
+PA ∶ 푛 = 푐1 − 1
+푐1
+푄1 = −푐2
+푐1
+퐴1 = 푐1 exp(푐0∕푐1
+)
+(12)
+PC ∶ 푛 = 푐1 − 1
+푐1
+푚 = −푐2
+푐1
+퐴2 = 푐1 exp(푐0∕푐1
+)
+(13)
+CM ∶ 푛 = 푐1 − 1
+푐1
+푄3 = −푐2
+푐1
+퐴3 = 푐1 exp(푐0∕푐1
+)
+(14)
+In this way, the rate constants can be expressed in terms of the fitting parameters of the annealing models as shown
+in Eqs. (PA2), (PC2), and (CM2) of Table 1. The values for the reaction order 푛 for the three models are 푛 ≈ −4, in
+agreement with a similar analysis carried out by Green et al. (1988) for the PA model. Therefore, the parallel models
+are not compatible with first-order annealing kinetics, meaning that the neighbor structure has a strong influence on
+the rate of defect recombination during annealing.
+There are no obvious expressions for the rate constant for the fanning models. A physicochemical analysis of their
+trends indicates that multiple concurring processes with different activation energies are occurring during the annealing
+of the fission tracks (Rufino and Guedes, 2022), in agreement with previous suggestions (Green et al., 1988; Tamer
+and Ketcham, 2020). Rufino and Guedes (2022) derived an expression from Eq. (7) to find the effective rate constant
+from the annealing model:
+푘푒푓(푡, 푇 ) = −
+1
+푓푟(푟)
+[휕푔(푟)
+휕푟
+]−1 휕푓(푡, 푇 )
+휕푡
+|||||푇
+(15)
+Eq. (15) provides a direct way to calculate this effective reaction rate constant from the model functions that fit the
+experimental annealing data, 푓(푡, 푇 ) and 푔(푟), and from the reaction function 푓푟(푟). The partial derivative in relation to
+time is taken because the annealing models were designed to describe constant temperature experiments. As a check,
+before applying Eq. (15) to the fanning models, one can show that Eqs. (PA2), (PC2), and (CM2) are found by the
+application of Eq. (15) respectively to Eqs. (PA1), (PC1), and (CM1), with 푔(푟) = ln(1 − 푟) and 푓푟(푟) = (1 − 푟)푛.
+The same procedure can be applied to Eq. (FA1) and (FC1) to find the effective reaction rates respectively for the
+Fanning Arrhenius (Eq. (FA2)) and Fanning Curvilinear (Eq. (FC2)) models. An alternative way to infer the effective
+rate constants for the FA and FC models is departing from the hypothesis that the Arrhenius activation energies and,
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 6 of 15
+
+therefore, the rate constants are dependent on the fission-track reduced length. Then, integration of Eq. (9), on the
+isothermal condition, results in
+− ∫
+푟
+1
+d푟
+푓푟(푟)푘푒푓(푟) = ∫
+푡
+0
+d푡 = 푡.
+(16)
+It can be shown that the primitive functions that make Eq. (16) true for the FA and FC models, with 푓푟(푟) = (1−푛)푛
+and 푔(푟) = ln(1 − 푟) are the ones with the effective rate constants given by Eqs. (FA2) and (FC2). This approach also
+illustrate how the incorporation of the time in the rate constant and, therefore, in the activation energies for the fanning
+models are implied from the dependence of the activation energies on the values of 푟.
+To obtain the reaction order 푛 for the FA and FC models, the effective reaction rate constants given by Eqs. (FA2)
+and (FC2) are integrated with Eq. (9) considering constant temperatures (isothermal experiments):
+∫
+푟
+1
+d푟
+(1 − 푟)푛 = − ∫
+푡
+0
+푐1
+1
+푅푇 − 푐3
+1
+푡 exp
+⎡
+⎢
+⎢⎣
+−(푛 − 1)
+⎛
+⎜
+⎜⎝
+푐0 + 푐1
+ln 푡 − 푐2
+1
+푅푇 − 푐3
+⎞
+⎟
+⎟⎠
+⎤
+⎥
+⎥⎦
+d푡
+(17a)
+∫
+푟
+1
+d푟
+(1 − 푟)푛 = − ∫
+푡
+0
+푐1
+ln
+(
+1
+푅푇
+)
+− 푐3
+1
+푡 exp
+⎡
+⎢
+⎢
+⎢⎣
+−(푛 − 1)
+⎛
+⎜
+⎜
+⎜⎝
+푐0 + 푐1
+ln 푡 − 푐2
+ln
+(
+1
+푅푇
+)
+− 푐3
+⎞
+⎟
+⎟
+⎟⎠
+⎤
+⎥
+⎥
+⎥⎦
+d푡
+(17b)
+With the necessary condition of 푛 < 1, the solution of the integral equation (17a) is
+(1 − 푟) = (−1)−1∕(푛−1) exp
+⎡
+⎢
+⎢⎣
+푐0 + 푐1
+ln(푡) − 푐2
+1
+푅푇 − 푐3
+⎤
+⎥
+⎥⎦
+(18)
+As the solution of this equation is to represent the shortening of the fission tracks, (1 − 푟) must be a real value
+between 0 and 1, which is true only if −1∕(푛 − 1) is an even and positive integer value 2푗. Then, the values of 푛 are
+restricted to
+푛 = 2푗 − 1
+2푗
+,
+(19)
+where 푗 = 1, 2, 3, .... With this condition and 푔(푟) = ln(1 − 푟), the FA model (Table 1, Eq.(FA1)) is recovered. The
+solutions for Eqs. (17a) and (17b) are similar, differing only on the logarithm of 1∕푅푇 for the FC model instead
+of the 1∕푅푇 for the FA model, which are both constants in this case. The previous analysis holds also for the FC
+model. The values of 푛 will be fractional for FA and FC (푛 = 1∕2, 3∕4, 5∕6, 7∕8, ...), according to Eq. (19). Fractional
+reaction orders are characteristics of multiple-step reactions or some more complex kinetic mechanism, as it has been
+explained for the decomposition of acetaldehyde (Laidler et al., 1965), a well know example of fractional reaction order
+in chemistry. However, for the fission tracks, where the displaced atoms and vacant sites take the role of reactants and
+the deformed track structure is the reaction medium, explanations of the kinetics of a single reactant via a mean-
+field approximation (MFA) may not be appropriate (Córdoba-Torres et al., 2003). Thus, for the effective reaction
+rate constant 푘푒푓 of the fanning annealing models, mechanistic modeling considering the intermediate steps, i.e.,
+recognizing the reaction order of each mechanism involved in annealing, would be desirable to elucidate the meaning
+of the fractional reaction order found (Koga et al., 1992). The rate constants for FA and FC are to be viewed as effective
+equations constraining the general behavior of annealing but that does not allow the description of the specifics of the
+annealing kinetics.
+In this physicochemical framework of the fission-track annealing, the effective reaction rate constant, 푘푒푓(푡, 푇 ),
+and the reaction function, 푓푟(푟), are the fundamental building blocks from which the fission-track annealing kinetics
+can be studied. The application to the constant temperature annealing made it possible to determine the rate constant
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 7 of 15
+
+parameters from the empirically determined parameters of the annealing equations. The calculation of the Arrhenius
+activation energies (퐸푎) for different models becomes possible through Eq. (3). The Arrhenius activation energies of
+the parallel annealing models (Eq. PA3, PC3 and CM3) will be constants with respect to the variable 푡. As for the
+fanning equations, 퐸푎 will vary with time and temperature (Eq. FA3 and FC3). However, the main advantage of this
+approach is the possibility of calculating the fission-track length reduction over any T-t path using Eq. (9), without
+recurring to the interactive application of the Principle of Equivalent Time.
+2.2. Fission track annealing under variable temperature thermal histories
+Fission-track thermal history inference is based on the Principle of Equivalent Time (PET) (Goswami et al., 1984;
+Duddy et al., 1988), which is an interactive method that allows the application of isothermal annealing models to
+variable temperature T-t paths. It is detailed in Appendix A. In general, a given variable temperature thermal history
+is divided into finite time intervals Δ푡푖, centered at times 푡푖 and temperatures 푇푖. At the time interval in which the
+population was born, a first reduced length is calculated by applying the annealing equation, using the temperature of
+the T-t path and the duration of the interval (푇푖, Δ푡푖). In the next interval, at a different temperature on T-t path, the
+annealing model is used to find an equivalent time capable of producing the same length shortening of the previous
+interval but at the new temperature. A new length shortening is then calculated by applying the annealing model to
+the period of time that is the sum of the equivalent time and the length of the time interval. This procedure is repeated
+and at any given temperature, 푇푖 on the T-t path, an equivalent time, 휏푖, which reproduces the length shortening at the
+previous interval, 푟푖−1, is determined, so that the new length shortening can be calculated as if the track had been at
+the same constant temperature from the beginning. The reduced length is updated (푟푖) by calculating it as a result of
+heating at 푇푖 for the duration 휏푖 + Δ푡푖. The hypothesis that the annealing kinetics does not depend on the previous
+thermal history of the track, but only on its current length so that any previous T-t path can be replaced with a constant
+temperature heating resulting in this length, is the basis of this procedure and defines the Principle of Equivalent Time.
+This means, in practice, that the track will have no memory of the material conditions of time and temperature of its
+previous shortenings. The equations for the application of the PET with the PA, PC, CM, FA, and FC models can be
+found in Table A1 in Appendix A.
+The physicochemical tool presented in the previous section provides an alternative way to access variable temper-
+ature annealing kinetics by solving the integral in the right side of Eq. (9) over a T-t path. Eq. (9) is solved as a line
+integral. A suitable parameterization is:
+푠 =
+{
+푇 = 푇 (푢)
+푡 = 푢
+,
+d푠 =
+√
+1 +
+(d푇
+d푢
+)2
+d푢.
+(20)
+Implementing the parameterized variables on the right side of the integral equation (Eq. 9),
+퐼 = ∫
+푡
+0
+푘푒푓 (푡(푢), 푇 (푢))
+d푠
+√
+1 +
+(
+d푇
+d푢
+)2
+⟹ 퐼 = ∫
+푡
+0
+푘푒푓 (푡(푢), 푇 (푢)) d푢.
+(21)
+Solving the left side of Eq. (9) for the 푓푟(푟) function given by Eq. (8) and the parameterized integral for the rate
+constant (Eq. (21)), the reduced length, after the track has experienced the thermal history given by the T-t path, is
+푟 = 1 −
+(
+(1 − 푛) ∫
+푡
+0
+푘푒푓 (푡(푢), 푇 (푢)) d푢
+)1∕1−푛
+,
+(22)
+in which 푛 < 1 as usual. At a first glance, the advantage of the Rate Constant path Integral (RCI, Eq. (22)) is that it is a
+one-shot calculation of the reduced track length. In addition, there was no need to restrict the form of the rate constant
+function and therefore the annealing mechanism it is related to.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 8 of 15
+
+3. Results and Discussion
+The RCI Eq. (22) can be applied to calculate the shortening in the reduced length of a single fission-track population
+submitted to any T-t path. The same calculation can be carried out using the interactive technique based on the Principle
+of Equivalent Time (PET). To compare the outcomes of the two methods, both calculations will be performed for the
+parallel (including CM) and fanning models. The case will be made for the linear cooling, 푇 (푡) = 푇0 − ̇푇 푡, where ̇푇
+is the cooling rate and 푇0 is the temperature at the time the track was generated. The temperature at the end of the T-t
+path (present time) was fixed to be 20 ◦C (293.15 K) for this analysis.
+3.1. Parallel models
+To solve the RCI, the effective rate constant functions for the parallel models (Table 1, Eqs. (PA2), (PC2), and
+(CM2)) are inserted in Eq. (22) with the variable 푇 replaced with 푇 (푡) = 푇0 − ̇푇 푡 wherever it appears. The analytical
+solutions for the reduced length shortening calculated for the PA, PC, and CM are
+푟푃 퐴 = 1 −
+⎛
+⎜
+⎜
+⎜
+⎜⎝
+푒푐0∕푐1
+(
+푐2Ei
+(
+푐2
+푐1푅(푇0− ̇푇 푡)
+)
+− 푐2Ei
+(
+푐2
+푐1푅푇0
+)
++ 푐1푅
+(
+( ̇푇 푡 − 푇0)푒
+푐2
+푐1푅(푇0− ̇푇 푡) + 푇0푒
+푐2
+푐1푅푇0
+))
+푐1 ̇푇 푅
+⎞
+⎟
+⎟
+⎟
+⎟⎠
+푐1
+,
+(23a)
+푟푃 퐶 = 1 −
+⎛
+⎜
+⎜
+⎜
+⎜⎝
+푐1푒푐0∕푐1
+(
+( ̇푇 푡 − 푇0)(푅(푇0 − ̇푇 푡))
+− 푐2
+푐1 + 푇0(푅푇0)
+− 푐2
+푐1
+)
+푐1 ̇푇 − 푐2 ̇푇
+⎞
+⎟
+⎟
+⎟
+⎟⎠
+푐1
+,
+(23b)
+푟퐶푀 = 1 − 2−푐1푒푐0 푐−2푐1
+1
+( ̇푇 푅)푐1
+[
+푐1푅
+(
+푇0푒
+푐2
+푐1푅푇0 (푐1푅푇0 + 푐2) − (푇0 − ̇푇 푡)푒
+푐2
+푐1푅푇0−푐1 ̇푇 푅푡 (푐1푅(푇0 − ̇푇 푡) + 푐2)
+)
++푐2
+2
+(
+Ei
+(
+푐2
+푐1푅푇0 − 푐1푅푡 ̇푇
+)
+− Ei
+(
+푐2
+푐1푅푇0
+))]푐1
+,
+(23c)
+where Ei is the exponential integral function. Eqs. (23a) - (23c) give the resulting reduced length 푟 for the parallel
+models, as functions of the three variables that characterize the thermal history: the duration of the T-t path (푡), the
+cooling rate ( ̇푇 ), and the temperature at the time when the track was born (푇0). The parameters 푐푖 are given in the last
+column of Table 1. The values of 푟 for the cooling path with the cooling rate ̇푇 = 1.0◦C/Ma calculated with the three
+parallel models are presented in Fig. 2. For each point, the value of 푟 is the length reduction after a linear cooling
+duration 푡 and measured in the present. Values of 푟 = 0 mean that the tracks have been erased before the present.
+Values obtained by the RCI solutions (Eqs. (23a) - (23c)) are represented as red curves marked with red circles and the
+values calculated using the PET are represented as blue curves marked with blue squares. RCI and PET calculations
+produce very close values of 푟 for the three parallel models (Figs. 2a, 2b, 2c).
+The temperature indexes Closure Temperature (푇퐶) and Total Annealing Temperature (푇퐴) were also calculated
+for the three parallel models, applying both methods of calculation, for cooling T-t paths with cooling rates of 1, 10,
+and 100 ◦C/Ma. 푇퐶 is, for a monotonic cooling thermal history, the temperature at the apparent sample age (Dodson,
+1973). 푇퐴 is the age of the oldest track that has not been erased and can be counted in the sample (Issler, 1996).
+Details for the method of calculation of the index temperatures can be found in Guedes et al. (2013). 푇퐶 and 푇퐴 are
+meaningful quantities that allow quantifying the impact of using the RCI instead of the interactive PET calculation.
+The uncertainties in 푇퐶 and in 푇퐴 were estimated by simple error propagation of apparent (푇퐶) or retention (푇퐴) ages
+and present time temperature. Results are shown in Table 2. Setting the PET results as the reference values, given
+that PET is the method established in the literature, a relative error analysis can be carried out to verify the internal
+consistency between PET and RCI calculations. The relative error between PA PET and PA RCI is on average 0.69%
+for 푇퐶 and 1.19% 푇퐴. The same trend is found for calculations of 푇퐶 and 푇퐴 with PC and CM: 0.66% and 0.58% (PC)
+and 0.7% and 1.19% (CM). All errors are well inside the estimated uncertainties for temperature index values and are
+most probably artifacts of the PET numerical calculation.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 9 of 15
+
+,
+(a)
+(b)
+(c)
+Figure 2: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.0◦C/Ma) starting at time 푡 and ending at
+the present time at a fixed temperature of 20◦C, calculated using the Parallel models: (a) Arrhenius, (b) Curvilinear, and
+(c) Carlson). The points that form the curve in red (circle marks) were calculated by applying the RCI (Eq. 23a, 23b, and
+23c). The values calculated using the PET are in blue (square marks).
+Table 2
+Comparison of thermal indexes for the models presented in this work, calculated using the PET and the RCI
+Principle of Equivalent Time (PET) thermal indexes
+Fit
+푇퐶1
+푇퐶10
+푇퐶100
+푇퐴1
+푇퐴10
+푇퐴100
+PA
+130(7)
+144(7)
+158(8)
+153(8)
+168(8)
+184(9)
+PC
+105(6)
+121(6)
+139(7)
+132(8)
+151(8)
+171(9)
+CM
+130(7)
+143(7)
+157(8)
+153(8)
+168(8)
+184(9)
+FA
+134(7)
+146(7)
+160(8)
+163(8)
+176(8)
+191(10)
+FC
+111(6)
+126(6)
+143(7)
+143(7)
+160(8)
+179(9)
+Rate constant integral on a path (RCI) thermal indexes
+Fit
+푇퐶1
+푇퐶10
+푇퐶100
+푇퐴1
+푇퐴10
+푇퐴100
+PA
+131(7)
+145(7)
+159(8)
+155(8)
+170(8)
+186(9)
+PC
+106(6)
+122(6)
+140(7)
+133(8)
+152(7)
+172(9)
+CM
+131(7)
+144(7)
+158(8)
+155(8)
+170(8)
+186(9)
+FA
+125(7)
+137(7)
+150(7)
+153(9)
+166(8)
+181(10)
+FC
+100(6)
+115(6)
+130(6)
+130(8)
+148(8)
+165(8)
+Notes: 1. Temperatures in ◦C. 2. The standard errors are given in parentheses. 3. 푇퐶: Closure Temperature. 4. 푇퐴:
+Total Annealing Temperature. 5. The numbers to the right of 푇퐶 and 푇퐴 are the cooling rates in ◦C/Ma.
+The PET was formulated under the hypothesis that the annealing of fission tracks is a single activation energy pro-
+cess Duddy et al. (1988). The internal consistency between PET and RCI values of 푟, 푇퐶, and 푇퐴 calculated with the
+parallel models is a check for the robustness of the physicochemical approach to deal with variable temperature thermal
+histories. It is to be noted that not only the PA model, in which the activation energy is temperature-independent (Ta-
+ble 1, Eq. (PA3)), but also the PC model, in which the activation energy is temperature-dependent (Table 1, Eqs. (FC3)),
+show such internal consistency. The same agreement is observed for CM. The CM activation energy may vary with
+temperature but, with the parameters shown in Table 1, its Arrhenius activation energy is approximately constant since
+the value of 푐2∕푐1 (54.9 kcal/mol) is much higher than typical values of 푅푇 (< 1.0 kcal/mol). Although the activation
+energies may vary with temperature, these models imply that at any given temperature, the recombination events are
+taking place with the same activation energy. This is a sufficient condition for the applicability of the PET.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 10 of 15
+
+1.0
+PA
+S0.8
+Reduction
+0.6
+Method
+Length F
+RCI
+0.4
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)1.0
+PC
+S0.8
+Reduction
+0.6
+Method
+Length F
+RCI
+0.4
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)1.0
+CM
+C0.8
+Reduction
+0.6
+Method
+Length F
+RCI
+0.4
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)3.2. Fanning models
+The values of 푟, 푇퐶, and 푇퐴 for the cooling T-t path were also calculated for the fanning models, using both the
+interactive PET and RCI methods. However, the RCI (Eq. (22)) could not be solved analytically for the FA and FC
+rate constants (Table 1, Eqs. (FA2) and (FC2)). The integrals were then solved numerically with the Wolfram Mathe-
+matica software (Wolfram-Research-Inc., 2021). For validation, the integrals for the parallel models were also solved
+numerically resulting in exactly the same values obtained with the analytical solutions (Eqs. (23a)-(23c)). Another
+feature to be considered is that there are certain fractional values allowed for the reaction order 푛, given by Eq. (19).
+The analysis will be limited to 푛 = 0.5, 푛 = 0.75 and 푛 = 0.9. The numerical method breaks down when 푛 > 0.95
+although its mathematical upper bound is 푛 < 1. The reduced length calculation results are shown in Fig. 3. Values
+calculated with the PET are shown in blue, with triangle marks, while values found by solving the RCI are shown,
+in red (푛 = 0.5), purple (푛 = 0.75), and light purple (푛 = 0.9), respectively with circle, square, and diamond marks.
+RCI 푟 curves are very close to each other but depart from the 푟 values calculated with the PET. Significant differences
+between RCI and PET 푟 values are observed for the FC (Fig. 3b) and for the FA (Fig. 3a) models.
+(a)
+(b)
+Figure 3: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.0◦C/Ma) starting at time 푡 and ending at
+the present time at a fixed temperature of 20◦C, calculated using the Fanning models: (a) Arrhenius and (b) Curvilinear.
+The points that form the curves in red (circle marks), purple (square marks), and light purple (diamond marks) were
+calculated by applying the RCI, respectively with 푛 = 0.5, 푛 = 0.75, and 푛 = 0.9. The values calculated using the PET are
+in blue.
+The values of 푇퐶 and 푇퐴 for the fanning models, calculated using the PET and the RCI (푛 = 0.5) are in Table 2. For
+the FA model, the mean relative errors between the PET and the RCI 푇퐶 and 푇퐴 calculations are respectively 6.25%
+and 5.68%. For the FC model, the same comparisons result in still more significant differences: 9.57% (푇퐶) and 8.48%
+(푇퐴). The deviations between the values calculated using the PET and the RCI are much more significant than the ones
+found for the parallel model calculations. One major issue is that the fanning models do not fulfill the single activation
+energy hypothesis on which the PET is founded. The fanning models emerge from multiple concurring processes with
+different activation energies (Tamer and Ketcham, 2020; Rufino and Guedes, 2022). The effective Arrhenius activation
+energies incorporate a time dependence (Table 1, Eqs. (FA3) and (FC3)) that is the consequence of their dependence
+on the current reduced fission-track length (different slopes of the isoretention curves in the pseudo-Arrhenius plot).
+On the other hand, the rate constant integral (Eq. (22)) was obtained in a physicochemical framework developed to
+deal with chemical reactions that did not fit the single activation energy Arrhenius law (Vyazovkin, 2015). It is by
+design suitable for complex activation energy systems like the ones pictured by the fanning models. Note also that
+the presented figures are particular of the fitting parameters in Table 1. A different set of parameters would result in
+different values without changing the conclusion that RCI and PET predictions deviate from each other.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 11 of 15
+
+1.0
+FA
+0.8
+Reduction
+0.6
+Method
+RCI (0.5)
+Length F
+RCI (0.75)
+0.4
+RCI (0.9)
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)1.0
+FC
+0.8
+Reduction
+0.6
+Method
+RCI (0.5)
+Length F
+RCI (0.75)
+0.4
+RCI (0.9)
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)3.3. Implications for the thermal history modeling
+The fanning models, especially the Fanning Curvilinear, have been shown to produce better fits to laboratory data
+and better geological extrapolation of annealing effects (Ketcham et al., 2007; Guedes et al., 2013). However, the
+application of the FC along with the PET is an approximation. Compared with the RCI formulation, it underestimates
+the annealing effect in about 10%, i.e., it predicts that higher temperatures are necessary for the same length shortening
+as calculated with the RCI for the tested cooling histories. In the context of the inverse problem of inferring T-t paths
+from the FT age and track length distribution of a mineral sample, it implies the requirement of a longer residence
+time in the partial annealing zone. For instance, compare, in Table 2, the FC 푇퐴 calculated with RCI for a cooling rate
+of 10◦C/Ma (148◦C) with the FC 푇퐴 calculated with PET for a cooling rate of 1◦C/Ma (143◦C). The same analysis
+applies to the FA model with less significant relative error figures (about 6%).
+The Parallel models (PA, PC, and CM), which can be safely applied along with the PET, have long been ruled out for
+FTT studies (Laslett et al., 1987; Guedes et al., 2013; Ketcham et al., 1999, 2007; Ketcham, 2019). Duddy et al. (1988)
+had argued that the FA deviated only slightly from the PA model and applied it along with the PET. The isoretention
+curves for the two models follow approximately the same trends (Fig. 1a). The same behavior is observed for the
+curvilinear models (Fig. 1b). FC and PC isoretention curves bend together towards lower temperatures. Their argument
+can be better appreciated in Fig. 4. All the PET and RCI predictions for reduced lengths after the track underwent the
+cooling history are gathered in the same plot. Note that the linear models (PA and FA) and the approximately linear CM
+form a cluster, while the curvilinear models (PC and FC) form a separate set. The predictions with the fanning models
+and PET are closer to the predictions of the parallel models for track populations born when the sample passed through
+intermediate temperatures (partial annealing zone), which results in closer 푇퐶 values (compare values in Table 2). For
+populations born at higher temperatures, the fanning-PET predictions depart from the parallel model ones, resulting in
+a more significant difference between calculated 푇퐴 values. Calculations with RCI approximate fanning and parallel
+model predictions for populations born at higher temperatures. Within this approximation, it could be possible to
+engineer fanning model parameters to make the model even closer to a parallel model.
+Figure 4: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.0◦C/Ma) starting at time 푡 and ending at
+the present time at a fixed temperature of 20◦C, calculated for the parallel and fanning models. Calculations using RCI
+are shown as solid geometric forms, while calculations using PET are represented by empty geometric forms.
+4. Concluding remarks
+Departing from Eq. (9), a physicochemical framework was built to deal with the effects of annealing in variable tem-
+perature T-t paths. The basic building blocks are the reaction function 푓푟(푟) and the effective rate constant, 푘푒푓(푡, 푇 ).
+The parallel models (PA, PC, and CM) were shown to be consistent with the single activation energy rate constants
+given by Eqs. (10a)-(10c) and with the reaction-order function (Eq. (8)). The fanning models (FA and FC) are the
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 12 of 15
+
+1.0
+0.8
+Reduction
+0.6
+RCI Method
+PET Method
+Length F
+0.4
+PA
+0
+PA
+PC
+FA
+CM
+CM
+FA
+口
+PC
+0.2
+FC
+FC
+0.0
+50
+100
+150
+200
+Thermal history duration (t)representation of multiple concurrent recombination processes with different activation energies (Rufino and Guedes,
+2022). The 푘푒푓(푡, 푇 ) functions were built (Eq. (15)) to be consistent with the reaction-order function (Eq.(8)). Obtain-
+ing FA and FC rate constants from first principle, i.e., a composition of rate constants for individual processes, and
+validating them experimentally is still an open issue that has to be dealt with. The Eq. (22) is the line integral related
+to Eq. (9) from which length shortening due to cooling T-t paths can be directly calculated, independently of whether
+the rate constant represents a single or multiple activation energy mechanism.
+The Principle of Equivalent Time, on the other hand, is only valid for single activation energy equations, which, for
+the fission-track system, are the parallel models. In these cases, the RCI-based calculations are in agreement with the
+PET ones (Fig. 2), indicating the robustness of RCI formulation. For the fanning models, the use of the PET has long
+been recognized as an approximation (Duddy et al., 1988). Deviations have indeed been observed between RCI and
+PET-based calculations (Fig. 3). Compared to the application of RCI, the PET calculation underestimates annealing
+effects in variable temperature T-t paths (Table 2).
+The PET along with FA or FC models is the calculation method used to infer most published thermal histories.
+This procedure introduces a systematic deviation that should be considered in the geological interpretation of the
+thermal history modeling. Alternatively, the rate constant integral (Eq. (22)) could be considered to substitute the
+PET in inversion thermal history codes (Ketcham, 2005; Gallagher, 2012). Computationally, solving an integral,
+even numerically, is a routine faster than the interactive steps necessary to apply the PET. More importantly, if the
+rate constants are representative of the track annealing kinetics, this framework results, in principle, in more accurate
+predictions of the annealing effects in samples submitted to variable temperature thermal histories.
+Appendix
+A. Equations for the application of the Principle of Equivalent Time
+The proposed method to deal with annealing in thermal histories with variable temperatures is also based on the
+Arrhenius equation and was first proposed by Goswami et al. (1984). The Principle of Equivalent Time (PET), on
+which this method is founded, states that the annealing rate of a track does not depend on its previous thermal history,
+but only on its current length.
+The procedure to solve this problem is to infer the magnitude of annealing recursively, dividing the thermal history
+into appropriate intervals Δ푡푖 and starting from a given value of 푟. For each step, annealing is carried out at constant
+temperature. In the first step, centered at (푡1, 푇1). The procedure will be shown for the Parallel Arrhenius model. The
+reduced fission-track length is calculated using Eq. (PA1), along with 푔(푟) = ln(1 − 푟):
+푟1 = 1 − (Δ푡1)푐1 exp
+(
+푐0 + 푐2
+푅푇1
+)
+(24)
+For the next step, the calculation of 푟2 incorporates the principle in the form that this new shortening is postulated
+to be independent of the previous one. Thus, one can find a length of time, 휏푟1, that will yield the length 푟1 but for
+heating at the temperature of the second interval, 푇2:
+휏푟1 = (1 − 푟1)−1∕푐1 exp
+(
+− 1
+푅푇2
+푐2
+푐1
++ 푐0
+푐1
+)
+.
+(25)
+The value of 푟2 is then found by the application of the annealing model to the interval 휏푟1 + Δ푡2:
+푟2 = 1 − (휏푟1 + Δ푡2)푐1 exp
+(
+푐0 + 푐2
+푅푇2
+)
+.
+(26)
+This procedure is interactively repeated for the entire T-t path. The last value of 푟 will be the reduced length of
+the population born in the first interval after experiencing the entire thermal history. For each interval 푗, the formulas
+above are:
+푟푗
+= 1 − (휏푟푗−1 + Δ푡푗)푐1 exp
+(
+푐0 +
+푐2
+푅푇푗
+)
+(27a)
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 13 of 15
+
+Table A1
+Equations for the application of the Principle of Equivalent Time
+FT models
+Equations
+Parameters (standard error)
+Parallel Arrhenius
+(PA)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓푃 퐴
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(PA4)
+ln
+(
+휏푟푗−1
+)
+=
+ln (1 − 푟푗−1
+)
+푐1
+−
+푐2
+푐1푅푇푗
+− 푐0
+푐1
+(PA5)
+푐0 = 5.631 (0.220)
+푐1 = 0.1865 (0.0066)
+푐2 = -10.46 (0.31) kcal/mol
+휒2
+휈 = 2.65
+Parallel curvilinear
+(PC)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓푃 퐶
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(PC4)
+ln
+(
+휏푟푗−1
+)
+=
+ln (1 − 푟푗−1
+)
+푐1
+− 푐2
+푐1
+ln
+(
+1
+푅푇푗
+)
+− 푐0
+푐1
+(PC5)
+푐0 = -4.910 (0.096)
+푐1 = 0.1944 (0.0060)
+푐2 = -9.610 (0.244)
+휒2
+휈 = 2.12
+Carlson Model
+(CM)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓퐶푀
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(CM4)
+ln
+(
+휏푟푗−1
+)
+=
+ln (1 − 푟푗−1
+)
+푐1
+−
+푐2
+푐1푅푇푗
+− 푐0
+푐1
+− ln (푅푇푗
+)
+(CM5)
+푐0 = 5.426 (0.2155)
+푐1 = 0.1867 (0.0066)
+푐2 = -10.25 (0.2994) kcal/mol
+휒2
+휈 = 2.63
+Fanning Arrhenius
+(FA)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓퐹퐴
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(FA4)
+ln
+(
+휏푟푗−1
+)
+=
+[ln (1 − 푟푗−1
+) − 푐0
+] [
+1
+푅푇푗 − 푐3
+]
+푐1
++ 푐2
+(FA5)
+푐0 = -8.518 (1.072)
+푐1 = 0.1266 (0.0191) mol/kcal
+푐2 = -20.99 (5.81)
+푐3 = 0.2985 (0.1026) mol/kcal
+휒2
+휈 = 1.66
+Fanning curvilinear
+(FC)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓퐹퐶
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(FC4)
+ln
+(
+휏푟푗−1
+)
+=
+[ln (1 − 푟푗−1
+) − 푐0
+] [
+ln
+(
+1
+푅푇푗
+)
+− 푐3
+]
+푐1
++ 푐2
+(FC5)
+푐0 = -9.449 (1.480)
+푐1 = 0.1627 (0.0298)
+푐2 = -24.58 (7.75)
+푐3 = -0.8626 (0.1549)
+휒2
+휈 = 1.88
+Notes: 1. For each fission-track annealing model (Eqs. (PA1), (PC1), (FA1), (FC1)), the length reduction 푟 (Eqs. (PA4),
+(PC4), (FA4), (FC4)) and equivalent time 휏 (Eqs. (PA5), (PC5), (FA5), (FC5)) were obtained using 푔(푟) = ln(1 − 푟).
+휏푟푗−1
+= (1 − 푟푗−1)−1∕푐1 exp
+(
+− 1
+푅푇푗
+푐2
+푐1 + 푐0
+푐1
+)
+(27b)
+The formulas for the PA, PC, CM, FA, and FC models are presented in Table A1.
+Acknowledgements
+This work has been funded by grant 308192/2019-2 by the National Council for Scientific and Technological
+Development (Brazil).
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+

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+Facilitating Contrastive Learning of Discourse Relational Senses by
+Exploiting the Hierarchy of Sense Relations
+Wanqiu Long†, and Bonnie Webber†
+† University of Edinburgh, Edinburgh, UK
+[email protected], [email protected]
+Abstract
+Implicit discourse relation recognition is a
+challenging task that involves identifying the
+sense or senses that hold between two adja-
+cent spans of text, in the absence of an explicit
+connective between them.
+In both PDTB-2
+(Prasad et al., 2008) and PDTB-3 (Webber
+et al., 2019), discourse relational senses are
+organized into a three-level hierarchy rang-
+ing from four broad top-level senses, to more
+specific senses below them.
+Most previous
+work on implicit discourse relation recogni-
+tion have used the sense hierarchy simply
+to indicate what sense labels were available.
+Here we do more — incorporating the sense
+hierarchy into the recognition process itself
+and using it to select the negative examples
+used in contrastive learning.
+With no addi-
+tional effort, the approach achieves state-of-
+the-art performance on the task.
+Our code
+is released in https://github.com/wanqiulong
+0923/Contrastive_IDRR.
+1
+Introduction
+Discourse relations are an important aspect of
+textual coherence. In some cases, a speaker or
+writer signals the sense or senses that hold between
+clauses and/or sentences in a text using an explicit
+connective. Recognizing the sense or senses that
+hold can be more difficult, in the absense of an
+explicit connective.
+Automatically identifying the sense or senses
+that hold between sentences and/or clauses can
+be useful for downstream NLP tasks such as text
+summarization (Cohan et al., 2018), machine trans-
+lation (Meyer et al., 2015) and event relation ex-
+traction (Tang et al., 2021). Recent studies on im-
+plicit discourse relation recognition have shown
+great success. Especially, pre-trained neural lan-
+guage models (Peters et al., 2018; Devlin et al.,
+2019; Liu et al., 2019) have been used and dramat-
+ically improved the performances of models (Shi
+and Demberg, 2019b; Liu et al., 2020; Kishimoto
+Level2
+Level1
+Temporal
+Asynchronous
+Level3
+Precedence
+Succession
+Root
+Comparison
+Contigency
+Expansion
+Synchronous
+Concession
+Contrast
+Cause
+...
+Purpose
+Cause+Belief
+...
+Reason
+Result
+Equivalence
+Manner
+Reason+Belief
+Result+Belief
+Figure 1: The PDTB-3 Sense Hierarchy
+et al., 2020). The senses available for labelling
+discourse relations in the PDTB-2 (and later in the
+PDTB-3) are arranged in a three-level hierarchy,
+with the most general senses at the top and more
+specific senses further down. In the PDTB-3, anno-
+tators could only choose senses at terminal nodes
+in the hierarchy – level-2 senses for symmetric re-
+lations such as EXPANSION.EQUIVALENCE and
+TEMPORAL.SYNCHRONOUS, and level-3 senses
+for asymmetric relations, with the direction of
+the relation encoded in its sense label such as
+SUBSTITUTION.ARG1-AS-SUBST (where the text
+labelled ARG1 substitutes for the denied text
+labelled ARG2) and SUBSTITUTION.ARG2-AS-
+SUBST (where the text labelled ARG2 substitutes
+for the denied text labelled ARG1). Early work on
+recognizing the implicit relations only used the hi-
+erarchy to choose a target for recognition (e.g., the
+senses at level-1 (classes) or those at level-2 (types).
+Recently, Wu et al. (2022) have tried to leverage the
+dependence between the level-1 and level-2 labels
+(cf. Section 2). The current work goes further, us-
+ing the whole three-level sense hierarchy to select
+the negative examples for contrastive learning.
+Contrastive learning, which aims to minimize
+the distance between similar instances (defined as
+positive examples) and widen the difference with
+dissimilar instances (negative examples), has been
+considered as effective in constructing meaning-
+ful representations (Kim et al., 2021; Zhang et al.,
+2021; Yan et al., 2021). Previous work on con-
+trastive learning indicates that it is critical to se-
+arXiv:2301.02724v1  [cs.CL]  6 Jan 2023
+
+lect good negative samples (Alzantot et al., 2018;
+Wu et al., 2020b; Wang et al., 2021). The insight
+underlying the current work is that the hierarchy
+of sense labels can enable the selection of good
+negative examples for contrastive learning.
+To
+see this, consider Examples 1-3 below from the
+PDTB-3. On the surface, they look somewhat sim-
+ilar, but in Examples 1 and 2, the annotators took
+the second sentence (Arg2) as providing more de-
+tail about the first sentence (Arg1) — the sense
+called EXPANSION.LEVEL-OF-DETAIL.ARG2-AS-
+DETAIL, while in Example 3, they took the sec-
+ond sentence as expressing a substitute for “Amer-
+ican culture” in terms of what is relevant – the
+sense called EXPANSION.SUBSTITUTION.ARG2-
+AS-SUBST.
+(1) “Valley National ”“isn’t out of the woods yet
+”. The key will be whether Arizona real es-
+tate turns around or at least stabilizes..
+(2) The House appears reluctant to join the sena-
+tors. A key is whether House Republicans
+are willing to acquiesce to their Senate col-
+leagues’ decision to drop many pet provi-
+sions..
+(3) Japanese culture vs. American culture is ir-
+relevant. The key is how a manager from
+one culture can motivate employees from
+another..
+In this work, we use a multi-task learning frame-
+work, which consists of classification tasks and a
+contrastive learning task. Unlike most previous
+work using one benchmark dataset (usually PDTB-
+2 or PDTB-3), we evaluate our systems on both
+PDTB-2 and PDTB-3. Besides, Wang et al. (2021)
+have shown that data augmentation can make rep-
+resentations be more robust, thereby enriching the
+data used in training. We thus follow Ye et al.
+(2021) and Khosla et al. (2020) in identifying a rel-
+evant form of data augmentation for our contrastive
+learning approach to implicit relation recognition.
+The main contributions of our work are as fol-
+lows:
+• We leveraged the sense hierarchy to get con-
+trastive learning representation, learning an
+embedding space in which examples from
+same types at level-2 or level-3 stay close to
+each other while sister types are far apart.
+• We explored and compared different methods
+of defining the negatives based on the sense
+hierarchies in PDTB-2 and PDTB-3, finding
+the approach which leads to the greatest im-
+provements.
+• Our proposed data augmentation method to
+generate examples is helpful to improve the
+overall performance of our model.
+• We demonstrate that implicit relation recogni-
+tion can benefit from a deeper understanding
+of the sense labels and their organization.
+2
+Related Work
+Implicit discourse relation recognition
+For
+this task, Dai and Huang (2018) considered
+paragraph-level context and inter-paragraph de-
+pendency. Recently, Shi and Demberg (2019b)
+showed that using the bidirectional encoder repre-
+sentation from BERT (Devlin et al., 2019) is more
+accurately to recognize Temporal.Synchrony, Com-
+parison.Contrast, Expansion.Conjunction and Ex-
+pansion.Alternative. Liu et al. (2020) showed that
+different levels of representation learning are all
+important to implicit relation recognition, and they
+combined three modules to better integrate con-
+text information, the interaction between two argu-
+ments and to understand the text in depth. How-
+ever, only two existing works leveraged the hier-
+archy in implicit relation recognition. Both Wu
+et al. (2020a) and Wu et al. (2022) first attempted
+to assign a Level-1 sense that holds between argu-
+ments, and then only considered as possible Level-
+2 senses, those that are daughters of the Level-1
+sense.
+Contrastive learning
+Recently, there has been
+a growing interest in applying contrastive learn-
+ing in both the pre-training and fine-tuning objec-
+tives of pre-trained language models. Gao et al.
+(2021) used a contrastive objective to fine-tune pre-
+trained language models to obtain sentence embed-
+dings, and greatly improves state-of-the-art sen-
+tence embeddings on semantic textual similarity
+tasks. Suresh and Ong (2021) proposed label-aware
+contrastive loss in the presence of larger number
+and/or more confusable classes, and helps models
+to produce more differentiated output distributions.
+Besides, many works have demonstrated that se-
+lecting good negative examples are very important
+for using contrastive learning (Schroff et al., 2015;
+Joshua et al., 2021; Cao et al., 2022). In our work,
+we integrate contrastive learning loss with super-
+
+vised losses and we use the structure of the sense
+hierarchy to guide the selection of negative exam-
+ples.
+3
+Learning Loss
+3.1
+Supervised Learning Loss
+The standard approach today for classification task
+is to use a standard cross-entropy loss:
+Lsup = 1
+N
+N
+�
+i=1
+−log
+eW T
+yisi
+�
+j eW T
+j si
+(1)
+Where N denotes the number of training examples,
+yi is the ground-truth class of the i-th class and Wj
+is the weight vector of the j-th class.
+3.2
+Contrastive Learning Loss
+In contrastive learning, each example can be treated
+as an anchor to get its positive and negative ex-
+amples. Contrastive learning can pull the anchor
+and its positive example together in the embedding
+space, while the anchor and negative samples are
+pushed apart. The contrastive learning loss was
+used by Chen et al. (2020); Suresh and Ong (2021)
+before. A set of N randomly sampled label pairs
+is defined as xk, yk, where x and y represent sam-
+ples and labels, respectively, k = 1, ..., N. Let i
+be the index of anchor sample and j is the index
+of a positive sample. where iϵ{1, ..., N}, i ̸= j.
+Contrastive loss is defined as:
+Lscl = −
+N
+�
+i=1
+esim(hj,hi)τ
+�
+i̸=k esim(hk,hi)τ
+(2)
+Here, h denotes the feature vector in the em-
+bedding space, and τ is the temperature parameter.
+Intuitively, the numerator computes the inner dot
+product between the anchor points i and its positive
+sample j. The denominator computes the inner dot
+product between all i and the inner dot product be-
+tween all negative samples. where a total of N − 1
+samples are computed.
+Supervised contrastive learning (Gunel et al.,
+2021) extends the equation.2 to the supervised sce-
+nario. In particular, given the presence of labels,
+the positive examples are all examples with the
+same label. The loss is defined as:
+Lscl =
+N
+�
+i=1
+−
+1
+Nyi − 1
+N
+�
+j=1
+1i̸=j1yi=yj
+log
+esim(hj,hi)τ
+�N
+k=1 1i̸=kesim(hk,hi)/τ
+(3)
+Nyjindicates the number of examples in a batch
+that have the same label as i, τ is the temperature
+parameter and h denotes the feature vector that is
+from the l2 normalized final encoder hidden layer
+before the softmax projection.
+4
+Our Approach
+Figure 2 shows the overall architecture of our
+method. As figure 2 illustrates, we firstly use a
+simple multi-task model based on RoBERTa-base
+(Liu et al., 2019), and then we develop a contrastive
+learning algorithm where the sense hierarchy is
+used to select positive and negative examples. De-
+tailed descriptions of our framework and our data
+augmentation method are given below.
+4.1
+Sentence Encoder
+Every annotated discourse relation consists of two
+sentences or clauses (its arguments) and one or
+more relational senses that the arguments bear to
+each other. We concatenate the two arguments
+of each example and input them into RoBERTa.
+Following standard practices, we add two special
+tokens to mark the beginning ([CLS]) and the end
+([SEP]) of sentences. We use the representation of
+[CLS] in the last layer as the representation of the
+whole sentences.
+4.2
+Data Augmentation
+To increase the number of training examples, we
+take advantage of meta-data recorded with each Im-
+plicit Discourse Relation in the PDTB (cf. (Webber
+et al., 2019), Section 8]). For each sense taken to
+hold between the arguments of that relation, anno-
+tators have recorded in the meta-data, an explicit
+connective that could have signalled that sense. In
+the past, this meta-data was used in implicit relation
+recognition by both Patterson and Kehler (2013)
+and Rutherford and Xue (2015). We have used
+it in a different way, shown in Figure 3, to create
+an additional training example for each connective
+that appears in the meta-data. In the added training
+example, this added connective becomes part of
+the second argument of the relation (i.e., appearing
+after the [SEP] character)
+Since there is at least one explicit connective
+recorded in the meta-data for each implicit dis-
+course relation and at most two 1, for a training
+batch of N tokens, there will be at least another
+1This is because the PDTB only allows for one or two
+senses per relation.
+
+...
+Encoder La�er
+... ...
+T�a��f���e�
+�a�e�
+...
+Encoder La�er
+Encoder La�er
+I����
+...
+...
+Le�e�-2
+C�a���f�e�
+Le�e�-1
+C�a���f�e�
+P���
+P���
+Se��e H�e�a�c��
+a�c���
+�������e
+�e�a���e
+[CLS]
+�1
+1
+�1
+�
+[SEP]
+�2
+1
+�2
+�
+[SEP]
+[CLS]
+�1
+1
+�1
+�
+[SEP]
+�2
+1
+�2
+�
+[SEP]
+Figure 2: The overall architecture of our model. When given an anchor, we search the positive and negative
+examples in a training batch based on the sense hierarchy of the PDTB. We narrow the distances among examples
+from the same types at level-2 or level-3 and enlarge the distances among examples from different types at level-2
+and level-3.
+N tokens introduced by this data augmentation
+method, increasing the training batch to at least 2N
+tokens.
+𝑒���:������������������������������������������������������������������������������
+𝑒���
+∗
+��������������������������������������������������������������������������������������������
+Figure 3: An example with inserted connective: the
+connective word is “In contrast”.
+4.3
+Positive Pair and Negative Pair
+Generation
+We use the structure of the sense hierarchy to iden-
+tify the positive and negative examples needed for
+contrastive learning. The only senses used in anno-
+tating discourse relations are ones at terminal nodes
+of the sense hierarchy. This is Level 2 for symmet-
+ric senses and Level 3 for asymmetric senses (i.e.,
+where the inverse of the sense that holds between
+Arg1 and Arg2 is what holds between Arg2 and
+Arg1. For example, CONTRAST and SIMILARITY
+are both symmetric senses, while MANNER and
+CONDITION are asymmetric, given that there is a
+difference between Arg2 being the manner of do-
+ing Arg1 or Arg1 being the manner of doing Arg2).
+In our work, when the lowest level of the senses
+is level-3, we directly used the level-3 labels in-
+stead of their parent at level-2. For example, under
+the level-2 label Temporal.asynchronous, there are
+two labels which are precedence and succession
+at level-3. For this case, we replaced the level-2
+label Temporal.asynchronous with the two labels
+precedence and succession at level-3.
+Although supervised contrastive learning in Eq.
+3 can be valid for different classes of positive ex-
+ample pairs, its negative examples come from any
+examples inside a batch except itself. We defined
+l1, l2, l3 as the first, second, and third level in the
+hierarchical structure respectively, and lϵli refers
+to the labels from level i.
+Instance e ∼ Same sub-level
+epos
+Given the representation of a sentence ei and its
+first, second and third level of label li
+1, li
+2, li
+3, we
+searched the set of examples with the same sec-
+ond level labels or the same third level labels (if
+the lowest level is level-3) as epos in each training
+batch:
+ei
+pos = {e ∈ ei
+pos : le
+2 == li
+2
+or
+le
+3 == li
+3}
+(4)
+E.g.
+If the label of the anchor is Tempo-
+ral.asynchronous.precedence, its positive examples
+would be the examples with the same label.
+Instance e ∼ Batch instance
+eneg
+Here, we would like to help the model discriminate
+the sister types at level-2 and level-3 (if the lowest
+level is level-3). We searched the set of examples
+with different level-2 labels or level-3 labels as eneg
+in each training batch.
+E.g.
+If the label of the anchor is Tempo-
+ral.asynchronous.precedence, its negative exam-
+ples would be its sister types at level-2 and level-
+3, namely Temporal.asynchronous.succession and
+Temporal.synchronous.
+ei
+neg = {e ∈ ei
+neg : le
+1 == li
+1
+&
+(le
+2 ̸= li
+2
+&
+le
+3 ̸= li
+3)}
+(5)
+
+4.4
+Loss Algorithms
+As described above, given the query ei with its
+positive pairs and negative pairs and based on the
+general contrastive learning loss (see Equation 2),
+the contrastive learning loss for our task and ap-
+proach is:
+Lscl =
+N
+�
+i=1
+−
+1
+|eipos| − 1
+2N
+�
+j=1
+1i̸=j1j∈eipos
+log
+wjesim(hj,hi)τ
+�2N
+k=1 1i̸=k1k∈eineg+eiposwkesim(hk,hi)/τ
+(6)
+where wj and wj are weight factors for differ-
+ent positive pairs and negative pairs respectively,
+sim(hi, hj) is cosine similarity and τ is a tempera-
+ture hyperparameter.
+Our overall training goal is:
+L = Ll1
+sup + Ll2
+sup + βLscl
+(7)
+As our classifications are done in the first level
+and second level for the same inputs, we used a
+standard cross-entropy loss to get supervised loss
+LL1
+sup and LL2
+sup. And β is the weighting factor for
+the contrastive loss.
+5
+Experiment Setting
+5.1
+Datasets
+Besides providing a sense hierarchy, the Penn Dis-
+course TreeBank (PDTB) also frequently serves as
+a dataset for evaluating the recognition of discourse
+relations. The earlier corpus, PDTB-2 (Prasad et al.,
+2008) included 40,600 annotated relations, while
+the later version, PDTB-3 (Webber et al., 2019)
+includes an additional 13K annotations, primarily
+intra-sentential, as well as correcting some incon-
+sistencies in the PDTB-2. The sense hierarchy used
+in the PDTB-3 differs somewhat from that used in
+the PDTB-2, with additions motivated by the needs
+of annotating intra-sentential relations and changes
+motivated by difficulties that annotators had in con-
+sistently using some of the senses in the PDTB-2
+hierarchy.
+Because of the differences in these two hierar-
+chies, we use the PDTB-2 hierarchy for PDTB-2
+data and the PDTB-3 hierarchy for PDTB-3 data
+respectively. We follow earlier work (Ji and Eisen-
+stein, 2015; Bai and Zhao, 2018; Liu et al., 2020;
+Xiang et al., 2022) using Sections 2-20 of the cor-
+pus for Training, Sections 0-1 for Validation, and
+Sections 21-22 for testing. With regard to those
+instances with multiple annotated labels, we also
+follow previous work (Qin et al., 2016). They are
+treated as separate examples during training. At
+test time, a prediction matching one of the gold
+types is taken as the correct answer. Implicit rela-
+tion recognition is usually treated as a classifica-
+tion task. While 4-way (Level-1) classification was
+carried out on both PDTB-2 and PDTB-3, more
+detailed 11-way (Level 2) classification was done
+only on the PDTB-2 and 14-way (Level 2) classifi-
+cation, only on the PDTB-3.
+5.2
+Baselines
+To exhibit the effectiveness of our proposed
+method, we compare our method with strong base-
+lines. As previous work usually used one dataset
+(PDTB-2 or PDTB-3) for evaluation, we use dif-
+ferent baselines for PDTB-2 and PDTB-3. Since
+PDTB-3 was not released until 2019, the baselines
+for PDTB-3 from 2016 and 2017 are from (Xiang
+et al., 2022). They reproduced those models which
+were originally used on PDTB-2 on PDTB-3.
+Baselines for PDTB-2:
+• (Dai and Huang, 2019): a neural model lever-
+aging external event knowledge and corefer-
+ence relations.
+• (Shi and Demberg, 2019a): a neural model
+that leverages the inserted connectives to learn
+better argument representations.
+• (Nguyen et al., 2019): a neural model which
+predicts the labels and connectives. simulta-
+neously.
+• (Guo et al., 2020): a knowledge-enhanced
+Neural Network framework.
+• (Kishimoto et al., 2020): a model applying
+three additional training tasks.
+• (Liu et al., 2020): a RoBERTa-based model
+which consists of three different modules.
+• (Jiang et al., 2021): a method that recognizes
+the relation label and generates the target sen-
+tence simultaneously.
+• (Dou et al., 2021): a method using conditional
+VAE to estimate the risk of erroneous sam-
+pling.
+• (Wu et al., 2022): a label dependence-aware
+sequence generation model.
+Baselines for PDTB-3:
+• (Liu and Li, 2016): a model that combines
+two arguments’ representation for stacked in-
+teractive attention.
+
+�1
+Acc
+β
+β
+(a) Top-level label classification 
+(b) Second-level label classification 
+PDTB3
+PDTB2
+Figure 4: Effects of β on the validation set.
+• (Chen et al., 2016a): a mixed generative-
+discriminative framework.
+• (Lan et al., 2017): a multi-task attention neu-
+ral network.
+• (Ruan et al., 2020): a propagative attention
+learning model.
+• (Xiang et al., 2022): a model that uses a Dual
+Attention Network (DAN).
+5.3
+Parameters Setting
+In our experiments,
+we use the pre-trained
+RoBERTa-base (Liu et al., 2019) as our Encoder.
+We adopt Adam (Kingma and Ba, 2015) with the
+learning rate of 3e−5 and the batch size of 256 to
+update the model. The maximum training epoch is
+set to 25 and the wait patience for early stopping
+is set to 10 for all models. We clip the gradient
+L2-norm with a threshold 2.0. For contrast learn-
+ing, the weight of positive examples is set to 1.6
+and the weight of negative examples is set to 1. All
+experiments are performed with 1× 80GB NVIDIA
+A100 GPU.
+5.4
+Evaluation Metrics
+We used Accuracy and Macro-F1 score as evalu-
+ation metrics, because PDTB datasets are imbal-
+anced and Macro-F1 score has been said to be an
+more appropriate assessment measure for imbal-
+anced datasets (Akosa, 2017; Bekkar et al., 2013).
+5.5
+Effects of the Coefficient β
+As shown in Equation 7, the coefficient β is an
+important hyperparameter that controls the relative
+importance of supervised loss and contrastive loss.
+Thus, we vary β from 0 to 2.4 with an increment of
+0.2 each step, and inspect the performance of our
+model using different β on the validation set.
+From Figure 4, we can find that, compared with
+the model without contrastive learning (β = 0), the
+performance of our model at any level is always
+improved via contrastive learning. For PDTB-2,
+when β exceeds 1.0, the performance of our model
+tends to be stable and declines finally. Thus, we
+directly set β = 1.0 for all PDTB-2 related exper-
+iments thereafter. For PDTB-3, the Acc and F1
+of the validation set reach the highest point at β =
+2.0. Therefore we choose β = 2.0 for all related
+experiments.
+We have considered three ways of investigat-
+ing why there is such a difference in the optimal
+weighting coefficient. First, compared with PDTB-
+2, the PDTB-3 contains about 6000 more implicit
+tokens annotated for discourse relations. Secondly,
+although the sense hierarchies of both the PDTB-
+2 and the PDTB-3 have three levels and have the
+same senses at level- 1, but many changes at level-2
+and level-3 due to difficulties found in annotating
+certain senses. Moreover, the intra-sentential im-
+plicit relations might be another reason. In PDTB-
+3, many more discourse relations are annotated
+within sentences. Liang et al. (2020) report quite
+striking difference in the distribution of sense re-
+lations inter-sententially vs. intra-sententially be-
+tween PDTB-2 and PDTB-3. Therefore, these ma-
+jor differences in the PDTB-3 and the PDTB-2
+might cause the fluctuation of the coefficient value.
+6
+Results and Analysis
+The results on PDTB-2 and PDTB-3 for Level-1
+and Level-2 are presented in Table 1 and Table 2
+respectively, where the best results are highlighted
+in bold. Classification performance on PDTB-2 in
+terms of Macro-F1 for the four general sense types
+at Level-1 and 11 sense types at Level-2 is shown
+in Table 3 and Table 4.
+These results demonstrate better performance
+than previous systems for both Level-1 and Level-2
+classification on both PDTB-2 and PDTB-3. In
+particular, the results clearly demonstrate benefits
+to be gained from contrastive learning. But there is
+more to be said: In Section 6.1, we discuss different
+ways of defining negative examples with respect to
+the sense hierarchy, and in Section 6.2, we discuss
+the relative value of the particular form of data
+augmentation we have used (cf. Section 4.2) as
+compared with our method of contrastive learning.
+6.1
+Comparisons with Other Negatives
+Selecting Methods
+There is not only one way to select negative ex-
+amples for contrastive learning based on PDTB
+
+75
+70
+65
+60
+55
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.460
+55
+50
+45
+40
+35
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.465
+62
+59
+56
+53
+50
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.475
+73
+71
+69
+67
+65
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4Model
+PDTB-2
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+Dai and Huang (2019)
+59.66
+52.89
+48.23
+33.41
+Shi and Demberg (2019a)
+61.42
+46.40
+47.83
+-
+Nguyen et al. (2019)
+-
+53.00
+49.95
+-
+Guo et al. (2020)
+57.25
+47.90
+-
+-
+Kishimoto et al. (2020)
+65.26
+58.48
+52.34
+-
+Liu et al. (2020)
+69.06
+63.39
+58.13
+-
+Jiang et al. (2021)
+-
+57.18
+-
+37.76
+Dou et al. (2021)
+70.17
+65.06
+-
+-
+Wu et al. (2022)
+71.18
+63.73
+60.33
+40.49
+Ours
+72.18
+69.60
+61.69
+49.66
+Table 1: Experimental results on PDTB-2.
+Model
+PDTB-3
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+Liu and Li (2016)
+57.67
+46.13
+-
+-
+Chen et al. (2016b)
+57.33
+45.11
+-
+-
+Lan et al. (2017)
+57.06
+47.29
+-
+-
+Ruan et al. (2020)
+58.01
+49.45
+-
+-
+Xiang et al. (2022)
+60.45
+53.14
+-
+-
+(BiLSTM)
+Xiang et al. (2022)
+64.04
+56.63
+-
+-
+(BERT)
+Ours
+75.31
+70.05
+64.68
+57.62
+Table 2: Experimental results on PDTB-3.
+Model
+Comp.
+Cont
+Exp.
+Temp.
+Nguyen et al. (2019)
+48.44
+56.84
+73.66
+38.60
+Guo et al. (2020)
+43.92
+57.67
+73.45
+36.33
+Liu et al. (2020)
+59.44
+60.98
+77.66
+50.26
+Jiang et al. (2021)
+55.40
+57.04
+74.76
+41.54
+Dou et al. (2021)
+55.72
+63.39
+80.34
+44.01
+Ours
+65.84
+63.55
+79.17
+69.86
+Table 3: The results for relation types at level-1 on
+PDTB-2 in terms of F1 (%) (top-level multi-class clas-
+sification).
+hierarchical structures. In addition to the method
+we adopt, we have explored another 4 different
+methods of defining positive and negative exam-
+ples by using the sense hierarchies, which can be
+shown in Figure 5. One can choose the level against
+which to select negative examples: method 2 be-
+low uses examples with different labels at level-2,
+while methods 1, 3 and 4 use examples with dif-
+ferent labels at level-1. With regard to the use of
+weight for method 3 and method 4, we aim to give
+more weight to more similar (potentially) positive
+examples based on the hierarchy. Specifically, we
+give more weight to the examples from the same
+level-2/level-3 type than their sister types at level-
+2/level-3 when all of the examples from the same
+level-1 are positive examples. Besides, method 4
+leverages level-3 labels, while method 1 to 3 only
+consider level-1 and level-2 labels. In our exper-
+iments for other negatives defining methods, we
+use the same hyperparameters as the experimental
+setup of our methods.For method 3 and method 4,
+Second-level Label
+Liu et al. (2020)
+Wu et al. (2022)
+Ours
+Temp.Asynchronous
+56.18
+56.47
+59.79
+Temp.Synchrony
+0.00
+0.00
+78.26
+Cont.Cause
+59.60
+64.36
+65.58
+Cont.Pragmatic cause
+0.0
+0.0
+0.00
+Comp.Contrast
+59.75
+63.52
+62.63
+Comp.Concession
+0.0
+0.0
+0.00
+Exp.Conjunction
+60.17
+57.91
+58.35
+Exp.Instantiation
+67.96
+72.60
+73.04
+Exp.Restatement
+53.83
+58.06
+60.00
+Exp.Alternative
+60.00
+63.46
+53.85
+Exp.List
+0.0
+8.98
+34.78
+Table 4: The results for relation types at level-2 on
+PDTB-2 in terms of F1 (%) (second-level multi-class
+classification).
+the weight of positive examples is set to 1.6 and
+1.3 and the weight of negative examples still is 1.
+It can be seen from table 5 and table 6 that our
+method is better than the above methods in both
+datasets for both level-1 and level-2 classification
+tasks. Compared with method 2, we utilize level-3
+labels, which indicated the level-3 label informa-
+tion is helpful for the approach. The greatest differ-
+ence between our method and other three methods
+is that our negative examples are only those sis-
+ter types at level-2 or level-3, not including the
+examples from different level-1. On the contrary,
+the negative examples in those three methods are
+examples from other level-1 types. We suppose
+that this might make a too strong assumption that
+examples from different level-1 are very dissimilar.
+In PDTB datasets, some examples have been anno-
+tated with multiple labels. We found that among all
+examples with multiple annotated labels, there are
+99.26% examples whose multiple labels are under
+different level-1. Moreover, some level-1 types of
+relation might be overlapped even if the annotators
+just annotate one label. For example, some exam-
+ples annotated as Temporal.asynchronous might
+have the sense of Contingency.cause as well. And
+Moens and Steedman (1988) have pointed out that
+when-clauses do not simply predicate a temporal re-
+lation, but a causal one as well, which can be called
+contingency. This shows up in the PDTB in terms
+of the variation in how particular tokens of when
+clauses have been annotated. But it also means
+that in choosing Negative examples, relations la-
+belled TEMPORAL.SYNCHRONOUS or TEMPO-
+RAL.ASYNCHRONOUS may closely resemble those
+labelled CONTINGENCY.CAUSE and therefore not
+be effective as negative examples. Specifically, for
+the following example:
+(4) when [they built the 39th Street bridge]1,
+[they solved most of their traffic problems]2.
+
+Model
+PDTB-2
+PDTB-3
+Top Level
+Second Level
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+Acc
+Macro-F1
+Acc
+Macro-F1
+Method 1
+68.91
+65.04
+58.61
+46.27
+73.25
+68.00
+61.17
+55.58
+Method 2
+69.39
+63.95
+58.33
+44.80
+73.53
+68.36
+61.93
+54.85
+Method 3
+69.39
+66.53
+58.61
+39.20
+72.49
+67.49
+60.77
+54.33
+Method 4
+69.10
+65.30
+57.07
+47.46
+71.26
+66.47
+59.53
+47.24
+Ours
+72.18
+69.60
+61.69
+49.66
+75.31
+70.05
+64.48
+57.62
+Table 5: Comparisons with other negatives defining methods.
+Positve: examples with same label at level-1.
+Negative: examples with different labels at level-1.
+(a) method 1
+Positve: examples with same label at level-2.
+Negative: examples with different labels at level-2.
+(b) method 2
+Positve: Examples with same label at level-1.     
+              More weight are given to the examples 
+              with same label at level-2.
+Negative: Examples with different labels at level-1.
+(c) method 3
+Positve: Examples with same label at level-1, 
+              more weight are given to   the examples 
+              with same label at level-2 or level-3. 
+Negative: Examples with different labels at level-1.
+(d) method 4
+Figure 5: Another four negative examples selected
+methods. orange ball represent anchor, green ball rep-
+resent negative examples, and blue ball represent posi-
+tive examples. Darker blue ball means more weight is
+given to more similar (potentially) positive examples.
+If the connective “when” is replaced with “be-
+cause”, the sentence still sounds not strange. There-
+fore, regarding all examples from different level-1
+as negative examples might have some negative
+impacts on learning the representations.
+6.2
+Ablation Study
+We wanted to know how useful our data augmen-
+tation method and our contrastive learning method
+are, so we have undertaken ablation studies for this.
+Model
+Comp.
+Cont
+Exp.
+Temp.
+Method 1
+63.26
+60.42
+76.78
+59.74
+Method 2
+60.78
+60.82
+77.89
+56.30
+Method 3
+59.85
+65.18
+76.43
+64.67
+Method 4
+57.25
+61.73
+77.30
+64.90
+Ours
+65.84
+63.55
+79.17
+69.86
+Table 6: The results of relation types at level-1 on
+PDTB-2 in terms of F1 (%) (top-level multi-class clas-
+sification).
+Effects
+of
+contrastive
+learning
+algorithm
+From Table 7, it can be seen that multi-task
+learning
+method
+where
+level-1
+and
+level-2
+labels are predicted simultaneously by using the
+same [CLS] representation perform better than
+separately predicting level-1 and level-2 labels,
+which verifies the dependency between different
+levels.
+Compared with the multi-task learning
+method, our model with a contrastive loss has
+better performance in PDTB-2 and PDTB-3, which
+means that our contrasting learning method is
+indeed helpful.
+Effects of data augmentation
+Table 8 compares
+the results with and without data augmentation for
+both PDTB-2 and PDTB-3. From the comparisons,
+it is clear that the data augmentation method is
+helpful to generate useful examples. Khosla et al.
+(2020) showed that having a large number of hard
+positives/negatives in a batch leads to better perfor-
+mance. Since we have many classes at the second
+level, 11 types for PDTB-2 and 14 types for PDTB-
+3. In a batch with the size of 256, it is difficult to
+guarantee that there are enough positive examples
+for each class to take full advantage of contrast
+learning. Therefore, without data augmentation,
+the performance of our method degrades consider-
+ably.
+7
+Limitations and Future work
+With regard to PDTB-2 and PDTB-3 annotation,
+there are two cases: (1) Annotators can assign mul-
+tiple labels to an example when they believe more
+than one relation holds simultaneously; (2) An-
+notators can be told (in the Annotation Manual)
+to give precedence to one label if they take more
+than one to hold. For example, they are told in
+the Manual (Webber et al., 2019) that examples
+that satisfy the conditions for both Contrast and
+Concession, should be labelled as concession. We
+over-simplified the presence of multiple labels by
+following Qin et al. (2017) in treating each label as
+
+Datasets
+Model
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+PDTB-2
+RoBERTa
+68.14
+64.87
+58.33
+48.37
+RoBERTa-MTL
+69.87
+65.39
+58.22
+45.21
+Ours
+72.18
+69.60
+61.69
+49.66
+PDTB-3
+RoBERTa
+72.02
+67.44
+60.56
+57.12
+RoBERTa-MTL
+72.63
+68.23
+60.56
+57.16
+Ours
+75.31
+70.05
+64.68
+57.62
+Table 7: Ablation study on PDTB-2 and PDTB-3.
+Model
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+PDTB-2
+Ours
+72.18
+69.60
+61.69
+49.66
+-augmentation
+71.70
+67.85
+59.19
+45.54
+PDTB-3
+Ours
+75.31
+70.05
+64.68
+57.62
+-augmentation
+73.32
+69.02
+63.24
+51.80
+Table 8: Effects of data augmentation.
+a separate example and did not consider the second
+case. Thus, our approach might be inadequate for
+dealing with the actual distribution of the data and
+can be extended or modified. It is worth exploring
+how to extend our approach to allow for examples
+with multiple sense labels and cases where one la-
+bel takes precedence over another. We believe that
+this will be an important property of the work.
+Another limitation is that we only use English
+datasets. There are PDTB-style datasets in other
+languages including a Chinese TED dicourse bank
+corpus (Long et al., 2020), a Turkish discourse
+Tree bank corpus (Zeyrek and Kurfalı, 2017) and
+an Italian Discourse Treebank (Pareti and Prodanof,
+2010). Moreover, Zeyrek et al. (2019) proposed
+a TED Multilingual Discourse Bank (TED-MDB)
+corpus, which has 6 languages. These datasets al-
+low us to assess the approach in languages other
+than English. Besides, there are datasets similar
+to PDTB-Style like Prague Dependency Treebank
+(Mírovský et al., 2014). The different datasets use
+essentially similar sense hierarchy, but two things
+need to be investigated (i) whether there are compa-
+rable differences between tokens that realise “sis-
+ter” relations, or (ii) whether tokens often have
+multiple sense labels, which would change what
+could be used as negative examples if leveraging
+our approach on them.
+In the future, we can also assess whether con-
+trastive learning could help in separating out En-
+tRel relations and AltLex relations from implicit
+relations or whether other methods would perform
+better.
+8
+Conclusions
+In this paper, we leverage the sense hierarchy to
+select the negative examples needed for contrastive
+learning for the task of implicit discourse relation
+recognition. Our method has better overall perfor-
+mance than achieved by previous systems, and com-
+pared with previous work, our method is better at
+learning minority labels. Moreover, we compared
+different methods of selecting the negative exam-
+ples based on the hierarchical structures, which
+shows some potential negative impacts might be
+produced when negative examples include those
+from other level-1 types. Moreover, we conduct ab-
+lation studies to investigate the effects of our data
+augmentation method and our contrastive learning
+method. Besides, the limitations and the future
+work are discussed.
+Acknowledgments
+This work was supported in part by the UKRI
+Centre for Doctoral Training in Natural Lan-
+guage Processing, funded by the UKRI (grant
+EP/S022481/1), the University of Edinburgh. The
+authors also gratefully acknowledge University of
+Edinburgh Huawei Laboratory for their support.
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+
+A
+Appendix
+A.1
+PDTB Hierarchy
+The hierarchies of both PDTB 2.0 and PDTB 3.0
+consist of three levels, but for implicit relation
+recognition, so far no classification for third level
+labels has been done. We also focus on the hier-
+archy between level-1 and level 2. The PDTB-3
+relation hierarchy simplifies and extends the PDTB-
+2 relation hierarchy. The PDTB 3.0 hierarchy not
+only simplifies the PDTB-2 relation hierarchy by
+restricting Level-3 relations to differences in di-
+rectionality and eliminating rare and/or difficult-
+to-annotate senses, but also augments the relation
+hierarchy. Figure 6 and Figure 7 show PDTB 2.0
+relation hierarchy and PDTB 3.0 relation hierarchy
+respectively.
+Asynchronous
+Synchronous
+Figure 6: The PDTB 2.0 Senses Hierarchy.
+Figure 7: The PDTB 3.0 Senses Hierarchy. The left-
+most column contains the Level-1 senses and the mid-
+dle column, the Level-2 senses. For asymmetric rela-
+tions, Level-3 senses are located in the rightmost col-
+umn.
+Model
+Comp.
+Cont
+Exp.
+Temp.
+Liu and Li (2016)
+29.15
+63.33
+65.10
+41.03
+Lan et al. (2017)
+30.10
+60.91
+64.03
+33.71
+Ruan et al. (2020)
+30.37
+61.95
+64.28
+34.74
+Chen et al. (2016b)
+27.34
+62.56
+64.71
+38.91
+Xiang et al. (2022)
+34.16
+65.48
+67.82
+40.22
+(BiLSTM)
+Xiang et al. (2022)
+35.83
+66.77
+70.00
+42.13
+(BERT)
+Ours
+63.30
+78.60
+79.91
+58.39
+Table 9: The results of different relations on PDTB-3
+in terms of F1 (%) (top-level multi-class classification).
+Second-level Label
+Ours
+Temp.Asynchronous
+66.35
+Temp.Synchrony
+41.38
+Cont.Cause
+71.38
+Cont.Cause+Belief
+0.0
+Cont.Condition
+74.07
+Cont.Purpose
+96.05
+Comp.Contrast
+56.91
+Comp.Concession
+60.11
+Exp.Conjunction
+61.70
+Exp.Equivalence
+11.43
+Exp.Instantiation
+69.83
+Exp.Level-of-detail
+55.34
+Exp.Manner
+78.43
+Exp.Substitution
+63.77
+Table 10: The results of different relations on PDTB-3
+in terms of F1 (%) (second-level multi-class classifica-
+tion).
+A.2
+The results on relation types on PDTB-3
+We also examine the classification performance on
+PDTB-3 in terms of Macro-F1 for the four main
+relation types at level-1 and 14 sense types at level-
+2. The results can be seen in Table 9 and Table 10.
+Our model has significantly better performance for
+all level-1 relations.
+As for level-2 sense types, because there are no
+results of previous systems, we just show the result
+of 14 level-2 sense types in PDTB-3 in terms of F1.
+
+Synchronous
+Temporal
+Precedence
+Asynchronous
+Succession
+Reason
+Cause
+Result
+Conjunction
+Disjunction
+Negative-result*
+Equivalence
+-
+Arg1-as-cond
+Condition
+Arg1-as-instance
+Contingency
+Arg2-as-cond
+Instantiation
+Arg2-as-instance
+Arg1-as-negcond
+Negative condition
+Arg1-as-detail
+Arg2-as-negcond
+Level-of-detail
+Expansion
+Arg2-as-detail
+Arg1-as-goal
+Arg1-as-subst
+Purpose
+Arg2-as-goal
+Substitution
+Arg2-as-negGoal
+Arg2-as-subst
+Arg1-as-excpt
+Exception
+Contrast
+Arg2-as-excpt
+Similarity
+Manner
+Arg1-as-manner
+Comparison
+Arg1-as-denier*
+Arg2-as-manner
+Concession
+Arg2-as-denierTEMPORAL
+COMPARISON
+ Asynchronous
+→Contrast
+→ Synchronous
+ juxtaposition
+> opposition
+precedence
+ succession
+Pragmatic Contrast
+→ Concession
+expectation
+contra-expectation
+CONTINGENCY
+Cause
+ Pragmatic Concession
+reason
+result
+EXPANSION
+Conjunction
+Pragmatic Cause
+Instantiation
+justification
+Restatement
+Condition
+ specification
+ hypothetical
+ equivalence
+general
+unreal present
+ generalization
+unreal past
+Alternative
+factual present
+ conjunctive
+ factual past
+→ disjunctive
+→ chosen alternative
+ Pragmatic Condition
+Exception
+relevance
+implicit assertion
+List

Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/2301.02041v1.pdf.txt

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+arXiv:2301.02041v1  [math.RA]  5 Jan 2023
+A Note On Square-free Commuting Probabilities
+of Finite Rings
+Andrew Mendelsohn
+Department of EEE, Imperial College London, London, SW7 2AZ, United Kingdom.
+[email protected]
+Abstract. It is shown that the commuting probability of a finite ring
+cannot be a fraction with square-free denominator, resolving a conjecture
+of Buckley and MacHale.
+1
+Introduction
+Let G be a finite group and U denote the uniform distribution. Let the commut-
+ing probability of G, PG, be denoted
+PG := Pra,b←U(G)(ab = ba).
+An alternative characterisation is
+PG =
+1
+|G|2 {(x, y) ∈ G2 : xy = yx}.
+Joseph made the following conjectures, where G is the set of all commuting
+probabilities of finite groups [1]:
+A. All limit points of G are rational.
+B. The set G is well ordered by >.
+C. The set 0 ∪ G is closed (that is, contains all its accumulation points).
+In [2], Eberhard resolved conjectures A and B, and in [3] conjecture C was
+resolved.
+One can extend the definition of commuting probability to finite rings: set
+PR =
+1
+|R|2 |{(x, y) ∈ R2 : xy = yx}| =
+1
+|R|2 |{(x, y) ∈ R2 : xy − yx = 0}|.
+In [4], the following conjectures were made, where R is the set PR over all finite
+rings R:
+1. 1/n ̸∈ R when n ∈ N is square-free.
+2. R ⊂ G.
+3. R coincides with the set of values of PG as G ranges over all finite nilpotent
+groups of class at most 2.
+4. All limit points of R are rational.
+
+2
+A. Mendelsohn
+5. For each 0 < t ≤ 1, there exists ǫt > 0 such that R ∩ (t − ǫt, t) = ∅.
+6. R does not contain any of its accumulation points.
+Note conjectures 4, 5, and 6 correspond to Joseph’s conjectures. Moreover, con-
+jectures 4 and 5 would follow from the veracity of conjectures 2 or 3, since
+Eberhard showed G has rational limit points and is well-ordered. In [5], conjec-
+ture 2 was in fact resolved, and thus conjectures 4 and 5. Moreover, conjecture
+3 was partially resolved: the authors obtained that R is a subset of the set of
+values of PG as G ranges over all finite nilpotent groups of class at most 2. We
+conclude that conjectures 1, 3, and 6 are open.
+In this work, we resolve conjecture 1.
+2
+Preliminaries and Prior Results
+Definition 1. Two finite groups G, H are called isoclinic if G/Z(G) ∼= H/Z(H)
+and G′ ∼= H′, and if the diagram below commutes:
+G/Z(G) × G/Z(G)
+H/Z(H) × H/Z(H)
+G′
+H′
+Isoclinism preserves nilpotency class and commuting probability [10]. A stem
+group is a group in a given isoclinism class of minimal order. It is well known
+that if G is a stem group, then Z(G) ≤ G′. For more on isoclinism, see [11].
+Below we state existing results in the literature we will need below.
+Lemma 1. [5] R ⊂ Gn,2, where Gn,2 is the set of commuting probabilities of all
+finite nilpotent groups of class at most 2.
+This statement is proved as follows: let R be a finite ring. We can turn
+R⊕R into a nilpotent ring of class 3 by endowing it with the multiplication rule
+(a, x)(b, y) = (0, ab). This ring can be turned into a nilpotent group GR of class
+at most 2 by endowing it with the binary operation a ◦ b = a + b + ab. Both
+of these transformations preserve the commuting probability. Thus the values of
+R \ {1} are a subset of the values PG, running over nilpotent groups G of class
+equal to 2. Note that if R has size n, then the resulting group GR has order n2,
+and if R is noncommutative then the resulting group is nonabelian.
+Lemma 2. [7] Pr(G) =
+1
+|G′|
+�
+1 + |G′|−1
+|G:Z(G)|
+�
+if and only if G is nilpotent.
+Lemma 3. [6] If G is a nilpotent group, then PG ̸= 1
+p.
+3
+Results
+Theorem 1.
+1
+p ̸∈ R for all p ∈ N≥2.
+
+On the Commuting Probability of Finite Rings
+3
+Proof. By Lemma 1, R is contained within the set of commuting probabilities
+of finite nilpotent groups of class at most 2. By Lemma 3, this latter set does
+not contain 1
+p for any prime p.
+Denote the set of commuting probabilities of rings of prime power order for
+some prime p by Rp.
+Proposition 1.
+1
+n ̸∈ Rp for any prime p ∈ N≥2 and n ∈ N>1.
+Proof. By Lemma 1, we need only consider commuting probabilities of finite
+nilpotent groups of class at most 2. By Lemma 2, we know a fortiori the com-
+muting probability of finite nilpotent groups of class at most 2 in terms of derived
+subgroups and centers. Suppose that for some n ∈ N≥2 we have
+1
+|G′|
+�
+1 +
+|G′| − 1
+|G : Z(G)|
+�
+= 1
+n.
+By the construction of [5] considered above, wlog let |G| = pe for some even
+positive integer e. Then Z(G) = pf and G′ = pg with 0 < g ≤ f < e (since G is
+at most class 2). Then
+1
+n = p−g
+�
+1 + pg − 1
+pe−f
+�
+= p−g + pg − 1
+pe−f+g
+(1)
+= pe−f + pg − 1
+pe−f+g
+.
+(2)
+For this to hold, some power ph must divide the numerator, with h > 0; but this
+cannot hold; for if so, then one must have pe−f + pg − 1 = kp, for some integer
+k ̸= 0. But then −1 ≡ 0 mod p, a contradiction.
+Theorem 2.
+ℓ
+n ̸∈ R for any squarefree n ∈ N>1 and ℓ < n with gcd(ℓ, n) = 1.
+Proof. Any finite ring can be turned into a nilpotent group of class at most
+2, such that the commuting probability of the ring is equal to the commuting
+probability of the group. The construction (outlined above) turns a commutative
+ring into a group of class 1, and a noncommutative ring into a nonabelian group of
+class at most 2, therefore of class equal to 2. The order of the group is the square
+of the order of the ring, so the Sylow subgroups of the group have order at least
+the square of a prime. Since the group is nilpotent, it can be written as a product
+of its Sylow subgroups, which are all of class at most 2, and the commuting
+probability of the group is the product of the commuting probabilities of its
+Sylow subgroups. Thus it remains to analyse the equation
+ℓ
+n =
+m
+�
+i=1
+pei−fi
+i
++ pgi
+i − 1
+pei−fi+gi
+i
+,
+for m > 1 and the pi distinct, where the ei, fi, and gi are as before. Via isoclinism,
+we may replace GR by a class two nilpotent (stem) group G with identical com-
+muting probability and minimal order. Thus we may assume that Z(GR) = G′
+R
+
+4
+A. Mendelsohn
+(note isoclinism preserves nilpotency class and GR is class two), and moreover
+that none of the Sylow subgroups are abelian. The above equality simplifies to
+ℓ
+n =
+m
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+,
+where the exponents e′
+i, f ′
+i correspond to the group G. We now proceed by in-
+duction on the number of prime factors of |GR|, denoted m.
+By Lemma 14 of [9], if PGR = ℓ
+n in lowest terms, the prime factors of n are
+precisely the prime factors of |GR|. If m = 1, it is known that
+ℓ
+n ̸∈ Rq for any
+prime q and square-free n (in fact, we know this to hold for m ≤ 69 by Theorem
+9 of [4]). Suppose the statement is true up to n = k − 1, and consider the case
+n = k; suppose, for a contradiction, that the commuting probability is equal to
+ℓ
+n, for some square-free integer n and ℓ ≤ n with gcd(ℓ, n) = 1, and without loss
+of generality that n has prime factors equal to the set of pi, i = 1, ..., k:
+ℓ
+n =
+k
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+.
+Rearrange for the following:
+ℓ · pe′
+k
+k
+n · (p
+e′
+k−f ′
+k
+k
++ p
+f ′
+k
+k − 1)
+=
+k−1
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+.
+Writing the left hand side in lowest terms, we have
+ℓ · pe′′
+k
+k
+n′ · (p
+e′
+k−f ′
+k
+k
++ p
+f ′
+k
+k − 1)
+=
+k−1
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+,
+where n′ is not divisible by pk. We have a commuting probability on the right
+hand side with k−1 prime factors; so by the induction hypothesis, the denomina-
+tor of the left hand side has no square factors. But we also have pe′
+k−f ′
+k
+k
++ pf ′
+k
+k − 1
+on the denominator of the left hand side, which is not divisible by pk, and by
+Lemma 14 of [9] must have prime factors equal to the set of pi, for i = 1, ..., k−1;
+moreover, there can be no cancellation between these factors and ℓ, by assump-
+tion on ℓ. But then for at least one index j, n′ · (pe′
+k−f ′
+k
+k
++ pf ′
+k
+k − 1) has a prime
+factor pj with multiplicity at least two, which is a contradiction.
+Remark 1. Since 1
+p is an accumulation point of Rp, 1
+n is an accumulation point
+of R for all n. The above result thus means that many accumulation points of
+R are not contained in R. As well as resolving the first conjecture stated at the
+beginning of this note, the result also makes progress on the sixth conjecture
+stated.
+
+On the Commuting Probability of Finite Rings
+5
+References
+1. Joseph,
+K.
+Several
+Conjectures
+on
+Commutativity
+in
+Algebraic
+Struc-
+tures.
+The
+American
+Mathematical
+Monthly.
+84,
+550-551
+(1977),
+https://doi.org/10.1080/00029890.1977.11994411.
+2. Eberhard, S. Commuting probabilities of finite groups. Bulletin Of The London
+Mathematical Society. 47, 796-808 (2015).
+3. Browning, T. Limit Points of Commuting Probabilities of Finite Groups. (arXiv,
+2022), https://arxiv.org/abs/2201.09402.
+4. Buckley,
+S.
+&
+MacHale,
+D.
+Contrasting
+the
+commut-
+ing
+probabilities
+of
+groups
+and
+rings.
+(arXiv,
+2014),
+https://archive.maths.nuim.ie/staff/sbuckley/Papers/bm_g-vs-r.pdf.
+5. Jur´a˘s, M. & Ursul, M. On commuting probabilities in finite groups and rings.
+Journal Of Algebra Combinatorics Discrete Structures And Applications. (2022),
+https://jacodesmath.com/index.php/jacodesmath/article/view/148.
+6. Castelaz, A. Commutativity Degree of Finite Groups. (Wake Forest University,
+2010), https://books.google.co.uk/books?id=QYxBnQAACAAJ.
+7. Nath, R. & Das, A. On a lower bound of commutativity degree. Rendiconti Del
+Circolo Matematico Di Palermo. 59 pp. 137-142 (2010, 4).
+8. Buckley,
+S.,
+MacHale,
+D.
+&
+Sh´e,
+A.
+Finite
+rings
+with
+many
+commuting
+pairs
+of
+elements.
+(2014),
+https://archive.maths.nuim.ie/staff/sbuckley/Papers/bms.pdf.
+9. Buckley,
+S.
+&
+MacHale,
+D.
+Groups
+with
+Pr(G)
+=
+1/3,
+https://archive.maths.nuim.ie/staff/sbuckley/Papers/bm_GpCP_1_3.pdf.
+10. Lescot, P. Isoclinism Classes and Commutativity Degrees of Finite Groups. Journal
+Of Algebra. 177, 847-869 (1995).
+11. Berkovich, Y. Groups of Prime Power Order, Volume 1. (De Gruyter, 2008),
+https://doi.org/10.1515/9783110208221.
+

Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf,len=188
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='02041v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='RA]  5 Jan 2023  A Note On Square-free Commuting Probabilities  of Finite Rings  Andrew Mendelsohn  Department of EEE, Imperial College London, London, SW7 2AZ, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='mendelsohn18@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='uk  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' It is shown that the commuting probability of a finite ring  cannot be a fraction with square-free denominator, resolving a conjecture  of Buckley and MacHale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  1  Introduction  Let G be a finite group and U denote the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Let the commut-  ing probability of G, PG, be denoted  PG := Pra,b←U(G)(ab = ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  An alternative characterisation is  PG =  1  |G|2 {(x, y) ∈ G2 : xy = yx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Joseph made the following conjectures, where G is the set of all commuting  probabilities of finite groups [1]:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' All limit points of G are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The set G is well ordered by >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The set 0 ∪ G is closed (that is, contains all its accumulation points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  In [2], Eberhard resolved conjectures A and B, and in [3] conjecture C was  resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  One can extend the definition of commuting probability to finite rings: set  PR =  1  |R|2 |{(x, y) ∈ R2 : xy = yx}| =  1  |R|2 |{(x, y) ∈ R2 : xy − yx = 0}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  In [4], the following conjectures were made, where R is the set PR over all finite  rings R:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 1/n ̸∈ R when n ∈ N is square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' R ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' R coincides with the set of values of PG as G ranges over all finite nilpotent  groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' All limit points of R are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Mendelsohn  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' For each 0 < t ≤ 1, there exists ǫt > 0 such that R ∩ (t − ǫt, t) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' R does not contain any of its accumulation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Note conjectures 4, 5, and 6 correspond to Joseph’s conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Moreover, con-  jectures 4 and 5 would follow from the veracity of conjectures 2 or 3, since  Eberhard showed G has rational limit points and is well-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' In [5], conjec-  ture 2 was in fact resolved, and thus conjectures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Moreover, conjecture  3 was partially resolved: the authors obtained that R is a subset of the set of  values of PG as G ranges over all finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We  conclude that conjectures 1, 3, and 6 are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  In this work, we resolve conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2  Preliminaries and Prior Results  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Two finite groups G, H are called isoclinic if G/Z(G) ∼= H/Z(H)  and G′ ∼= H′, and if the diagram below commutes:  G/Z(G) × G/Z(G)  H/Z(H) × H/Z(H)  G′  H′  Isoclinism preserves nilpotency class and commuting probability [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' A stem  group is a group in a given isoclinism class of minimal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' It is well known  that if G is a stem group, then Z(G) ≤ G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' For more on isoclinism, see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Below we state existing results in the literature we will need below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' [5] R ⊂ Gn,2, where Gn,2 is the set of commuting probabilities of all  finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  This statement is proved as follows: let R be a finite ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We can turn  R⊕R into a nilpotent ring of class 3 by endowing it with the multiplication rule  (a, x)(b, y) = (0, ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' This ring can be turned into a nilpotent group GR of class  at most 2 by endowing it with the binary operation a ◦ b = a + b + ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Both  of these transformations preserve the commuting probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Thus the values of  R \\ {1} are a subset of the values PG, running over nilpotent groups G of class  equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Note that if R has size n, then the resulting group GR has order n2,  and if R is noncommutative then the resulting group is nonabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' [7] Pr(G) =  1  |G′|  �  1 + |G′|−1  |G:Z(G)|  �  if and only if G is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' [6] If G is a nilpotent group, then PG ̸= 1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  3  Results  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  1  p ̸∈ R for all p ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  On the Commuting Probability of Finite Rings  3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 1, R is contained within the set of commuting probabilities  of finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 3, this latter set does  not contain 1  p for any prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Denote the set of commuting probabilities of rings of prime power order for  some prime p by Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  1  n ̸∈ Rp for any prime p ∈ N≥2 and n ∈ N>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 1, we need only consider commuting probabilities of finite  nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 2, we know a fortiori the com-  muting probability of finite nilpotent groups of class at most 2 in terms of derived  subgroups and centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Suppose that for some n ∈ N≥2 we have  1  |G′|  �  1 +  |G′| − 1  |G : Z(G)|  �  = 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  By the construction of [5] considered above, wlog let |G| = pe for some even  positive integer e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Then Z(G) = pf and G′ = pg with 0 < g ≤ f < e (since G is  at most class 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Then  1  n = p−g  �  1 + pg − 1  pe−f  �  = p−g + pg − 1  pe−f+g  (1)  = pe−f + pg − 1  pe−f+g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  (2)  For this to hold, some power ph must divide the numerator, with h > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' but this  cannot hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' for if so, then one must have pe−f + pg − 1 = kp, for some integer  k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' But then −1 ≡ 0 mod p, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  ℓ  n ̸∈ R for any squarefree n ∈ N>1 and ℓ < n with gcd(ℓ, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Any finite ring can be turned into a nilpotent group of class at most  2, such that the commuting probability of the ring is equal to the commuting  probability of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The construction (outlined above) turns a commutative  ring into a group of class 1, and a noncommutative ring into a nonabelian group of  class at most 2, therefore of class equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The order of the group is the square  of the order of the ring, so the Sylow subgroups of the group have order at least  the square of a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Since the group is nilpotent, it can be written as a product  of its Sylow subgroups, which are all of class at most 2, and the commuting  probability of the group is the product of the commuting probabilities of its  Sylow subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Thus it remains to analyse the equation  ℓ  n =  m  �  i=1  pei−fi  i  + pgi  i − 1  pei−fi+gi  i  ,  for m > 1 and the pi distinct, where the ei, fi, and gi are as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Via isoclinism,  we may replace GR by a class two nilpotent (stem) group G with identical com-  muting probability and minimal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Thus we may assume that Z(GR) = G′  R  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Mendelsohn  (note isoclinism preserves nilpotency class and GR is class two), and moreover  that none of the Sylow subgroups are abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The above equality simplifies to  ℓ  n =  m  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  ,  where the exponents e′  i, f ′  i correspond to the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We now proceed by in-  duction on the number of prime factors of |GR|, denoted m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  By Lemma 14 of [9], if PGR = ℓ  n in lowest terms, the prime factors of n are  precisely the prime factors of |GR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' If m = 1, it is known that  ℓ  n ̸∈ Rq for any  prime q and square-free n (in fact, we know this to hold for m ≤ 69 by Theorem  9 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Suppose the statement is true up to n = k − 1, and consider the case  n = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' suppose, for a contradiction, that the commuting probability is equal to  ℓ  n, for some square-free integer n and ℓ ≤ n with gcd(ℓ, n) = 1, and without loss  of generality that n has prime factors equal to the set of pi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=', k:  ℓ  n =  k  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Rearrange for the following:  ℓ · pe′  k  k  n · (p  e′  k−f ′  k  k  + p  f ′  k  k − 1)  =  k−1  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Writing the left hand side in lowest terms, we have  ℓ · pe′′  k  k  n′ �� (p  e′  k−f ′  k  k  + p  f ′  k  k − 1)  =  k−1  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  ,  where n′ is not divisible by pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We have a commuting probability on the right  hand side with k−1 prime factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' so by the induction hypothesis, the denomina-  tor of the left hand side has no square factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' But we also have pe′  k−f ′  k  k  + pf ′  k  k − 1  on the denominator of the left hand side, which is not divisible by pk, and by  Lemma 14 of [9] must have prime factors equal to the set of pi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=', k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  moreover, there can be no cancellation between these factors and ℓ, by assump-  tion on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' But then for at least one index j, n′ · (pe′  k−f ′  k  k  + pf ′  k  k − 1) has a prime  factor pj with multiplicity at least two, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Since 1  p is an accumulation point of Rp, 1  n is an accumulation point  of R for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The above result thus means that many accumulation points of  R are not contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' As well as resolving the first conjecture stated at the  beginning of this note, the result also makes progress on the sixth conjecture  stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  On the Commuting Probability of Finite Rings  5  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Joseph,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Several  Conjectures  on  Commutativity  in  Algebraic  Struc-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  The  American  Mathematical  Monthly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  84,  550-551  (1977),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='1080/00029890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='11994411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Eberhard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Commuting probabilities of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Bulletin Of The London  Mathematical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 47, 796-808 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Browning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Limit Points of Commuting Probabilities of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (arXiv,  2022), https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='09402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Buckley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  &  MacHale,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Contrasting  the  commut-  ing  probabilities  of  groups  and  rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  (arXiv,  2014),  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='nuim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ie/staff/sbuckley/Papers/bm_g-vs-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Jur´a˘s, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' & Ursul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' On commuting probabilities in finite groups and rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Journal Of Algebra Combinatorics Discrete Structures And Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (2022),  https://jacodesmath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='com/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='php/jacodesmath/article/view/148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Castelaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Commutativity Degree of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (Wake Forest University,  2010), https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='uk/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='id=QYxBnQAACAAJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Nath, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' & Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' On a lower bound of commutativity degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Rendiconti Del  Circolo Matematico Di Palermo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 59 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 137-142 (2010, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Buckley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=',  MacHale,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  &  Sh´e,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Finite  rings  with  many  commuting  pairs  of  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  (2014),  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='nuim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ie/staff/sbuckley/Papers/bms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Buckley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  &  MacHale,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Groups  with  Pr(G)  =  1/3,  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='nuim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ie/staff/sbuckley/Papers/bm_GpCP_1_3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Lescot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Isoclinism Classes and Commutativity Degrees of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Journal  Of Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 177, 847-869 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Berkovich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Groups of Prime Power Order, Volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (De Gruyter, 2008),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='1515/9783110208221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}

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+Laser Field Initiation of Higher Order Poles of S-Matrix-Optical Realization of
+Field Theoretic Models*
+G.S. Agarwal1,2
+1Physical Research Laboratory, Navrangpura, Ahmedabad, India
+2Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore, India
+We discuss the possibility of converting a simple pole in the radiative decay of a state into a pole of
+higher order by using resonant electromagnetic fields. This process of creation of higher order pole
+is controllable by the intensity of the laser field. We use density matrix and Liouville space and
+present the modification of the Lorentzian line shapes (Breit-Wigner formula) for example to ones
+involving square of Lorentzian and derivatives of Lorentzians.
+INTRODUCTION
+In a classic paper Goldberger and Watson [1] con-
+sidered the possibility that the decay law for an un-
+stable particle can be more complex than a simple
+exponential. They showed the possibility of the ex-
+istence of the poles of S-matrix which were not nec-
+essarily simple poles. Since then, higher order poles
+have been extensively studied. Recently, there is re-
+vival [2,3] of interest in such studies and in particular
+Bhamathi and Sudarshan have analyzed several field
+theoretic models like Friedrich-Lee model, cascade
+model and their extensions. They examine the spec-
+trum (complex) of eigenvalues for such models. A
+related question is how the Breit-Wigner line shape
+formula is modified if S-matrix possess higher order
+poles.
+In this paper we examine the possibility of cre-
+ation of the higher order poles using laser fields. We
+consider the decay of say excited state of an atom.
+Normally this decay is described by the Wigner-
+Weisskopf theory which leads to exponential decay
+law.
+We next discuss the case when the excited
+state is coupled to another state by a resonant elec-
+tromagnetic field. In such a case we show that for
+appropriate value of the intensity of the laser field
+the corresponding spectral function has a pole of or-
+der two. We calculate the resulting line shape and
+discuss the line narrowing etc. We emphasize that
+we work within the framework of density matrices
+and hence we work in Liouville space rather than
+in Hilbert space. We present optical realization of
+various field theoretic models.
+Consider the decay of the state |1⟩ into the states
+|3⟩ and |2⟩ at the rates 2γ1 and 2γ2 respectively as
+shown in Fig.1 (with Gl = 0, △l = 0).
+It is well
+known that the rate of decay of the population in
+|1⟩ is given by
+ρ11(t) = ρ11(0)exp(−2(γ1 + γ2)t).
+(1)
+Here ρ is the density matrix of the atom. The spec-
+trum of the spontaneously emitted photons will con-
+sist of two Lorentzians centered at ω13 and ω12 with
+a half width (γ1 + γ2). Let us concentrate on the
+emission on the transition |1⟩ ↔ |3⟩. The spectrum
+will be described by the well-known form
+S(ω) =
+γ1/π
+(γ1 + γ2)2 + (ω − ω13)2 .
+(2)
+Note that γ2 will be zero if the decay channel |1⟩ →
+|2⟩ is not allowed.
+We will discuss how the laser
+fields could be used to modify significantly the re-
+sults predicted by (1) and (2).
+LIOUVILLE SPACE FORMULATION OF
+DECAY
+We next recall how the spectrum is calculated in
+the density matrix framework [4]. We have included
+this material for completeness so that our discussion
+in subsequent sections can be followed by the non-
+Quantum optics practitioners.
+Consider a system
+with two states |1⟩ and |3⟩ interacting with the vac-
+uum of the electromagnetic field. The Hamiltonian
+can be written in the form
+H = ¯hω13 |1⟩ ⟨1| +
+�
+ks
+¯hωksa†
+ksaks + V13
+V13 =
+�
+ks
+(¯hgksa†
+ks |1⟩ ⟨3| + h.c.).
+(3)
+The vacuum modes are characterized by the propa-
+gation index −→k and the polarization index s. The
+aks, a†
+ks represent annihilation and creation opera-
+tors for the mode −→k s. The V13 describes the decay
+of |1⟩ to |3⟩. The gks is the coupling constant be-
+tween the field mode and the atom.
+We use the
+weak coupling assumption and the flat nature of the
+density of states of the electromagnetic vacuum to
+eliminate the degrees of freedom associated with the
+field vacuum. We derive an equation for the den-
+sity matrix of the atomic system alone which can be
+written in the form
+∂ρ
+∂t = Lρ
+(4)
+or in terms of the components as
+˙ρ11 = −2γ1ρ11,
+˙ρ13 = −iω13ρ13 − γ1ρ13,
+˙ρ33 = 2γ1ρ11,
+etc., 2γ1 =
+�
+ks
+|gks|2δ(ω13 − ωks).
+(5)
+This yields steady state as well as transient behavior.
+The spectrum of radiation is related to the Fourier
+transform of the two time dipole correlation func-
+tion, for example in the above case to
+S(ω) = 1
+π Re[S(z)|z=+iω],
+(6)
+S(z) ≡
+� ∞
+0
+dτe−zτ ⟨A13(t + τ)A31(t)⟩ ,
+A13 = A†
+31 = |1⟩ ⟨3| .
+(7)
+The poles of S(z) determine the spectrum. For the
+standard problem S(z) has simple poles.
+The two time correlation function is calculated
+from the solution of (4) and by using the quantum
+———————————————————————————————————————————————————————
+*Published in “Frontiers of Quantum Optics and Laser Physics”, p.155-165, ed. S.Y. Zhu, M.S. Zubairy and M.O. Scully
+(Springer, 1997). This work on higher order poles of S matrix has close connection to the exceptional point physics. Thus this
+work brings out how the exceptional point physics in active systems can be manipulated by laser field. See also G. S. Agarwal,
+Quantum Optics, Cambridge University Press, 2012, Section 17.3.1.
+arXiv:2301.05179v1  [physics.optics]  12 Jan 2023
+
+2
+regression theorem. For completeness, we state what
+it means. We write the solution of (4) as
+ραβ(t + τ) =
+�
+m,n
+Gαβ,mn(τ)ρmn(t).
+(8)
+It should be borne in mind that in the Liouville space
+ραβ is an element of the column matrix.
+We can
+rewrite (8) as
+⟨Aβα(t + τ)⟩ =
+�
+m,n
+Gαβ,mn(τ) ⟨Anm(t)⟩ ,
+(9)
+then the quantum regression theorem leads to two
+time correlation function:
+⟨Aβα(t + τ)Apq(t)⟩ ≡
+�
+m,n
+Gαβ,mn(τ) ⟨Anm(t)Apq(t)⟩
+=
+�
+m,n
+Gαβ,mn(τ) ⟨Anq(t)⟩ δmp
+=
+�
+m,n
+Gαβ,mn(τ)δmpρqn(t).
+(10)
+On using (10) in (6) it is clear that S(z) is related to
+the Laplace transform of G(τ) or to (z −L)−1. Gen-
+erally, the Liouvilliean matrix relevant for the cal-
+culation of (10) decomposes in block diagonal form
+and only a part of L determines the decay or the
+spectral line shapes. For the two level example, the
+correlation function is essentially determined by a
+single equation for ρ13. If there is more than one
+decay channel, then additional terms appear in (5),
+for example, for the case shown in Fig.1, γ1 should
+be replaced by (γ1 +γ2) in the two first equations in
+(5).
+Figure 1. Schematic illustration of the scheme that leads
+to the creation of poles of order two in the decay of the
+state |1⟩; which could be pumped in two different ways ei-
+ther from the state |3⟩ or from a state outside the system.
+This provides the realization of the extended Friedrich-
+Lee model.
+CREATION OF A DOUBLE POLE
+We next demonstrate how by using external elec-
+tromagnetic fields we can convert simple poles of L
+into poles of higher order. For this purpose, we con-
+sider the application of an electromagnetic field that
+is tuned close to the transition frequency ω12 [Fig. 1.
+Λ0 = 0, Λ ̸= 0, Gl ̸= 0]. The Hamiltonian describing
+this system can be written as
+H = ¯hω13 |1⟩ ⟨1|+¯h(ω13−ω12) |2⟩ ⟨2|+Hext+V12+V13,
+(11)
+where Vαβ describes the decay on the transition
+|α⟩ → |β⟩ and where
+Hext = −¯h(Gle−iωlt |1⟩ ⟨2| + h.c.),
+(12)
+Gl = (−→d 12 · −→
+E l/¯h).
+(13)
+The parameter 2Gl is the Rabi frequency of the field
+and is a measure of the strength of the laser field
+applied on the transition |1⟩ ↔ |2⟩.
+The Hamil-
+tonian (11) is time-dependent.
+However one can
+make a canonical transformation to reduce it to a
+time-independent Hamiltonian. In the special case
+V12 → 0 the model (11) is equivalent to the ex-
+tended Friedrich-Lee model.
+We have thus pro-
+duced a realization of a field-theoretic model in the
+context of atoms interacting with laser fields.
+In
+our case lasers are used to control the decay pro-
+cess. Note that we have two control parameters ωl
+and Gl, to manipulate the nature of the poles of
+L. The situation shown in Fig. 1 is realizable in
+many atoms, molecules dopants in solid matrices,
+etc. For example, in 87Rb vapor, the states |1⟩, |2⟩
+and |3⟩ could be the states 5P 3
+2 , 5S 1
+2 , F = 2 and
+5S 1
+2 , F = 1, respectively.
+We eliminate the opti-
+cal frequencies by making canonical transformations
+ρ13 → ρ13e−iω13t, ρ12 → ρ12e−iωlt etc. After canon-
+ical transformations and after eliminating vacuum
+degrees of freedom using the master equation tech-
+niques the density matrix equations read [5]
+˙ρ11 = −2(γ1 + γ2 + Λ)ρ11 + 2Λρ33 + iGlρ21 − iG∗
+l ρ12,
+˙ρ22 = 2γ2ρ11 − iGlρ21 + iG∗
+l ρ12,
+˙ρ21 = −(Γ21 − i∆l)ρ21 − iG∗
+l ρ22 + iG∗
+l ρ11,
+˙ρ31 = −Γ31ρ31 − iG∗
+l ρ32,
+˙ρ32 = −(Γ32 + i∆l)ρ32 − iGlρ31.
+(14)
+Here we have also included a pumping parameter λ
+to pump the population from the level |3⟩ to |1⟩.
+The Γ
+′s
+αβ give the decay of off-diagonal elements ρ
+′s
+αβ
+of the density matrix and are given by
+Γ31 = γ1 + γ2 + 2Λ, Γ32 = Λ,
+Γ21 = γ1 + γ2 + Λ, ∆2 = ω12 − ωl.
+(15)
+From (14) and the quantum regression theorem we
+derive coupled equations for two time atomic corre-
+lation functions
+�
+d
+dτ +
+�
+Γ31
+iG∗
+l
+iGl
+Γ32 + i∆2
+�� �
+⟨A13(t + τ)A31(t)⟩
+⟨A23(t + τ)A31(t)⟩
+�
+= 0.
+(16)
+These are to be solved subject to initial conditions
+⟨A13A31⟩ = ρ11, ⟨A23A31⟩ = ρ12,
+(17)
+which in turn are determined from the steady state
+solution of (14). Clearly the poles of L that deter-
+mine the spectral characteristics are given by
+P(z) = (z + Γ31)(z + Γ32 + i∆l) + |Gl|2.
+(18)
+The zeroes of (18) for ∆l = 0 are shown in Fig. 2.
+The conditions under which P(z) has double zero
+are
+∆l = 0, (Γ32 − Γ31)2 = 4|Gl|2.
+(19)
+The double zero z0 occurs at the bifurcation point
+in Fig. 2
+z0 = −1
+2(Γ31 + Γ32).
+(20)
+We therefore conclude [6] that a simple pole can be
+converted into a double pole in a laboratory experi-
+ment by applying an electromagnetic field resonant
+with the transition |1⟩ ↔ |2⟩ and with Rabi fre-
+quency equal to |Γ31 − Γ32|.
+
+[1)
+V
+W1, G
+Y1
+[2)
+[3]3
+Figure 2. Motion of the zeroes of (18) for Γ31 = 1, Γ32 =
+0.2.
+Note the presence of the bifurcation point.
+This
+is precisely the point where we create a pole of order
+two. The solid curve represents Im(z) + 0.6 whereas the
+dashed curve gives Re(z).
+LINE SHAPES AND DOUBLE POLES
+The line shape can be calculated from the solution
+of (16) and (6):
+S(ω) ≡ ρ11Re[
+(γ2 + Γ32 − iδ)
+(Γ31 − iδ)(Γ32 − iδ) + |Gl|2 ]
+(21)
+which under the double pole condition 2|Gl| = |Γ31−
+Γ32| reduces to
+S(ω) = ρ11Re[γ2 + Γ32 − iδ
+(−iδ + γ0)2 ]
+= ρ11
+δ2(γ1 + 2Λ) + γ2
+0(γ2 + Λ)
+(δ2 + γ2
+0)2
+,
+γ0 = 1
+2(γ1 + γ2 + 3Λ).
+(22)
+This is the modification of the line shape formula.
+Note the double hump structure of the line shape.
+Note further the sensitiveness of S(ω) to the pump-
+ing parameter Λ. In the limit γ2 → 0 and Λ ≪ γ1,
+(22) reduces to
+S(ω) ≡ ρ11
+γ1(δ2 + γ1
+4 Λ)
+(δ2 + γ2
+1
+4 )2
+(23)
+It is also interesting to note, that the scale param-
+eter is now (γ1/2) rather than γ1. Thus the total
+line shape is a sum of (a) Square of the Lorentzian
+(b) derivative of the Lorentzian (ζ/(ζ + γ0)2 ≡
+−ζ ∂
+∂ζ (
+1
+ζ+γ )).
+It is possible to consider an alternate model of
+pumping obtained by setting Λ = 0 in Fig. 1. As-
+suming that γ2 = 0, one can show that instead of
+(23) the spectral line shape is now given by
+S(ω) ≡
+γ1ρ11δ2
+(δ2 + γ2
+1
+4 )2 = (−δ ∂
+∂δ ) (γ1/2)ρ11
+(δ2 + γ2
+1/4)
+(24)
+which is shown in Fig.
+3.
+The figure also shows
+for comparison the Breit-Wigner formula (2) Note
+the double hump structure of the line shape. The
+maxima now occur at δ = ±γ1/2. From Eq. (14)
+we can also compute the time dependence of ρ11(t)
+under the condition of a double pole. The result is
+ρ11(t) = (1 − γ1t
+2 )2e−γ1t
+(25)
+It is again interesting to note that the time scale
+is governed by γ1/2 rather than γ1.
+The basic idea presented above is easily extended
+to more complex situations.
+For example, two-
+photon decay in the system as shown in figure 4
+Figure 3.
+The modified line shape (24) (dashed) as
+a function of δ/γ1 and its comparison with the Breit-
+Wigner line shape (solid).
+which is easily realizable atoms and molecules. The
+full Hamiltonian for this system can be written as
+H = ¯hω13 |1⟩ ⟨1| + ¯hω23 |2⟩ ⟨2| + ¯hω43 |4⟩ ⟨4|
+− ¯h(Gle−iωlt |4⟩ ⟨2| + h.c.)
++
+�
+ks
+¯hωksa†
+kxaks + V12 + V23 + V42
+(26)
+where the meaning of different terms is obvious.
+Again a canonical transformation will change the
+above H into a time-independent H. For V42 → 0,
+the above Hamiltonian becomes identical to the one
+for the quantum field theoretic extended cascade
+field model. We thus have a simple atomic realiza-
+tion of the field-theoretic model. As shown recently
+[7], this system exhibits very interesting two photon
+absorption characteristics. Clearly, the electromag-
+netic coupling between the levels |2⟩ and |4⟩ can pro-
+duce a double pole in the decay of the system. It is
+interesting that a system equivalent to this has been
+studied by Bhamathi and Sudarshan [2].
+Figure 4. A scheme involving laser coupling the inter-
+mediate state |2⟩ which will create pole of order two in
+the two-photon decay. This provides an analog of the
+extended cascade model.
+DOUBLE POLES AND INTERFERENCE
+EFFECTS
+The existence of double poles and the possibility
+of a line shape which is a derivative of Lorentzian
+suggest that the quantum interferences must be
+crucial.
+This is indeed the case as can be seen
+from the following considerations. The electromag-
+netic coupling of |1⟩ and |2⟩ produced dressed states
+|ψ±⟩ =
+1
+√
+2(± |1⟩ + |2⟩) with eigenvalues ±Gl. Since
+Gl ∼ γ, the two states are within the radiative
+line width. We pump the population in the state
+|1⟩ which is equivalent to pumping in both |ψ±⟩ as
+
+2.0
+1.5
+1.0
+.......
+.=
+0.5
+0.0
+-0.5
+-1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Gt1.0
+1
+-
+-
+0.8
+-
+-
+-
+-
+-
+1
+1
+0.6
+-
+1
+1
+S
+1
+-
+1
+1
+1
+0.4
+1
+1
+1
+0.2
+I:
+0.0
+-2
+0
+2
+4
+S|1)
+14)
+Laser Coupling
+[2)
+ 3)4
+|1⟩ = (|ψ+⟩ + |ψ−⟩)/
+√
+2. Both states |ψ±⟩ can de-
+cay to |3⟩ as |ψ±⟩ involve admixtures of |1⟩ and |2⟩.
+These two decays will not be independent [8,9] as
+−→d +3 · −→d ∗
+−3 ̸= 0 and as Gl ∼ γ.
+EXPONENTIAL DECAY RECOVERED
+We also examine the initial conditions for our sys-
+tem which would result in exponential decay. From
+Eq. (16) it is seen that
+d
+dτ ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩
++ γ1
+2 ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩ = 0 (27)
+if Gl = γ1
+2 , γ2 → 0. Thus the correlation function
+defined in terms of the vector ˜ψ =
+1
+√
+2(|1⟩ + i |2⟩)
+obeys simple exponential decay law with a time scale
+governed by γ1/2 rather than γ1:
+�
+A ˜
+ψ3(t + τ)A3 ˜
+ψ(t)
+�
+= e−γ1τ/2 �
+A ˜
+ψ ˜
+ψ(t)
+�
+(28)
+Thus a pumping of the system to the state ˜ψ rather
+than |1⟩ will result in exponential decay[10].
+Thus, in conclusion, we have shown how higher-
+order poles in the decay of states can be produced
+by using resonant electromagnetic fields. We demon-
+strated this by creating a pole of order two. Clearly,
+the technique is quite versatile and by using combi-
+nations of electromagnetic fields we can create poles
+of higher order.
+I thank George Sudarshan for discussions on
+higher order poles of S-Matrix and R.P. Singh for
+help in preparation of this paper.
+[1] M.L. Goldberger and K.M. Watson, Phys. Rev. 136,
+B 1472 (1964); J, S. Bell and C.J. Goebel, Phys,
+Rev. 138, B 1198 (1965)
+[2] G. Bhamathi and E.C.G. Sudarshan, Int. J. Mod
+Phys. B 10, 1531 (1996); see also E.C.G Sudarshan,
+Phys. Rev. A 50, 2006 (1994); E.C.G. Sudarshan,
+C.B. Chiu and G. Bhamathi, Phys. Rev. D 46, 3508
+(1992).
+[3] A. Bohm, S. Maxson and M. Loewe, Physica A, in
+press; A. Mondragon and E. Her- nandez, J. Phys.
+A 26, 5595 (1993); C. Puntmann, paper presented
+at the International Colloquium on Group Theory,
+Goslar, Germany 1996.
+[4] G.S. Agarwal, Quantum Optics (Springer-Verlag,
+Berlin, 1974).
+[5] G.S. Agarwal, Phys. Rev. A 54, Rapid Commun.
+3734 (1996).
+[6] An almost trivial case occurs when Γ23 = 0, γ2 = 0
+and no pumping (Λ = 0). The atom can start in
+state |1⟩. Then one can work with a nonhermitian
+Hamiltonian
+�
+−iγ1 G
+G
+0
+�
+which has identical real
+eigenvalues if 2G = γ1. This case has been pre-
+viously considered in literature (H. Steudel, Ann.
+Physik 22, 113 (1969).
+[7] G.S. Agarwal and W. Harshawardhan, Phys. Rev,
+Lett. 77, 1039 (1996).
+[8] G.S. Agarwal, Quantum Optics (Springer-Verlag,
+Berlin, 1974) pp. 94-96.
+[9] A. Imamoglu, Phys. Rev. A 40, 2835 (1989); S.Y.
+Zhu and M.O. Scully, Phys. Rev. Lett. 76, 388
+(1996); D.A. Cardimona, M.G. Raymer and C.R.
+Stroud Jr., J. Phys. B 15, 55 (1982).
+[10] Pumping the system in the state ≫ is possible using
+an excitation pulse with phase switching at appro-
+priate instant (cf. Y.S. Bai, A.G. Yodh and T.W.
+Mossberg, Phys. Rev. Lett. 55, 1277 (1984)).
+

aNE4T4oBgHgl3EQfnw2m/content/tmp_files/load_file.txt

@@ -0,0 +1,289 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf,len=288
+page_content='Laser Field Initiation of Higher Order Poles of S-Matrix-Optical Realization of  Field Theoretic Models*  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal1,2  1Physical Research Laboratory, Navrangpura, Ahmedabad, India  2Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore, India  We discuss the possibility of converting a simple pole in the radiative decay of a state into a pole of  higher order by using resonant electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This process of creation of higher order pole  is controllable by the intensity of the laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We use density matrix and Liouville space and  present the modification of the Lorentzian line shapes (Breit-Wigner formula) for example to ones  involving square of Lorentzian and derivatives of Lorentzians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  INTRODUCTION  In a classic paper Goldberger and Watson [1] con-  sidered the possibility that the decay law for an un-  stable particle can be more complex than a simple  exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' They showed the possibility of the ex-  istence of the poles of S-matrix which were not nec-  essarily simple poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Since then, higher order poles  have been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Recently, there is re-  vival [2,3] of interest in such studies and in particular  Bhamathi and Sudarshan have analyzed several field  theoretic models like Friedrich-Lee model, cascade  model and their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' They examine the spec-  trum (complex) of eigenvalues for such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A  related question is how the Breit-Wigner line shape  formula is modified if S-matrix possess higher order  poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  In this paper we examine the possibility of cre-  ation of the higher order poles using laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We  consider the decay of say excited state of an atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Normally this decay is described by the Wigner-  Weisskopf theory which leads to exponential decay  law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We next discuss the case when the excited  state is coupled to another state by a resonant elec-  tromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' In such a case we show that for  appropriate value of the intensity of the laser field  the corresponding spectral function has a pole of or-  der two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We calculate the resulting line shape and  discuss the line narrowing etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We emphasize that  we work within the framework of density matrices  and hence we work in Liouville space rather than  in Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We present optical realization of  various field theoretic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Consider the decay of the state |1⟩ into the states  |3⟩ and |2⟩ at the rates 2γ1 and 2γ2 respectively as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='1 (with Gl = 0, △l = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  It is well  known that the rate of decay of the population in  |1⟩ is given by  ρ11(t) = ρ11(0)exp(−2(γ1 + γ2)t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (1)  Here ρ is the density matrix of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The spec-  trum of the spontaneously emitted photons will con-  sist of two Lorentzians centered at ω13 and ω12 with  a half width (γ1 + γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Let us concentrate on the  emission on the transition |1⟩ ↔ |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The spectrum  will be described by the well-known form  S(ω) =  γ1/π  (γ1 + γ2)2 + (ω − ω13)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (2)  Note that γ2 will be zero if the decay channel |1⟩ →  |2⟩ is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We will discuss how the laser  fields could be used to modify significantly the re-  sults predicted by (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  LIOUVILLE SPACE FORMULATION OF  DECAY  We next recall how the spectrum is calculated in  the density matrix framework [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We have included  this material for completeness so that our discussion  in subsequent sections can be followed by the non-  Quantum optics practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Consider a system  with two states |1⟩ and |3⟩ interacting with the vac-  uum of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The Hamiltonian  can be written in the form  H = ¯hω13 |1⟩ ⟨1| +  �  ks  ¯hωksa†  ksaks + V13  V13 =  �  ks  (¯hgksa†  ks |1⟩ ⟨3| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (3)  The vacuum modes are characterized by the propa-  gation index −→k and the polarization index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The  aks, a†  ks represent annihilation and creation opera-  tors for the mode −→k s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The V13 describes the decay  of |1⟩ to |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The gks is the coupling constant be-  tween the field mode and the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We use the  weak coupling assumption and the flat nature of the  density of states of the electromagnetic vacuum to  eliminate the degrees of freedom associated with the  field vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We derive an equation for the den-  sity matrix of the atomic system alone which can be  written in the form  ∂ρ  ∂t = Lρ  (4)  or in terms of the components as  ˙ρ11 = −2γ1ρ11,  ˙ρ13 = −iω13ρ13 − γ1ρ13,  ˙ρ33 = 2γ1ρ11,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=', 2γ1 =  �  ks  |gks|2δ(ω13 − ωks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (5)  This yields steady state as well as transient behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The spectrum of radiation is related to the Fourier  transform of the two time dipole correlation func-  tion, for example in the above case to  S(ω) = 1  π Re[S(z)|z=+iω],  (6)  S(z) ≡  � ∞  0  dτe−zτ ⟨A13(t + τ)A31(t)⟩ ,  A13 = A†  31 = |1⟩ ⟨3| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (7)  The poles of S(z) determine the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For the  standard problem S(z) has simple poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The two time correlation function is calculated  from the solution of (4) and by using the quantum  ———————————————————————————————————————————————————————  Published in “Frontiers of Quantum Optics and Laser Physics”, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='155-165, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Zubairy and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Scully  (Springer, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This work on higher order poles of S matrix has close connection to the exceptional point physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Thus this  work brings out how the exceptional point physics in active systems can be manipulated by laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' See also G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal,  Quantum Optics, Cambridge University Press, 2012, Section 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='05179v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='optics]  12 Jan 2023  2  regression theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For completeness, we state what  it means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We write the solution of (4) as  ραβ(t + τ) =  �  m,n  Gαβ,mn(τ)ρmn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (8)  It should be borne in mind that in the Liouville space  ραβ is an element of the column matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We can  rewrite (8) as  ⟨Aβα(t + τ)⟩ =  �  m,n  Gαβ,mn(τ) ⟨Anm(t)⟩ ,  (9)  then the quantum regression theorem leads to two  time correlation function:  ⟨Aβα(t + τ)Apq(t)⟩ ≡  �  m,n  Gαβ,mn(τ) ⟨Anm(t)Apq(t)⟩  =  �  m,n  Gαβ,mn(τ) ⟨Anq(t)⟩ δmp  =  �  m,n  Gαβ,mn(τ)δmpρqn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (10)  On using (10) in (6) it is clear that S(z) is related to  the Laplace transform of G(τ) or to (z −L)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Gen-  erally, the Liouvilliean matrix relevant for the cal-  culation of (10) decomposes in block diagonal form  and only a part of L determines the decay or the  spectral line shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For the two level example, the  correlation function is essentially determined by a  single equation for ρ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' If there is more than one  decay channel, then additional terms appear in (5),  for example, for the case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='1, γ1 should  be replaced by (γ1 +γ2) in the two first equations in  (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Schematic illustration of the scheme that leads  to the creation of poles of order two in the decay of the  state |1⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' which could be pumped in two different ways ei-  ther from the state |3⟩ or from a state outside the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  This provides the realization of the extended Friedrich-  Lee model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  CREATION OF A DOUBLE POLE  We next demonstrate how by using external elec-  tromagnetic fields we can convert simple poles of L  into poles of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For this purpose, we con-  sider the application of an electromagnetic field that  is tuned close to the transition frequency ω12 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Λ0 = 0, Λ ̸= 0, Gl ̸= 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The Hamiltonian describing  this system can be written as  H = ¯hω13 |1⟩ ⟨1|+¯h(ω13−ω12) |2⟩ ⟨2|+Hext+V12+V13,  (11)  where Vαβ describes the decay on the transition  |α⟩ → |β⟩ and where  Hext = −¯h(Gle−iωlt |1⟩ ⟨2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='),  (12)  Gl = (−→d 12 · −→  E l/¯h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (13)  The parameter 2Gl is the Rabi frequency of the field  and is a measure of the strength of the laser field  applied on the transition |1⟩ ↔ |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The Hamil-  tonian (11) is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  However one can  make a canonical transformation to reduce it to a  time-independent Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' In the special case  V12 → 0 the model (11) is equivalent to the ex-  tended Friedrich-Lee model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We have thus pro-  duced a realization of a field-theoretic model in the  context of atoms interacting with laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  In  our case lasers are used to control the decay pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Note that we have two control parameters ωl  and Gl, to manipulate the nature of the poles of  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The situation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 1 is realizable in  many atoms, molecules dopants in solid matrices,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For example, in 87Rb vapor, the states |1⟩, |2⟩  and |3⟩ could be the states 5P 3  2 , 5S 1  2 , F = 2 and  5S 1  2 , F = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We eliminate the opti-  cal frequencies by making canonical transformations  ρ13 → ρ13e−iω13t, ρ12 → ρ12e−iωlt etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' After canon-  ical transformations and after eliminating vacuum  degrees of freedom using the master equation tech-  niques the density matrix equations read [5]  ˙ρ11 = −2(γ1 + γ2 + Λ)ρ11 + 2Λρ33 + iGlρ21 − iG∗  l ρ12,  ˙ρ22 = 2γ2ρ11 − iGlρ21 + iG∗  l ρ12,  ˙ρ21 = −(Γ21 − i∆l)ρ21 − iG∗  l ρ22 + iG∗  l ρ11,  ˙ρ31 = −Γ31ρ31 − iG∗  l ρ32,  ˙ρ32 = −(Γ32 + i∆l)ρ32 − iGlρ31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (14)  Here we have also included a pumping parameter λ  to pump the population from the level |3⟩ to |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The Γ  ′s  αβ give the decay of off-diagonal elements ρ  ′s  αβ  of the density matrix and are given by  Γ31 = γ1 + γ2 + 2Λ, Γ32 = Λ,  Γ21 = γ1 + γ2 + Λ, ∆2 = ω12 − ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (15)  From (14) and the quantum regression theorem we  derive coupled equations for two time atomic corre-  lation functions  �  d  dτ +  �  Γ31  iG∗  l  iGl  Γ32 + i∆2  �� �  ⟨A13(t + τ)A31(t)⟩  ⟨A23(t + τ)A31(t)⟩  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (16)  These are to be solved subject to initial conditions  ⟨A13A31⟩ = ρ11, ⟨A23A31⟩ = ρ12,  (17)  which in turn are determined from the steady state  solution of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Clearly the poles of L that deter-  mine the spectral characteristics are given by  P(z) = (z + Γ31)(z + Γ32 + i∆l) + |Gl|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (18)  The zeroes of (18) for ∆l = 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The conditions under which P(z) has double zero  are  ∆l = 0, (Γ32 − Γ31)2 = 4|Gl|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (19)  The double zero z0 occurs at the bifurcation point  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 2  z0 = −1  2(Γ31 + Γ32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (20)  We therefore conclude [6] that a simple pole can be  converted into a double pole in a laboratory experi-  ment by applying an electromagnetic field resonant  with the transition |1⟩ ↔ |2⟩ and with Rabi fre-  quency equal to |Γ31 − Γ32|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [1)  V  W1, G  Y1  [2)  [3]3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Motion of the zeroes of (18) for Γ31 = 1, Γ32 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Note the presence of the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  This  is precisely the point where we create a pole of order  two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The solid curve represents Im(z) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='6 whereas the  dashed curve gives Re(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  LINE SHAPES AND DOUBLE POLES  The line shape can be calculated from the solution  of (16) and (6):  S(ω) ≡ ρ11Re[  (γ2 + Γ32 − iδ)  (Γ31 − iδ)(Γ32 − iδ) + |Gl|2 ]  (21)  which under the double pole condition 2|Gl| = |Γ31−  Γ32| reduces to  S(ω) = ρ11Re[γ2 + Γ32 − iδ  (−iδ + γ0)2 ]  = ρ11  δ2(γ1 + 2Λ) + γ2  0(γ2 + Λ)  (δ2 + γ2  0)2  ,  γ0 = 1  2(γ1 + γ2 + 3Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (22)  This is the modification of the line shape formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Note the double hump structure of the line shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Note further the sensitiveness of S(ω) to the pump-  ing parameter Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' In the limit γ2 → 0 and Λ ≪ γ1,  (22) reduces to  S(ω) ≡ ρ11  γ1(δ2 + γ1  4 Λ)  (δ2 + γ2  1  4 )2  (23)  It is also interesting to note, that the scale param-  eter is now (γ1/2) rather than γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Thus the total  line shape is a sum of (a) Square of the Lorentzian  (b) derivative of the Lorentzian (ζ/(ζ + γ0)2 ≡  −ζ ∂  ∂ζ (  1  ζ+γ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  It is possible to consider an alternate model of  pumping obtained by setting Λ = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' As-  suming that γ2 = 0, one can show that instead of  (23) the spectral line shape is now given by  S(ω) ≡  γ1ρ11δ2  (δ2 + γ2  1  4 )2 = (−δ ∂  ∂δ ) (γ1/2)ρ11  (δ2 + γ2  1/4)  (24)  which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The figure also shows  for comparison the Breit-Wigner formula (2) Note  the double hump structure of the line shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The  maxima now occur at δ = ±γ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' (14)  we can also compute the time dependence of ρ11(t)  under the condition of a double pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The result is  ρ11(t) = (1 − γ1t  2 )2e−γ1t  (25)  It is again interesting to note that the time scale  is governed by γ1/2 rather than γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The basic idea presented above is easily extended  to more complex situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  For example, two-  photon decay in the system as shown in figure 4  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The modified line shape (24) (dashed) as  a function of δ/γ1 and its comparison with the Breit-  Wigner line shape (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  which is easily realizable atoms and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The  full Hamiltonian for this system can be written as  H = ¯hω13 |1⟩ ⟨1| + ¯hω23 |2⟩ ⟨2| + ¯hω43 |4⟩ ⟨4|  − ¯h(Gle−iωlt |4⟩ ⟨2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=')  +  �  ks  ¯hωksa†  kxaks + V12 + V23 + V42  (26)  where the meaning of different terms is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Again a canonical transformation will change the  above H into a time-independent H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For V42 → 0,  the above Hamiltonian becomes identical to the one  for the quantum field theoretic extended cascade  field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We thus have a simple atomic realiza-  tion of the field-theoretic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' As shown recently  [7], this system exhibits very interesting two photon  absorption characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Clearly, the electromag-  netic coupling between the levels |2⟩ and |4⟩ can pro-  duce a double pole in the decay of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' It is  interesting that a system equivalent to this has been  studied by Bhamathi and Sudarshan [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A scheme involving laser coupling the inter-  mediate state |2⟩ which will create pole of order two in  the two-photon decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This provides an analog of the  extended cascade model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  DOUBLE POLES AND INTERFERENCE  EFFECTS  The existence of double poles and the possibility  of a line shape which is a derivative of Lorentzian  suggest that the quantum interferences must be  crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  This is indeed the case as can be seen  from the following considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The electromag-  netic coupling of |1⟩ and |2⟩ produced dressed states  |ψ±⟩ =  1  √  2(± |1⟩ + |2⟩) with eigenvalues ±Gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Since  Gl ∼ γ, the two states are within the radiative  line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We pump the population in the state  |1⟩ which is equivalent to pumping in both |ψ±⟩ as  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  Gt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='8 1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='6 1  1  S  1 1  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='4  1  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='2  I:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  2  0  2  4  S|1)  14)  Laser Coupling  [2)  3)4  |1⟩ = (|ψ+⟩ + |ψ−⟩)/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Both states |ψ±⟩ can de-  cay to |3⟩ as |ψ±⟩ involve admixtures of |1⟩ and |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  These two decays will not be independent [8,9] as  −→d +3 · −→d ∗  −3 ̸= 0 and as Gl ∼ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  EXPONENTIAL DECAY RECOVERED  We also examine the initial conditions for our sys-  tem which would result in exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' From  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' (16) it is seen that  d  dτ ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩  + γ1  2 ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩ = 0 (27)  if Gl = γ1  2 , γ2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Thus the correlation function  defined in terms of the vector ˜ψ =  1  √  2(|1⟩ + i |2⟩)  obeys simple exponential decay law with a time scale  governed by γ1/2 rather than γ1:  �  A ˜  ψ3(t + τ)A3 ˜  ψ(t)  �  = e−γ1τ/2 �  A ˜  ψ ˜  ψ(t)  �  (28)  Thus a pumping of the system to the state ˜ψ rather  than |1⟩ will result in exponential decay[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Thus, in conclusion, we have shown how higher-  order poles in the decay of states can be produced  by using resonant electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We demon-  strated this by creating a pole of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Clearly,  the technique is quite versatile and by using combi-  nations of electromagnetic fields we can create poles  of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  I thank George Sudarshan for discussions on  higher order poles of S-Matrix and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Singh for  help in preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Goldberger and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Watson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 136,  B 1472 (1964);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' J, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bell and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Goebel, Phys,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 138, B 1198 (1965)  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bhamathi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Sudarshan, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Mod  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' B 10, 1531 (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' see also E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G Sudarshan,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A 50, 2006 (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Sudarshan,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Chiu and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bhamathi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' D 46, 3508  (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bohm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Maxson and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Loewe, Physica A, in  press;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Mondragon and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Her- nandez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  A 26, 5595 (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Puntmann, paper presented  at the International Colloquium on Group Theory,  Goslar, Germany 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal, Quantum Optics (Springer-Verlag,  Berlin, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A 54, Rapid Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  3734 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [6] An almost trivial case occurs when Γ23 = 0, γ2 = 0  and no pumping (Λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The atom can start in  state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Then one can work with a nonhermitian  Hamiltonian  �  −iγ1 G  G  0  �  which has identical real  eigenvalues if 2G = γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This case has been pre-  viously considered in literature (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Steudel, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Physik 22, 113 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Harshawardhan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev,  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 77, 1039 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal, Quantum Optics (Springer-Verlag,  Berlin, 1974) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 94-96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Imamoglu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A 40, 2835 (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Zhu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Scully, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 76, 388  (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Cardimona, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Raymer and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Stroud Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' B 15, 55 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [10] Pumping the system in the state ≫ is possible using  an excitation pulse with phase switching at appro-  priate instant (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Yodh and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Mossberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 55, 1277 (1984)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}

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@@ -0,0 +1,1056 @@
+Towards Multi-robot Exploration:
+A Decentralized Strategy for UAV Forest Exploration
+Luca Bartolomei, Lucas Teixeira and Margarita Chli
+Vision For Robotics Lab, ETH Z¨urich, Switzerland
+Abstract— Efficient exploration strategies are vital in tasks
+such as search-and-rescue missions and disaster surveying.
+Unmanned Aerial Vehicles (UAVs) have become particularly
+popular in such applications, promising to cover large areas at
+high speeds. Moreover, with the increasing maturity of onboard
+UAV perception, research focus has been shifting toward higher-
+level reasoning for single- and multi-robot missions. However,
+autonomous navigation and exploration of previously unknown
+large spaces still constitutes an open challenge, especially when
+the environment is cluttered and exhibits large and frequent
+occlusions due to high obstacle density, as is the case of forests.
+Moreover, the problem of long-distance wireless communication
+in such scenes can become a limiting factor, especially when
+automating the navigation of a UAV swarm. In this spirit,
+this work proposes an exploration strategy that enables UAVs,
+both individually and in small swarms, to quickly explore
+complex scenes in a decentralized fashion. By providing the
+decision-making capabilities to each UAV to switch between
+different execution modes, the proposed strategy strikes a
+great balance between cautious exploration of yet completely
+unknown regions and more aggressive exploration of smaller
+areas of unknown space. This results in full coverage of forest
+areas of variable density, consistently faster than the state of the
+art. Demonstrating successful deployment with a single UAV as
+well as a swarm of up to three UAVs, this work sets out the basic
+principles for multi-root exploration of cluttered scenes, with
+up to 65% speed up in the single UAV case and 40% increase in
+explored area for the same mission time in multi-UAV setups.
+I. INTRODUCTION
+The growing interest in Unmanned Aerial Vehicles (UAVs)
+has led to their extensive deployment in tasks such as
+inspection and search-and-rescue missions. In these appli-
+cations, the capacity of the robot to quickly explore and
+map unknown environments autonomously is fundamental.
+The literature on this topic is extensive, and many different
+approaches have been proposed throughout the years [1]–
+[5]. However, one of the biggest challenges in the explo-
+ration of unknown environments is the capacity to achieve
+a good trade-off between the competing goals of shorter
+exploration times of an area of interest (i.e. pushing for
+high-speed navigation) and safety, which requires caps on
+the velocity of each robot. In fact, navigating in the vicinity
+of the boundaries between known and unknown space is
+challenging, as the robot can get stuck in dead ends, or
+needs to perform complex dodging maneuvers to avoid
+collisions. Consequently, to maintain the safety of both the
+platform and its surroundings, most path planners generate
+conservative start-and-stop motions, not fully exploiting the
+This work was supported by NCCR Robotics, the Amazon Research
+Awards, and the HILTI group.
+Fig. 1: 3D-view of the proposed system guiding safe and successful
+exploration of a UAV in a digital model of a real-world forest
+[6]. The planner is able to avoid collisions between the UAV and
+the obstacles, clearing frontiers on-the-go by balancing cautious
+navigation with aggressive exploitation of known, free space, in a
+bid to maximize the efficiency of the exploration.
+capacity of a UAV to fly at high speeds. This effect is
+exacerbated when the environment to explore is particularly
+cluttered, as is the case in forests, leading to inefficient and
+incomplete coverage. By design, these methods generally
+drive the exploration process by biasing exploration towards
+large areas of unexplored space. While this strategy could be
+advantageous in open and wide spaces, it can be detrimental
+when exploring cluttered scenes. In fact, the main pitfall of
+such strategies is that, while the exploration process attempts
+to cover as much unknown space as possible, when this
+is deployed in environments with many obstacles, thinner
+trails of unknown space are left unexplored (e.g. due to
+occlusions), imposing the need for a second sweep of the
+environment over mostly explored areas.
+Aiming to mitigate these issues, pushing for faster cov-
+erage of the areas of interest, multi-robot extensions for
+exploration have also been proposed [7]–[10]. However,
+these focus on the problem of coordination at the system-
+level and, and while they can perform better from a global
+planning point of view, they suffer from the same limitations
+as the single-UAV case in obstacle-dense environments.
+Motivated by these challenges, in this work we propose an
+exploration strategy for autonomous UAV robots aiming to
+explore forests of increasing tree density, as they pose some
+of the most difficult challenges for exploration planning. Our
+objective is to exploit the platform’s dynamics to the fullest
+despite the high density of obstacles, in order to achieve the
+complete coverage of the environment efficiently. To this end,
+the proposed strategy enables switching between two differ-
+arXiv:2301.08537v1  [cs.RO]  20 Jan 2023
+
+ent behaviors for each robot; namely, cautious exploration
+of unknown space and more aggressive maneuvers when
+navigating in already explored areas to clear smaller portions
+of unknown space caused by occlusions. We evaluate the
+proposed approach in a series of challenging experiments in
+simulation, both in randomly generated forests and in a 3D
+reconstruction of a real forest (Fig 1). Benchmarking against
+the state of the art reveals superior efficiency for the proposed
+approach achieving higher overall UAV speeds and lower
+exploration times. Finally, aiming to set out the scaffolding
+toward decentralized multi-robot exploration planning, we
+show how the proposed strategy can accommodate more than
+one UAV in the exploration mission, and we demonstrate that
+our method performs comparably to or better than a map-
+splitting centralized approach.
+In summary, the contributions of this work are as follows:
+• the design of an exploration strategy, able to strike
+an effective balance between cautious exploration and
+aggressive exploitation of the explored map,
+• the extension of the single-robot design to a multi-robot
+decentralized approach, and
+• extensive evaluations in simulation, demonstrating bet-
+ter performance than the state of the art.
+II. RELATED WORKS
+Autonomous exploration of unknown environments with
+UAVs has been an active field of research over the past
+few decades. The most popular approach to exploring an
+area of interest is to use frontiers, defined as the boundary
+between known and unknown space [11]. These can be
+utilized to identify potentially informative spatial regions
+in order to drive the exploration process efficiently until
+no new frontiers are found and the exploration process can
+be considered complete. There are different criteria used to
+decide which frontier to explore next, such as their proximity
+to the current field of view, following a greedy selection
+strategy, or having global planning dictate the selection [12].
+However, while frontier-based approaches have been proven
+to yield satisfactory performances, especially in terms of
+coverage [2], [3], they generally lead to inefficient motions
+and sub-optimal action selection. This is mostly caused by
+the sensing modalities used to generate the map of the
+environment to explore, as the most common sensors, such as
+RGB-D and stereo cameras, have a limited detection range.
+Consequently, UAVs need to fly cautiously to ensure safety.
+Cieslewski et al. [4] tackle this limitation, by proposing
+an exploration strategy that generates velocity commands
+based on newly detected frontiers, in a bid to maximize
+the UAV’s speed. This method is shown to outperform
+classical methods [11], but focuses only on local frontiers.
+Instead, FUEL [5] proposes a hierarchical planner which
+generates efficient global paths, while encouraging safe and
+agile local maneuvers for high-speed exploration. FUEL’s
+strategy performs better than [4] and [11] in scenes with
+low obstacle densities. However, it is more computationally
+demanding, as it needs to maintain a list of active frontiers,
+as well as to compute accurate distances between them. This
+additional bookkeeping becomes prohibitive and impractical
+in more cluttered and complex environments such as forests.
+In fact, in this type of scenery, the number of frontiers
+quickly increases due to occlusions caused by tree trunks,
+branches, and shrubs.
+Another line of research focuses instead on sampling-
+based path planning to generate viewpoints to explore the
+space [13], [14], by guiding the robot along possible trails
+of sampled configurations. The best path is generally found
+using a greedy approach [14], bringing to complete explo-
+ration or accurate surface reconstruction [2] depending on
+the information gain formulation. Nonetheless, these sampled
+routes may generate trajectories that deviate from the shortest
+paths, without taking into consideration the robot’s dynamics.
+Consequently, this causes the UAV to navigate in zigzag
+patterns, leading to inefficient, slow motions and conservative
+maneuvers.
+To tackle the limitations of frontier- and sampling-based
+methods, also hybrid approaches have been proposed [1],
+[15]. Such methods compute global paths towards the most
+informative frontiers while generating local trajectories using
+sampling-based planners. However, they do not exploit the
+full dynamics of the platform and generate sub-optimal
+routes.
+To boost the efficiency in exploration, various multi-robot
+cooperative frontier-based methods have also been proposed
+in the literature, both in centralized [16] and decentralized
+formats [17]. In this spirit, the work in [18] greedily assigns
+view configurations, while [19] distributes the workload
+between agents using a Voronoi-based partitioning of the
+area to explore. Nevertheless, these solutions suffer from
+the same limitations as in the single-robot case. Instead, the
+approach in [20] is able to generate efficient trajectories for
+3D reconstruction, tackling the multi-robot coordination with
+a centralized architecture. However, this method requires
+a prior overhead flight over the area of interest, making
+it unsuitable for the exploration of forests. The approach
+proposed in [7] puts more focus on the problem of navigating
+forests, but the emphasis is more on state estimation rather
+than on path planning.
+Motivated by these limitations, in this work, we propose
+a strategy that allows a robot to explore complex forest-
+like environments while flying at high speeds, thanks to
+the freedom and flexibility that our planner provides to
+each UAV to switch between different navigation modes
+online. While slower, cautious exploration is performed
+using a frontier-based approach, we efficiently clear trails of
+unexplored space caused by occlusions by employing a more
+aggressive local exploration strategy, boosting the efficiency
+of the mission and pushing the overall time to cover a given
+area of interest down. Moreover, we demonstrate that the
+proposed pipeline can also be extended to the multi-robot
+setting in a decentralized fashion.
+III. METHODOLOGY
+The overall problem considered in this work is to ex-
+plore unknown cluttered environments, such as forests, in
+
+(a) Time-instant 1
+(b) Time-instant 2
+(c) Time-instant 3
+(d) Time-instant 4
+Fig. 2: A schematic example demonstrating the problem with greedy frontier-based exploration, at progressive time-instants, generating
+islands of unknown space surrounded by free regions. The field of view of the robot is depicted as a light-gray shaded area delimited by
+black solid lines, while the obstacles and the unexplored space are in black and dark gray, respectively (a). The robot navigates towards
+the most informative frontiers (b); however, due to the limited sensor range, the space occluded by the obstacles is not cleared (c).
+Consequently, since the exploration process is biased towards larger, more informative frontiers, the na¨ıve planner flies the UAV robot
+ignoring the smaller portion of unexplored space (d).
+the minimum time possible. We assume that the robot is
+equipped with a front-looking depth camera with a limited
+sensing range and that the robot’s odometry information is
+available at a constant rate. However, forest-like scenes are
+characterized by a high number of obstacles in a variety
+of dimensions (e.g. trunks, leaves, and branches) that make
+standard frontier-based exploration approaches inefficient.
+In fact, during the exploration process, many islands of
+unknown space are usually left behind, as illustrated in
+Fig. 2, necessitating subsequent passes of exploration on a
+nearly completely explored map. To tackle this limitation,
+we propose an exploration pipeline that can change the ex-
+ploratory behavior of the robot depending on the frontiers in
+its vicinity. In particular, we propose to define two different
+modes of operation for the robot, namely the Explorer and
+the Collector modes. In the Explorer state, the robot is driven
+by frontiers and it is tasked to explore large unknown areas.
+Consequently, it predominantly operates on the most external
+boundaries between known and unknown areas. Conversely,
+the robot in the Collector mode clears small islands of
+unknown space generated by occlusions, that are left behind
+during the exploratory phase. The objective of a Collector is
+to clear these portions of space on the go, avoiding the need
+for subsequent revisits of the map, at the expense of short
+local detours. However, notice that these can be performed at
+high speed, since, when in Collector mode, the robot operates
+in mostly explored areas. By allowing a robot to switch
+between these two different modes and by finding the right
+trade-off between map exploration and exploitation, we can
+quickly reach full coverage of large cluttered environments.
+In the following, we first give an overview of the proposed
+system and our exploration strategies, and then we illustrate
+how it can be extended to multiple robots.
+A. System Overview
+As shown in Fig. 3, the pipeline is composed of three main
+components: a mapping system, a mode selector, and a path
+planner.
+Given input depth and odometry information, a voxel
+grid map M of the environment is generated. At every
+update, frontiers are extracted from M and clustered. For
+each cluster, we adopt the sampling strategy from [5] to
+generate viewpoints covering the frontiers, and we use them
+as possible target poses during the exploration process.
+Moreover, each cluster undergoes a binary classification
+step, where unconnected islands of frontiers, or trails, are
+identified. This is necessary to identify those regions that are
+likely to require an additional revisiting phase towards the
+end of the mission if a traditional frontier-based exploration
+method is utilized. Here, a cluster is considered a trail if its
+convex hull is surrounded by free space, or when it has only
+another neighboring cluster. This implies that most clusters
+at the corners of the area to be explored are classified as
+trails. We motivate this design choice by arguing that corners
+are generally problematic for exploration due to their low
+informative value. In fact, they are rarely covered in a first
+sweep of the map, implying the need for a revisiting step.
+The labeled clusters are then utilized by the Mode Selector to
+choose the best exploration strategy for the robot, deciding
+whether it has to persevere in its current mode, or transit
+to Explorer or Collector. The mode assignment is regulated
+according to the frontiers in the vicinity of the UAV. Given
+that our objective is to clear trails locally to avoid large
+detours on the map, we assign the role of Collector if a
+minimum number of trails is close to the robot. Instead, we
+adopt a more exploratory strategy once all smaller islands of
+unknown space are cleared, or when the trails are far away
+from the drone. Once a strategy is selected, the viewpoint of
+the most promising cluster is selected as the new target pose.
+This is fed to a path planner [21] that generates the trajectory
+flying the UAV toward its destination. We now describe the
+different exploration modes in more detail.
+B. Exploration Strategies
+1) Explorer: Driven by frontiers, the objective of an
+Explorer is to cover large areas of previously unknown space.
+Similarly to [4], we process the incoming clusters of frontiers
+C from the most recent map update and extract the one with
+the lowest cost JE. Notice that these clusters are mostly
+aligned with the direction of the UAV’s motion, implying
+that, if one of these is selected as the target, the robot
+avoids abrupt changes in the flight direction or aggressive
+
+Mapping
+3D Map
+Frontiers Classification
+Frontiers Extraction
+Odometry 
+Depth
+Target Pose Selector
+Trajectory Generation
+Path Planning
+Mode Selector
+Low-level 
+Controller
+Fig. 3: Schematic overview of the proposed exploration pipeline for a single agent. The inputs to the system are the robot’s odometry and
+depth information. These are used to generate a 3D grid-based map of the environment, from which frontiers are extracted and clustered.
+The trails of frontiers are identified and used to select the adequate exploration mode for the agent. Then, the next target pose is chosen
+and a trajectory towards the goal pose is generated using [21].
+maneuvers.
+The cost associated to the viewpoint ξc := {xc, γc}
+covering cluster c ∈ C is defined as
+JE(ξc) := ωDJD(ξc) + ωV JV (ξc) + ωLJL(c),
+(1)
+where xc ∈ R3 is the position of the viewpoint and γc ∈
+R its orientation. The cost JD is the length of the path in
+M between the current robot’s position and xc, and it is
+calculated using the A* algorithm. Instead, JV is associated
+with the change in direction of travel, while JL to the label
+of cluster c. The terms ωD, ωV and ωL are constant weights.
+The cost JV (ξc) is calculated as
+JV (ξc) := acos(vT
+R
+xc − xR
+||xc − xR||2
+),
+(2)
+where vR ∈ R3 and xR ∈ R3 are the robot’s current velocity
+and position, respectively. This cost is directly associated
+with the angle between the velocity and the direction vector
+towards the candidate position xc covering cluster c. How-
+ever, it may happen that the cluster is labeled as trails, e.g. in
+the case of occlusions caused by thin obstacles, such as tree
+trunks. Since an Explorer should focus on actual frontiers,
+we assign a penalty to these clusters:
+JL(c) =
+�
+0
+if c is frontier
+ptrail
+if c is trail
+,
+(3)
+where ptrail is the constant penalty associated with trails.
+We then select as target pose the next best viewpoint
+ξc∗ := {xc∗, γc∗} covering the cluster c∗ with the lowest
+cost:
+ξc∗ := arg min
+ξc ∀c∈C JE(ξc).
+(4)
+In case the UAV is trapped in a dead-end, or if no new
+clusters are available in front of the robot, we ignore the cost
+associated with the robot’s velocity and we employ a greedy
+approach to select the new target pose. We find the best
+cluster in the vicinity of the robot at a maximum distance
+dmax using the same cost function as in Eq. 1, with JV set
+to zero:
+ξc∗ := arg min
+ξc ∀c∈C ωDJD(ξc) + ωLJL(c)
+s.t. ||xc − xR||2 ≤ dmax.
+(5)
+2) Collector: The objective of a Collector is to clear as
+many trails as possible, in order to avoid the need for a
+revisiting step in poorly explored regions of the map at the
+end of the mission. Since this task implies a detour from
+the main direction of exploration, the UAV’s speed needs
+to be maximized in order to go back to Explorer mode as
+soon as possible. To reach this objective, we sort the set of
+trails Ctrails ⊆ C by associating a cost JC to each cluster
+c ∈ Ctrails:
+JC(ξc) := ωP JP (ξc) + ωAJA(ξc),
+(6)
+where JP is associated with the time to reach xc and JA with
+the time to cover the angular change between the current
+robot’s yaw and the viewpoint’s orientation. Instead, ωP and
+ωA are constant weights.
+Given the path πR
+c
+from xR to xc and the maximum
+allowed velocity vmax, JP is computed as
+JP (ξc) := length(πR
+c )
+vmax
+.
+(7)
+Similarly, given the robot current’s heading γR, the view-
+point’s orientation γc and the maximum allowed yaw rate
+˙γmax, JA is computed as
+JA(ξc) := ∠(γR, γc)
+˙γmax
+,
+(8)
+where ∠(γR, γc) indicates the angular difference between γR
+and γc.
+The robot then selects the target trail in a step-by-step
+greedy procedure and behaves as a Collector until all close-
+by trails are cleared. Since the trails are surrounded by free
+known space, we double the maximum velocity compared to
+when in Explorer mode. Consequently, the UAV is able to
+maximize its velocity, leading to fast motions that allow it
+to quickly cover all the viewpoints associated with the trails.
+C. Extension to Multi-Robot
+1) System Architecture: The proposed exploration strat-
+egy can be easily extended to the multi-robot case. The
+extended pipeline is shown in Fig. 4. Assuming that the
+agents can localize in a common reference frame, they
+exchange local sub-maps, as well as odometry informa-
+tion, current target pose, and execution mode. Notice that
+here we propose a decentralized architecture. Centralized
+approaches generally assume infinite-range communication
+between agents and with a ground station. However, standard
+Wi-Fi communications have a limited range and, when
+navigating in cluttered environments such as forests, com-
+munication lines can be potentially obstructed and signal
+can be lost. In this work, we propose to use a more flexible
+point-to-point strategy, assuming that there exists a maximum
+range of communication between each pair of agents. Our
+
+Mapping
+Path Planning
+Mode Selector
+Agent 1
+Odometry 
+Depth
+Mapping
+Path Planning
+Mode Selector
+Agent 2, ..., N
+Odometry 
+Depth
+Mode
+Target Poses
+Local 
+Sub-maps
+Mode
+Fig. 4: Overview of the pipeline in the multi-robot setting, where
+the UAVs are tasked to collaboratively build a complete map of the
+area of interest. To fulfill the objective, agents exchange odometry
+information and local sub-maps, as well as current execution mode
+and target poses. We assume there exists a maximum communi-
+cation range between robots. If the distance between two agents
+exceeds this limit, communication is lost and information is not
+exchanged anymore, leading to poor coordination.
+design targets to keep the agents at a valid communication
+distance. If the distance between agents is higher than the
+maximum range, communication is lost and information is
+not exchanged anymore, leading to poor inter-agent coor-
+dination and sub-optimal decision-making. We also assume
+that, when communication is lost and successively regained,
+agents can synchronize their maps.
+2) Multi-robot Coordination: We encourage coordination
+between pairs of agents only if they perform compatible
+actions from an exploration point of view. This implies that,
+if two UAVs are operating in the same mode, they can
+collaborate in order to either explore more unknown spaces
+(Explorers) or clear trails (Collectors). On the contrary, coor-
+dination between an Explorer and a Collector should not be
+encouraged, since their tasks are intrinsically different. In this
+situation, we propose a leader-follower paradigm, where the
+Explorer (leader) explores unknown areas regardless of the
+position of the second agent, while the Collector (follower)
+follows the leader and clears the trails left unexplored. This
+design choice allows more flexibility during the execution
+of a mission, thanks to the possibility to change execution
+modes online. Notice that, if communication between all
+agents is lost, their exploration strategy falls back to the
+single-robot case.
+Our collaboration strategy within a robotic team is encour-
+aged as a soft constraint by modifying the cost functions in
+Eq. 1 and Eq. 6. If we consider robot i with position xi
+R,
+given the positions X i
+R := {xk
+R}N−1
+k=0 of the other N − 1
+robots in the team with k ̸= i, and their current target
+positions GR = {xk
+c∗}N−1
+k=0 , we modify the cost functions
+JE and JC for robot i as follows:
+JE(ξc, X i
+R, GR) := ωDJD(ξc) + ωV JV (ξc)+
+ωLJL(c) + ωF JF (ξc, X i
+R, GR)
+(9)
+and
+JC(ξc, X i
+R, GR) := ωP JP (ξc) + ωAJA(ξc)+
+ωF JF (ξc, X i
+R, GR),
+(10)
+where
+ωF
+is
+a
+constant
+weight.
+The
+cost
+function
+JF (ξc, X i
+R, GR) is defined as
+JF (ξc, X i
+R, GR) := Jatt
+F (X i
+R) + Jrep
+F
+(ξc, X i
+R, GR),
+(11)
+where Jatt
+F
+aims at keeping the agents i and k in com-
+munication range, while Jrep
+F
+ensures a minimum distance
+between them to avoid collisions. Moreover, Jrep
+F
+encourages
+map splitting, by assigning a high cost to candidate target
+positions close to other agents’ current goals.
+In more details, the function Jatt
+F
+for agent i is defined as
+follows:
+Jatt
+F (X i
+R) :=
+N−1
+�
+k=0,k̸=i
+I(i, k) · 1
+2kA||xi
+R − xk
+R||2,
+(12)
+where kA is constant factor and I(i, k) is an indicator
+function that embeds our coordination strategy:
+I(i, k) :=
+�
+0
+if i Explorer and k Collector
+1
+otherwise
+.
+(13)
+This indicates that agent i is attracted toward agent k only if
+they are in a compatible execution mode. On the contrary, in
+leader-follower mode, i.e. when robot i is an Explorer and
+robot k is a Collector, the leader ignores the follower, and
+Jatt
+F
+goes to zero. Notice that instead, if i is Collector and
+k an Explorer, I(i, k) = 1 and Jatt
+F
+̸= 0.
+Instead, Jrep
+F
+is computed as follows:
+Jrep
+F
+(ξc, X i
+R, GR) :=
+N−1
+�
+k=0,k̸=i
+Jrep
+ik (xi
+R, xk
+R) + Jrep
+ik (xi
+c, xk
+c∗),
+(14)
+where, given the Euclidean distance dAB := ||xA − xB||2,
+Jrep
+AB(xA, xB) :=
+�
+�
+�
+�
+�
+kR(dc − d0)2 dcd0
+d0−dc
+if dAB ≤ d0
+kR(dAB − d0)2
+if dc ≤ dAB ≤ d0
+0
+otherwise
+.
+(15)
+The parameter kR is a constant weight, while d0 represents
+the minimum distance between positions A and B to have
+a collision. The parameter dc represents the distance after
+which the positions should not approach any closer. This
+can be selected on the basis of the safety distance required
+between the UAVs. Notice that Jrep
+ik
+is not influenced by
+the roles of agents i and k, as safety and minimum distance
+requirements need to be always met.
+IV. EXPERIMENTS
+We evaluate the proposed exploration pipeline in both
+single- and multi-UAV setups in simulation. In particular,
+we benchmark our method on a series of realistic, randomly
+generated forests of increasing tree densities [22], as well as
+on a 3D reconstruction of a real forest [6].
+In the single-agent setup, we compare the proposed
+method against FUEL [5], while in the experiments with
+multiple robots we test against a centralized strategy based
+on map-splitting. In all tests, we use grid map resolutions of
+0.10 m or 0.15 m depending on the map size, while we set the
+dynamic limits to vmax = 1.5 m/s and ˙γmax = 0.9 rad/s for
+all planners. We simulate a depth camera with a fixed range
+
+Ours
+FUEL [5]
+REAL FOREST
+Completion Time [s]
+500.7 ± 14.8
+757.7 ± 47.9
+Travelled Distance [m]
+645.0 ± 20.0
+533.2 ± 11.0
+Velocity [m/s]
+1.3 ± 0.5
+0.7 ± 0.4
+SPARSE FOREST (0.05 TREES / m2)
+Completion Time [s]
+665.4 ± 32.7
+1114.1 ± 97.4
+Travelled Distance [m]
+860.9 ± 34.0
+758.1 ± 57.6
+Velocity [m/s]
+1.3 ± 0.6
+0.7 ± 0.5
+AVERAGE-DENSITY FOREST (0.10 TREES / m2)
+Completion Time [s]
+779.6 ± 110.9
+954.1 ± 28.8
+Travelled Distance [m]
+910.3 ± 63.7
+713.5 ± 33.1
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.5
+DENSE FOREST (0.15 TREES / m2)
+Completion Time [s]
+613.2 ± 16.2
+1130.2 ± 28.8
+Travelled Distance [m]
+789.7 ± 16.8
+791.0 ± 37.7
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.5
+VERY DENSE FOREST (0.20 TREES / m2)
+Completion Time [s]
+658.2 ± 57.2
+904.1 ± 109.5
+Travelled Distance [m]
+802.0 ± 52.7
+680.6 ± 49.9
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.5
+TABLE I: Results of the experiments for a single agent. We report
+the average completion time over 3 runs, as well as the average
+travelled distance and velocity. The best performance is in bold.
+The relatively high standard deviation in the timings to complete
+the missions, in particular in the cases of tree densities of 0.10 and
+0.20 trees/m2, is caused by the complex nature of the map and the
+large number of occlusions.
+0
+200
+400
+600
+Time [s]
+0
+2000
+Explored Volume [m3]
+Ours
+FUEL
+Fig. 5: The average exploration rate during the experiments in
+the model REAL FOREST. The shaded region shows the standard
+deviation. The proposed method reaches complete coverage in less
+time than FUEL [5].
+of 4.5 m using the Vulkan-based renderer of [23] and the
+same physical simulator as in [5]. We report the planners’
+performance in terms of the time needed to complete the
+exploration of the scene, the total travelled distance, and the
+average velocity of the UAVs during each experiment.
+A. Single-robot Experiments
+In the single-robot experiments, the models of the syn-
+thetic forests are of size 50 m × 50 m × 2 m, while the 3D
+reconstruction of the REAL FOREST has dimensions 40 m ×
+40 m×2 m. The map resolution is set to 0.10 m. As reported
+in Table I, the proposed planner outperforms FUEL [5] across
+all scenes in terms of the time taken to reach full coverage
+(Fig. 5), thanks to our adaptive exploration policy that leads
+0
+200
+400
+600
+Time [s]
+0.5
+1.0
+1.5
+Velocity [m/s]
+Ours
+FUEL
+Fig. 6: The average UAV velocity during the experiments in the
+REAL FOREST. The shaded region indicates the standard deviation.
+The proposed strategy is able to fly the UAV at higher velocities
+than FUEL leading to improved mission efficiency (i.e. time to
+mission completion). Towards the end of the mission, the UAV
+following the FUEL strategy also speeds up, as by then the map
+is mostly explored, and smaller trails are cleared, resembling the
+Collector mode in the proposed strategy.
+Fig. 7: Top view of an exploration mission, where a team of two
+UAVs is tasked to map a randomly generated forest with tree
+density 0.05 trees/m2, illustrated with the black dots here. The
+initial positions of the UAVs are denoted as colored blobs and the
+final positions as enlarged drone models for clearer visualization.
+The map is represented as a 2D occupancy grid obtained by slicing
+the 3D model at 1.5 m from the ground.
+to consistently higher UAV velocity throughout each mission,
+as illustrated in Fig. 6. These results demonstrate the benefit
+of using the proposed adaptive exploration strategy over
+a fixed-mode method. However, notice that the proposed
+design leads to longer travelled distances, albeit guaranteeing
+that there are no small unexplored areas left. In fact, decision-
+making both in Explorer and Collector modes is done on a
+local-map level, and this may cause the UAV to fly longer
+routes, deviating from the shortest path. Nonetheless, in the
+proposed strategy we compensate for this shortcoming by
+encouraging decisions leading to higher UAV velocities, and
+thus shorter mission times.
+B. Multi-robot Experiments
+In the multi-robot experiments, the proposed planning
+strategy is tested in a variety of models with fixed, homoge-
+neous obstacle density, as well as in a randomly generated
+forest with tree density varying across different regions of
+the map.
+1) Maps with a fixed tree density: The results of the multi-
+robot collaborative exploration strategy of maps with fixed
+tree densities with two agents are shown in Table II, where
+
+0
++Ours
+Split Map (FUEL [5])
+SPARSE FOREST (0.05 TREES / m2)
+Completion Time [s]
+780.3 ± 32.2
+834.3 ± 64.3
+Travelled Distance [m]
+958.8 ± 9.2
+595.7 ± 51.8
+958.7 ± 60.4
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.4
+1.3 ± 0.7
+AVERAGE-DENSITY FOREST (0.10 TREES / m2)
+Completion Time [s]
+838.5 ± 64.4
+848.1 ± 98.5
+Travelled Distance [m]
+915.3 ± 68.4
+616.1 ± 40.7
+906.2 ± 58.3
+Velocity [m/s]
+1.2 ± 0.5
+0.8 ± 0.4
+1.1 ± 0.6
+DENSE FOREST (0.15 TREES / m2)
+Completion Time [s]
+786.3 ± 40.7
+754.0 ± 23.9
+Travelled Distance [m]
+839.0 ± 26.1
+586.7 ± 14.2
+882.2 ± 32.0
+Velocity [m/s]
+1.2 ± 0.7
+0.8 ± 0.4
+1.3 ± 0.7
+VERY DENSE FOREST (0.20 TREES / m2)
+Completion Time [s]
+803.7 ± 52.8
+705.8 ± 73.2
+Travelled Distance [m]
+873.8 ± 47.0
+580.3 ± 73.3
+912.6 ± 42.7
+Velocity [m/s]
+1.2 ± 0.7
+0.7 ± 0.5
+1.2 ± 0.8
+TABLE II: Results in randomly generated forests with fixed tree
+densities explored with two UAVs, averaged over 3 runs. For the
+proposed strategy we report the average travelled distance and
+velocity per agent. The best performance is highlighted in bold.
+0
+200
+400
+600
+800
+Time [s]
+0
+2500
+Explored Volume [m3]
+Agent 1
+Agent 2
+Fig. 8: The average exploration rate per agent with the proposed
+approach using two UAVs during the experiments in a random
+forest with density 0.10 trees/m2. The shaded region shows the
+standard deviation. The explored volume is shown to be consistently
+balanced across the two agents.
+a maximum connection distance of 50 m for data exchange
+between agents is assumed (Fig. 7). Here, experiments are
+performed in forest models of size 100 m×50 m×2 m with
+a map resolution of 0.10 m. The proposed approach is com-
+pared against a centralized strategy we devised, employing
+the FUEL planner [5] and assigning the UAVs to explore
+maps of equal sizes (i.e. using map-splitting). Note that this
+strategy assumes homogeneous forest maps, and knowledge
+of the original map size, which renders this unsuitable for
+realistic deployment unlike the proposed approach; however,
+comparisons are presented for the sake of benchmarking.
+The proposed strategy reaches comparable results with
+respect to the centralized approach using FUEL. Similarly
+to single-robot exploration, the proposed strategy flies the
+UAVs at higher speeds, incurring longer travel distances.
+Moreover, our strategy enables automatic load balancing of
+the exploration mission, yielding similar exploration rates
+per agent as illustrated in Fig. 8. However, in denser for-
+est models, the performance of the proposed approach is
+seen to degrade, as the UAVs are tasked to fly within a
+connection range to each other, resulting in limited freedom
+of movement. This is exacerbated by the increased number
+of obstacles and occlusions in smaller maps, leading to
+lower exploration rates. Nevertheless, as aforementioned, the
+proposed approach is realistically deployable in contrast to
+the baseline strategy using FUEL.
+2) Map with non-homogeneous tree density: The results
+of the experiments in a map with non-homogeneous obstacle
+densities with a team of two agents are shown in Table III.
+We utilize a model with size 100 m × 200 m × 2 m, with
+different tree densities (0.2, 0.3 and 0.5 trees/m2) across
+distinct map regions. The occupancy grip map resolution is
+set to 0.15 m, with a maximum inter-agent communication
+range of 200 m. We perform the same analysis as in Sec.
+IV-B.1, assuming a realistic maximum flight time of 1500 s
+for the UAVs. As none of the tested planners is able to fully
+explore the environment within the allowed time, we report
+the total team coverage at fixed timestamps. The proposed
+approach consistently outperforms the solution based on
+map-splitting using FUEL [5] as the planning back-end. The
+gain in performance is related to the capacity of our strategy
+to fly the UAVs faster in regions with lower obstacle density
+and to explore cautiously more cluttered areas while clearing
+smaller frontier trails on the go. This leads to a more efficient
+exploration process, able to better exploit the capacity of
+UAVs to perform highly dynamic flights.
+Finally, we report the performance of our method when
+a team of three robots is deployed. As shown in Table
+IV, a larger team size yields higher total coverage. This
+demonstrates that this work presents an effective strategy for
+peer-to-peer sharing of the responsibility of exploration of
+a large forest area and that the extension to larger teams
+of multiple UAVs can be realized following this paradigm,
+pushing for scalable, decentralized multi-robot planning for
+exploration.
+V. CONCLUSION AND FUTURE WORK
+In this work, we propose an exploration pipeline for
+autonomous UAVs operating in complex, cluttered environ-
+ments, with a particular focus on forests. We choose this type
+of environment as one of the inherently most challenging for
+effective planning due to the increased number of obstacles
+and occlusions that they exhibit. The proposed strategy
+allows each UAV to switch between different exploratory
+behaviors, autonomously balancing cautious exploration of
+unknown space and more aggressive maneuvers, exploiting
+already mapped space within a mission. This leads to faster
+completion times due to higher-speed flights and, conse-
+quently, to more efficient and faster map coverage than the
+state of the art. Moreover, we show how the proposed method
+can be extended to three, and potentially more robots in a
+decentralized fashion, demonstrating automatic and effective
+load balancing across the participating agents. Following the
+
+Ours
+Split Map (FUEL [5])
+Travelled Distance [m]
+1481.9 ± 467.0
+735.2 ± 500.1
+1487.0 ± 440.8
+697.1 ± 365.6
+Velocity [m/s]
+1.3 ± 0.6
+0.7 ± 0.4
+1.2 ± 0.7
+0.6 ± 0.5
+Explored Volume [m3] - 300 s
+542.3 ± 23.3
+514.6 ± 313.0
+565.9 ± 31.4
+632.0 ± 184.8
+Explored Volume [m3] - 600 s
+1062.8 ± 81.3
+840.0 ± 580.7
+1011.6 ± 65.3
+728.3 ± 163.5
+Explored Volume [m3] - 900 s
+1400.0 ± 265.3
+1145.3 ± 814.7
+1302.6 ± 110.5
+798.8 ± 146.2
+Explored Volume [m3] - 1200 s
+1626.7 ± 431.2
+1330.1 ± 905.3
+1466.5 ± 185.4
+910.4 ± 83.9
+Explored Volume [m3] - 1500 s
+1816.0 ± 550.8
+1437.8 ± 971.8
+1637.0 ± 299.1
+1040.5 ± 124.3
+TABLE III: Results in a randomly generated forest with non-
+homogeneous tree densities when explored with two UAVs, av-
+eraged over 3 runs. We report the average travelled distance and
+velocity per agent at 1500 s, as well as the total explored volume
+by the team at different timestamps. The best performance is
+highlighted in bold.
+Timestamp
+Two Agents
+Three Agents
+300 s
+1108.2 ± 54.7 m3
+1208.0 ± 91.6 m3
+600 s
+2074.3 ± 144.8 m3
+2098.4 ± 143.8 m3
+900 s
+2702.6 ± 375.9 m3
+2748.3 ± 111.9 m3
+1200 s
+3093.2 ± 616.6 m3
+3223.5 ± 127.9 m3
+1500 s
+3453.0 ± 849.9 m3
+3621.8 ± 321.0 m3
+TABLE IV: Results in a randomly generated forest with varying
+tree densities across different regions of the model, averaged over
+3 runs. Here, the map is explored with teams composed of two and
+three UAVs. We report the total volume covered by the team at
+different timestamps. The best performance is highlighted in bold.
+push for automating higher-level decision-making in robotic
+missions, this work constitutes a key milestone towards
+effective exploration planning for robotic teams.
+The natural next step for this work is to address the inte-
+gration and deployment of the proposed pipeline onboard real
+platforms, while further investigations will push advancing
+coordination strategies in larger multi-robot teams.
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+

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@@ -0,0 +1,694 @@
+Diffusion Model based Semi-supervised Learning
+on Brain Hemorrhage Images for Efficient
+Midline Shift Quantification
+Shizhan Gong1, Cheng Chen1, Yuqi Gong1, Nga Yan Chan2, Wenao Ma1,
+Calvin Hoi-Kwan Mak3, Jill Abrigo2, Qi Dou1
+1Department of Computer Science and Engineering,
+The Chinese University of Hong Kong, Hong Kong, China
+2Department of Imaging and Interventional Radiology,
+The Chinese University of Hong Kong, Hong Kong, China
+3Queen Elizabeth Hospital, Hong Kong, China
+Abstract. Brain midline shift (MLS) is one of the most critical factors
+to be considered for clinical diagnosis and treatment decision-making for
+intracranial hemorrhage. Existing computational methods on MLS quan-
+tification not only require intensive labeling in millimeter-level measure-
+ment but also suffer from poor performance due to their dependence on
+specific landmarks or simplified anatomical assumptions. In this paper,
+we propose a novel semi-supervised framework to accurately measure the
+scale of MLS from head CT scans. We formulate the MLS measurement
+task as a deformation estimation problem and solve it using a few MLS
+slices with sparse labels. Meanwhile, with the help of diffusion models,
+we are able to use a great number of unlabeled MLS data and 2793 non-
+MLS cases for representation learning and regularization. The extracted
+representation reflects how the image is different from a non-MLS image
+and regularization serves an important role in the sparse-to-dense refine-
+ment of the deformation field. Our experiment on a real clinical brain
+hemorrhage dataset has achieved state-of-the-art performance and can
+generate interpretable deformation fields.
+Keywords: Computer-aided diagnosis · Semi-supervised learning · Dif-
+fusion models · Intracranial hemorrhage
+1
+Introduction
+Intracranial hemorrhage (ICH) refers to brain bleeding within the skull, a se-
+rious medical emergency that would cause severe disability or even death [1].
+A characteristic symptom of severe ICH is brain midline shift (MLS), which
+is the lateral displacement of midline cerebral structures (see Fig. 1). MLS is
+an important and quantifiable indicator of the severity of mass effects and the
+urgency of intervention [2,3,9]. For instance, the 5 millimeters (mm) threshold
+of MLS is frequently used to determine whether immediate intervention and
+close monitoring is required [4]. MLS quantification demands high accuracy and
+arXiv:2301.00409v1  [cs.CV]  1 Jan 2023
+
+2
+S.Z. Gong et al.
+posterior falx 
+anterior falx 
+(a) No MLS
+falx 
+mls
+(b) MLS on falx
+septum
+pellucidum
+mls
+(c) MLS on septum
+pellucidum
+The third 
+ventricle 
+mls
+(d) MLS on the
+third ventricle
+Fig. 1: Examples of head CT scans to illustrate how radiologists measure MLS.
+Dash red line connecting the anterior falx and posterior falx denotes a hypothet-
+ical normal midline. Blue circles denote the shifted landmarks. Perpendicular
+red lines from the shifted landmarks to normal midline are measured as MLS
+scale.
+efficiency, which is difficult to achieve with manual quantification, especially in
+emergencies, due to the variability in shift regions, unclear landmark boundaries,
+and non-standard scanning pose. An automated MLS quantification algorithm
+that can immediately and accurately quantify MLS is highly desirable to identify
+urgent patients for timely treatment.
+To measure MLS, clinicians usually first identify a few CT slices with large
+shifts and then measure and identify the maximum deviation of landmarks such
+as the septum pellucidum, third ventricle, or falx from their normal counterpart
+as the final MLS distance (see examples in Fig. 1). Such a clinical fashion of
+MLS quantification can be difficult to be translated into a well-defined automa-
+tion process. Currently, there are only limited studies on automated MLS quan-
+tification, using different strategies and varied labeling requirements. Nguyen
+et al. proposed a landmark-based method that relies on anatomical markers to
+determine the location of the deformed midline [9]. However, this method can
+only apply to cases where MLS appears on these specific marker regions. Liao et
+al. adopted a symmetric-based method to seek a curve connecting all deformed
+structures [10], which is difficult to generalize due to over-simplified anatom-
+ical assumptions and sensitivity to patients’ scan poses. A few recent works
+try to overcome these limitations by using stronger supervision with dense la-
+beling. Some studies formulated MLS quantification as a midline segmentation
+task [5,6,7], by delineating the intact midline as labels to supervise the train-
+ing of segmentation models. Another study designed a hemisphere segmentation
+task to quantify MLS [8], which requires pixel-wise annotation for each slice.
+However, obtaining such dense annotations is very costly and time-consuming,
+while may not be necessary for MLS quantification.
+To tackle these limitations, we propose to fit MLS quantification into a de-
+formation prediction problem by using semi-supervised learning (SSL) with only
+limited annotations. Our framework avoids the strong dependency on specific
+
+Diffusion model based ICH midline shift quantification
+3
+landmarks or over-simplified assumptions in previous methods while not increas-
+ing the labeling efforts. We aim to use only sparse and weak labels as ground
+truth supervisions, which are just one shifted landmark and its normal counter-
+part on a limited number of slices provided by radiologists, but we try to fully
+exploit the unlabeled slices and non-MLS data to impose extra regularization
+for the sparse-to-dense extension. Existing SSL methods typically use a partially
+trained model with labeled data to generate pseudo labels for unlabeled data,
+assuming that labeled and unlabeled data are generally similar. These meth-
+ods can be sub-optimal in our case as labeled slices of MLS usually present the
+largest deformation while unlabeled slices contain only minor or no deformation.
+Instead, we propose our SSL strategy by generating a corresponding non-MLS
+image for each unlabeled MLS slice with generative models and regularizing
+that the deformation field should warp the generated non-MLS images into the
+original MLS ones. However, as we only have volume-wise labels for MLS and
+non-MLS classification, it can be difficult to train a slice-wise discriminator as
+required by many generative models such as GANs [12]. Fortunately, the recently
+proposed diffusion models [15], which prove to have strong power in both dis-
+tribution learning and image generation without dependency on discriminators,
+can be a potentially good solution.
+In this work, we propose a novel semi-supervised learning framework based
+on diffusion models to quantify the brain MLS from head CT images with defor-
+mation prediction. Our method effectively exploits supervision and regulariza-
+tion from all types of available data including MLS images with sparse ground
+truth labels, MLS images without labels, and non-MLS images. We validate our
+method on a real clinical head CT dataset, showing effectiveness of each proposed
+component. Our contributions include: (1) innovating an effective deformation
+strategy for brain MLS quantification, (2) incorporating diffusion models as a
+representation learner to extract features reflecting where and how an MLS im-
+age differs from a non-MLS image, and (3) proposing a diffusion model-based
+semi-supervised framework that can effectively leverage massive unlabelled data
+to improve the model performance.
+2
+Methods
+Fig. 2 illustrates our diffusion model-based semi-supervised learning framework
+for MLS quantification via deformation prediction. In Sec. 2.1, we introduce our
+deformation prediction by using only sparse supervision. In Sec. 2.2, we propose
+to incorporate non-MLS data for representation learning. In Sec. 2.3, we describe
+how to utilize unlabeled MLS images for sparse-to-dense regularization.
+2.1
+MLS Quantification through Deformation Estimation
+Our proposed deformation strategy for brain MLS quantification aims to find
+an optimal deformation field φ so that an MLS image can be regarded as a
+
+4
+S.Z. Gong et al.
+◼ Semi-supervised framework
+𝑥𝑢
+𝑥𝑙
+Deformation 
+estimation
+Image 
+generation
+𝜙(𝑥′𝑢)
+Weak label
+Deformation network
+Conditional 
+diffusion 
+network 𝐺𝜃𝑐
+Unconditional 
+diffusion 
+network 𝐺𝜃𝑢
+Ƹ𝜖𝜃𝑐 − Ƹ𝜖𝜃𝑢
+𝑥𝑡
+𝑥0
+◼ Deformation estimation
+𝜙
+◼ Image generation
+Denoising with 
+𝜆𝐺𝜃𝑐 − (1 − 𝜆)𝐺𝜃𝑢
+𝑥0
+𝑥𝐿
+𝑥0
+,
+𝑙𝑚𝑠𝑒 + 𝑙𝑐𝑒𝑖𝑙
+𝑙ℎ𝑢𝑏𝑒𝑟
++ 𝑙𝑠𝑚𝑜𝑜𝑡ℎ
+⊕ ⊕
+𝜖
+𝜖
+warp
+𝜙𝑙
+𝜙𝑢
+𝑥′𝑢
+Fig. 2: The pipeline of our proposed semi-supervised deformation strategy for
+MLS quantification. The labeled image xl is supervised by sparse labels and the
+unlabeled image xu is self-supervised with generated negative image.
+hypothetically non-MLS image warped with this deformation field. The defor-
+mation field can be parameterized by a function with high complexity so that
+it does not explicitly rely on a single landmark or over-simplified symmetric as-
+sumptions, which naturally overcomes the limitations of existing methods. We
+apply a learning-based framework to parameterize the deformation field with a
+U-Net shape neural network. The output of the network is the stationary veloc-
+ity field v. The diffeomorphic deformation field φ is then calculated through the
+integration of the velocity field, similarly to VoxelMorph [11] for image regis-
+tration. The learning process is supervised by sparse deformation ground truth.
+For each labeled slice, we have the ground truth y = (y1, y2), which is a two-
+dimensional vector directing from shifted landmark point toward its presumably
+normal location (the red arrow in Fig. 2). The predicted deformation ˆy is bi-
+linearly interpolated at the shifted landmark point from the deformation field,
+which is also a two-dimensional vector. To alleviate the influence of a few ex-
+tremely large deformation points and increase model’s robustness, we use Huber
+loss to measure the similarity between the predicted deformation and the label:
+lhuber(yd, ˆyd) =
+�
+�
+�
+|yd − ˆyd|,
+|yd − ˆyd| ≥ c,
+(yd − ˆyd)2 + c2
+2c
+,
+|yd − ˆyd| < c.
+(1)
+where d ∈ {1, 2}. The hyperparameter c defines the range for absolute error or
+squared error. We also encourage a smooth deformation field with a diffusion
+regularizer on the spatial gradients of deformation φ to avoid a discontinuous
+
+Diffusion model based ICH midline shift quantification
+5
+deformation field:
+lsmooth =
+�
+j
+�
+k
+∥φjk − φ(j−1)k∥2 + ∥φjk − φj(k−1)∥2,
+(2)
+As the deformation can be extremely large in our case, and meanwhile to force a
+smooth transition between the deformation peak and its adjacent pixels, we use
+a coarse-to-fine manner, where velocity fields are generated through upsampling
+with skip connection to progressively aggregate features of different scales.
+2.2
+Learning Negative Patterns from Non-MLS Images
+In order to learn a deformation field to warp a non-MLS image into MLS one,
+ideally we would need a pair of non-MLS and MLS images for network train-
+ing, which however does not exist in practice. Lacking such information makes
+the network difficult to learn. A naive solution is to generate a corresponding
+non-MLS image. However, generated images entail some randomness and can
+often lack important details. Depending too much on such fake inputs can lead
+to poor robustness. Inspired by the score-matching interpretation of diffusion
+models [17], we propose to learn the non-MLS distribution from massive amount
+of negative cases. Given an MLS image, we can evaluate which parts of the im-
+age make it different from a non-MLS image. This deviation can serve as latent
+features that help the deformation network with deformation prediction.
+Diffusion models, especially DDPM [14], define a forward diffusion process as
+the Markov process progressively adding random Gaussian noise to a given image
+and then trying to approximate the reverse process by a Gaussian distribution.
+The forward process can be simplified by a one-step sampling: xt = √αtx0 +
+√1 − αtϵ, where αt := �t
+s=0 1 − βt, and βt are predefined variance schedule.
+ϵ is sampled from N(0, I). The mean µθ(xt, t) and variance Σθ(xt, t) of the
+reverse process can be parameterized by neural networks. A popular choice is
+to re-parameterize µθ(xt, t) so that ˆϵθ(xt, t) instead of µθ(xt, t) is estimated by
+neural networks to approximate the noise ϵ. Moreover, the output of the diffusion
+network ϵ(xt, t) is actually a scaled score function ∇ log p(xt) as it moves the
+corrupted image towards the opposite direction of the corruption. [18].
+As a result, through pre-training one unconditional diffusion model trained
+with all data (denoted as U) and one conditional diffusion model trained with
+only non-MLS data (denoted as C), the subtraction of two outputs
+ˆϵθU (xt, t) − ˆϵθC(xt, t) ∝ ∇ log p(xt|n) − ∇ log p(xt) = ∇ log p(n|xt),
+(3)
+can be regarded as the gradient of class prediction (n = 1 for non-MLS and 0
+otherwise) w.r.t to the input image, which reflects how the input images devi-
+ate from a non-MLS image. This latent contains information regarding how to
+transform the MLS positive image into a non-MLS one and therefore is helpful
+for training the deformation network. Moreover, this feature representation ex-
+hibits less fluctuation toward the randomness of the additive noise because the
+stochastic parts are eliminated through subtraction. It is more stable than the
+
+6
+S.Z. Gong et al.
+predicted noise or generated MSL negative images. For training, we randomly
+sample t from 0 to the diffusion steps Ttrain, while for inference we fix it to be a
+certain value. We examine the effects of this value in Section 3.4.
+2.3
+Semi-Supervised Deformation Regularization
+Deformation estimation is a dense prediction problem, while we only have sparse
+supervision. This can lead to flickering and poor generalizability if the deforma-
+tion lacks certain regularization. On the other hand, we have a significant amount
+of unlabeled data from the MLS volumes that is potentially helpful. Therefore,
+we propose to include these unlabeled data during training in a semi-supervised
+manner, so that unlabeled data can provide extra regularization for training or
+produce additional training examples based on noisy pseudo labels. Many exist-
+ing semi-supervised methods seek to use the prediction for unlabeled data given
+by the same or a twin network as pseudo-labels and then supervise the model or
+impose some regularization with these pseudo-labels. However, these methods
+hold a strong assumption that labeled and unlabeled data are drawn from the
+same distribution, which is not true in our case because most labeled data are
+with large deformation while unlabeled data are with minor or no deformation.
+Therefore, we want to find another type of pseudo-label to bypass the distribu-
+tion assumption. As the deformation field is assumed to warp a hypothetically
+normal image into an MLS one, we generate hypothetically non-MLS images x′
+0
+using pre-trained diffusion models through classifier-free guidance [16]:
+ˆϵ(xt, t) = γˆϵθC(xt, t) + (1 − γ)ˆϵθU (xt, t),
+(4)
+where γ is a hyper-parameter controlling the strength of the condition. We com-
+pare x′
+0 warped with the deformation field φ(x′
+0) and calculate its similarity with
+the original x0 through MSE loss. As it can be difficult for the generated image
+to be fully faithful to the original image because the generative process entails a
+lot of random sampling, this lmse can only serve as noisy supervision. Therefore,
+instead of generating x′
+0 ahead of deformation network training, we generate it
+in an ad-hoc way so that the noisy effects can be counteracted.
+The final MLS measurement is estimated by calculating the length of the
+maximum displacement vector from the predicted deformation field, so it is
+more sensitive to over-estimation. And our results also show most of the errors
+come from over-estimation. As for unlabelled slices, we still have the prior that
+its MLS cannot be larger than the MLS of that specific volume δ, we propose to
+incorporate an additional ceiling loss to punish the over-estimation:
+lceil =
+�
+j
+�
+k
+max(0, ||φjk|| − δ).
+(5)
+Overall, the loss term is a combination of supervised loss and unsupervised loss,
+with a weight term controlling the relative importance of each loss term:
+l = lhuber + w1lsmooth + u(i)(lmse + w2lceil),
+(6)
+
+Diffusion model based ICH midline shift quantification
+7
+where w1 and w2 are two fixed weight terms and u(i) is a time-varying weight
+term that is expected to gradually increase as the training iteration i progresses
+so that the training can converge quickly through strong supervision first and
+then refine and enhance generalizability via unsupervised loss.
+3
+Experiments and Results
+3.1
+Data Acquisition and Preprocessing
+We retrospectively collected anonymous thick-slice, non-contrast head CT of pa-
+tients who were admitted with head trauma or stroke symptoms and diagnosed
+with various subtypes of intracranial hemorrhage, including epidural hemor-
+rhage, subdural hemorrhage, subarachnoid hemorrhage, intraventricular hem-
+orrhage, and intraparenchymal hemorrhage, between July 2019 and December
+2019 in the Prince of Wales Hospital, a public hospital under the Hospital Au-
+thority of Hong Kong. The ethics approval was obtained from the Joint Chinese
+University of Hong Kong-New Territories East Cluster Clinical Research ethics
+committee. The eligible patients comprised 2793 CT volumes, among them 124
+are MLS positive cases. The MLS ranges between 2.24mm and 20.12mm, with
+mean value of 8.34mm and medium value of 8.73mm. The annotation was per-
+formed by two trained physicians and verified by one experienced radiologist
+(with over 10 years of clinical experience on ICH). The labeling process followed
+a real clinical measurement pipeline, where the shifted landmark, anterior falx
+point, and posterior falx point were pointed out, and the length of the vertical
+line from the landmark to the line connecting the anterior falx point and the
+posterior falx point was the measured MLS value. For each volume, a few slices
+with large deformation were separately measured and annotated while the shift
+of the largest one served as the case-level label. On average, 4 out of 30 slices
+of each volume were labeled. We discarded the first 8 and the last 5 slices as
+they are mainly structures irrelevant to MLS. For pre-processing, we adjusted
+the pixel size of all images to 0.86mm and then cropped or padded the resulting
+images to the resolution of 256 × 256 pixels. The HU window was set to 0 and
+80. We applied intensity clipping (0.5 and 99.5 percentiles) and min-max nor-
+malization (between -1 and 1) to each image. Random rotation between −15◦
+and 15◦ was used for data augmentation.
+3.2
+Implementation Details
+For the diffusion network, we use the network architecture designed in DDPM [15]
+and set the noise level from 10−4 to 2 × 10−2 by linearly scheduling with
+Ttrain = 1000. For non-MLS image generation, we apply the Denoising Diffu-
+sion Implicit Model (DDIM) [13] with 50 steps and set the noise scale to 15 to
+shorten the generative time. We set the hyper-parameters as α = 1, β = 1, c = 3
+and γ = 2. u(i) is set from 1 to 10 with the linear schedule. The diffusion models
+are trained by the AdamW optimizer with an initial learning rate of 1 × 10−4,
+
+8
+S.Z. Gong et al.
+Table 1: Comparison of different methods with 5-fold cross-validation.
+Methods
+Training data
+Volume-wise
+Slice-wise
+Labeled
+Unlabeled
+MAE↓
+(mm)
+RMSE↓
+(mm)
+MAE↓
+(mm)
+RMSE↓
+(mm)
+Regression
+✓
+3.91
+4.90
+3.56
+4.16
+Deformation
+✓
+3.80
+4.47
+2.51
+3.17
+Mean-Teacher [19]
+✓
+✓
+2.89
+3.67
+2.43
+3.22
+CPS [20]
+✓
+✓
+2.72
+3.42
+2.38
+3.15
+Ours
+✓
+✓
+2.43
+3.17
+2.25
+3.09
+batch size 4, for 2 × 105 iterations. We up-sample the MLS positive data by 10×
+when training the unconditional diffusion model. The deformation network is
+trained by the AdamW optimizer with an initial learning rate of 1×10−4, batch
+size 16, for 100 epochs. All models are implemented with PyTorch 1.12.1 using
+one Nvidia GeForce RTX 3090 GPU.
+3.3
+Quantification Accuracy and Deformation Quality
+We evaluate the performance of our quantification strategy through mean abso-
+lute error (MAE) and root mean square error (RMSE). For volume-wise evalua-
+tion, we measure the maximum deformation of each slice of the whole volume and
+select the largest one as the final result. We also report the slice-wise evaluation,
+which is calculated based on labeled slices. This error can reflect how the models
+perform on slices with relatively large deformation. Since existing MLS estima-
+tion methods require different types of labels from ours, it is difficult to directly
+compare with those methods. We therefore first compare our deformation-based
+strategy with a regression-based strategy, which uses DenseNet-121 [21] to di-
+rectly predict the slice-wise MLS. We also compare our proposed semi-supervised
+learning approach with two popular semi-supervised learning methods, that are
+Mean-Teacher [19] and Cross Pseudo Supervision (CPS) [20], which are imple-
+mented into our deformation framework. The results are given in Table 1, which
+are based on 5-fold cross-validations.
+From the results, we can see that when only using labeled MLS slices for
+model learning, our deformation strategy already shows better performance than
+the regression model. This may attribute to that our deformation model learns
+the knowledge of both MLS values and locations while a regression model only
+captures the MLS value information. This difference can be further enlarged if
+we consider slice-wise performance. Moreover, all three semi-supervised learn-
+ing methods, i.e., Mean-Teacher, CPS, and ours, consistently improve the per-
+formance of deformation prediction, showing the benefits and importance of
+
+Diffusion model based ICH midline shift quantification
+9
+true:
+pred:
+11.00
+11.43
+true:
+pred:
+5.43
+6.02
+true:
+pred:
+6.23
+6.78
+true:
+pred:
+7.48
+8.87
+(a) Examples of predicted deformation on MLS images
+true:
+pred:
+0.00
+1.60
+true:
+pred:
+0.00
+1.75
+true:
+pred:
+0.00
+1.87
+true:
+pred:
+0.00
+1.69
+(b) Examples of predicted deformation on non-MLS images
+Fig. 3: Predicted deformation on (a) MLS images. (b) non-MLS images. The
+regions with the largest deformation are highlighted. Slice-wise predicted MSL
+and ground truth are provided.
+incorporating unlabeled data into model learning. Our semi-supervised learn-
+ing method based on diffusion models achieves better quantification results
+than Mean-Teacher and CPS, significantly reducing the volume-wise MAE from
+3.80mm to 2.43mm. An interesting observation is that the unlabeled data con-
+tribute more to the volume-wise evaluation than the slice-wise evaluation. By
+inspecting the prediction, we find that the deformation prediction trained with
+labeled data tends to overestimate the deformation of slices with little or no
+deformation, which makes the volume-wise prediction error-prone. As most un-
+labeled data are slices with minor shifts, incorporating these data for semi-
+supervised learning can impose constraints to avoid large deformation, which
+greatly improves the model’s robustness.
+We also visualize the predicted deformation field of several sample cases.
+From Fig. 3 (a), we can see the model can well posit the location where the
+maximum shift appears and push it to its hypothetically normal counterpart.
+The largest deformation happens exactly at the site with the maximum shift. To
+validate the robustness of our model, we also select several patients diagnosed
+with no MLS and plot the predicted deformation of these samples. As can be
+seen in Fig. 3 (b), our method is able to provide a reasonable prediction for
+non-MLS images by outputting much smaller values than that for MLS images.
+Our model’s predictions for non-MLS images are not exactly zero are caused on
+one hand by that even for a completely healthy person, the midline cannot be
+perfectly aligned due to multiple factors such as scan pose, on the other hand, our
+
+O10
+S.Z. Gong et al.
+Methods
+MAE↓
+(mm)
+RMSE↓
+(mm)
+Fully-supervised
+3.61
+4.47
++ Representation
+3.22
+3.69
+Semi-supervised
+2.61
+3.24
++ Representation
+2.45
+3.05
+Table 2: Effects of the representation.
+0
+200
+400
+600
+800
+t
+2.6
+2.8
+3.0
+3.2
+3.4
+Effects of noise level t
+MAE
+RMSE
+Fig. 4: Effects of the noise level.
+models tend to overestimate the shift because we are calculating the maximum
+deformation as final measurement.
+3.4
+Ablation Study
+We conduct several ablation experiments to study the effects of several compo-
+nents in our proposed framework on the model performance. The volume-wise
+results reported are trained on four folders and tested on one folder.
+Effects for representation learning. We first conduct ablation studies to
+verify that the latent feature extracted from the two diffusion models is truly
+useful for deformation prediction. To this end, we select two deformation models,
+one trained with only labeled data and the other using semi-supervised learning,
+and compare their performance with and without the extracted representation
+as input. The results are given in Table 2. As expected, incorporating the rep-
+resentation can improve the model performance in both cases.
+The noise level is an important component of diffusion models. Only with a
+proper noise level, can the model accurately estimate the deviation of the image
+toward the negative sample space. Therefore, we do inference with multiple noise
+levels and compare its effect on model performance. The results are shown in
+Fig. 4. Our model is very robust towards this hyper-parameter. As long as t is
+not too small, the model gives very similar performances. The best performance
+appears in the middle when t = 600. This is reasonable as small noise fails to
+corrupt the original image thus degenerating the performance of score estimation
+while large noise may obscure too many details of the original image.
+Quantity of unlabeled images. To verify the usefulness of unlabeled images,
+we conduct ablation studies on the number of unlabeled images used. For each
+experiment, we randomly sample 20%, 40%, 60%, and 80% volumes, and we
+incorporate unlabeled slices of these volumes for semi-supervised training. For
+the rest volumes, we are only using the labeled slices. We also do one experiment
+that completely removes the uses of unlabeled images. For each experiment, the
+pre-trained diffusion models are the same, which uses all the data. In other
+words, these unlabeled images somehow still contribute to the model training.
+The results are shown in Fig. 5 (a). As can be seen, the model performance
+and robustness can be enhanced as we incorporate more unlabeled images. This
+
+Diffusion model based ICH midline shift quantification
+11
+(a)
+(b)
+Fig. 5: Results of our ablation experiments in terms of: (a) proportion of unla-
+beled data used, and (b) proportion of negative data used.
+provides strong evidence for our claim that our model truly learns valuable
+information from unlabeled data.
+Quantity of non-MLS images. To further measure the benefits of includ-
+ing non-MLS cases, we conduct another ablation study on the proportion of
+non-MLS cases. Non-MLS cases are used to train diffusion models. As currently,
+the amount of non-MLS cases is much higher than MLS cases, we upsample the
+MLS cases so that their quantities are approximately the same when training the
+unconditional diffusion model. For ablation, we first downsample the non-MLS
+data so that their quantity is 1×, 5×, and 10× that of the MLS cases, and then
+upsample the MLS cases to make them balanced. From the results in Fig. 5 (b),
+we find model performance improves with more non-MLS cases incorporated. In-
+creasing non-MLS cases can help train diffusion models and further improve the
+quality of generated images and extracted feature representations. However, this
+effect will soon be saturated as the amount of MLS cases is relatively small. This
+can be a bottleneck for effectively using the non-MLS cases as it is challenging
+to train unconditional diffusion models with such imbalanced datasets.
+4
+Conclusions and Future Work
+In this paper, we propose a novel framework based on deformation field esti-
+mation to automatically measure the brain MLS. The labels we are using are
+sparse which can greatly alleviate the labeling workload. We also propose a
+semi-supervised learning strategy based on diffusion models which significantly
+improves the model performance. Experiments on a clinic dataset show our meth-
+ods can achieve satisfying performance. We also verify that using unlabeled data
+and non-MLS cases can truly help improve the model’s performance.
+Our methods have several limitations. First, the model performance highly
+relies on pre-trained diffusion models. Training diffusion models with extremely
+imbalanced data requires great effort. Second, the measurement results exhibit
+randomness due to noise corruption. Finally, the measurement results are prone
+to overestimation. Our future work will figure out solutions for these limitations.
+
+Effects of unlabeled data
+4.5
+MAE
+RMSE
+4.0
+3.5
+3.0
+2.5
+2.0
+0
+20%
+40%
+60%
+80%
+100%
+proportion of unlabeled image usedEffects of negative cases
+5.0
+MAE
+4.5
+RMSE
+4.0
+3.5
+3.0
+2.5
+2.0
+0
+1X
+5x
+10x
+all
+ratio of negative cases / positive cases12
+S.Z. Gong et al.
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+

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+arXiv:2301.01830v1  [hep-ph]  4 Jan 2023
+Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the λφ4
+4 Theory
+Robert Shrock
+C. N. Yang Institute for Theoretical Physics and
+Department of Physics and Astronomy
+Stony Brook University, Stony Brook, NY 11794
+We investigate whether the seven-loop beta function of the λφ4
+4 theory exhibits evidence for
+an ultraviolet zero. In addition to a direct analysis of the beta function, we calculate and study
+Pad´e approximants and discuss effects of scheme transformations on the results. Confirming and
+extending our earlier studies of the five-loop and six-loop beta functions, we find that in the range
+of λ where the perturbative calculation of the seven-loop beta function is reliable, the theory does
+not exhibit evidence for an ultraviolet zero.
+I.
+INTRODUCTION
+In this paper we consider the renormalization-group
+(RG) behavior of the λφ4 field theory in d = 4 spacetime
+dimensions, where φ is a real scalar field. This theory,
+commonly denoted φ4
+4, is described by the Lagrangian
+L = 1
+2(∂νφ)(∂νφ) − m2
+2 φ2 − λ
+4! φ4 .
+(1.1)
+The Lagrangian (1.1) is invariant under the global dis-
+crete Z2 symmetry φ → −φ. Quantum loop corrections
+lead to a dependence of the physical quartic coupling
+λ = λ(µ) on the Euclidean energy/momentum scale µ
+at which this coupling is measured. The dependence of
+λ(µ) on µ is described by the RG beta function of the
+theory, βλ = dλ/dt, or equivalently, βa = da/dt, where
+dt = d ln µ [1] and
+a ≡
+λ
+(4π)2 .
+(1.2)
+(The argument µ will often be suppressed in the nota-
+tion.) Since we will investigate the properties of the the-
+ory for large µ in the ultraviolet (UV), the value of m2
+will not play an important role in our analysis. For tech-
+nical convenience, we assume that m2 is positive. At a
+reference scale µ0, the quartic coupling λ(µ0) is taken to
+be positive for the stability of the theory. The one-loop
+term in this beta function has a positive coefficient, so
+that for small λ, βλ > 0 and hence as µ → 0, the cou-
+pling λ(µ) → 0, i.e., the theory is infrared (IR)-free. This
+perturbative result is in agreement with nonperturbative
+approaches [2]; some reviews include [3, 4].
+The beta function βa has the series expansion
+βa = a
+∞
+�
+ℓ=1
+bℓ aℓ .
+(1.3)
+The n-loop (nℓ) beta function, denoted βa,nℓ, is given by
+Eq. (1.3) with the upper limit of the loop summation in-
+dex ℓ = n instead of ℓ = ∞. The one-loop and two-loop
+terms in βa are independent of the scheme used for regu-
+larization and renormalization, while terms of loop order
+ℓ ≥ 3 are scheme-dependent [5, 6]. For the O(N) λ|⃗φ|4
+theory with an N-component field, ⃗φ = (φ1, ..., φN), the
+coefficients b1, b2, and b3 were calculated in [5]. Higher-
+loop coefficients bℓ with ℓ ≥ 3 have been computed using
+the MS minimal subtraction scheme [7, 8]. A calculation
+of b5 and discussion of earlier computations of b4 and
+b5 (e.g., [9–11]) was given in [4, 12]. The coefficient b6
+was calculated for N = 1 in [13] and for general N in
+[14]. Most recently, the seven-loop coefficient b7 was cal-
+culated in [15]. In analyzing the series expansion (1.3),
+one recalls that it is an asymptotic expansion and the
+large-order behavior has been the subject of extensive
+study [16], including [17] and references therein.
+An interesting question is whether, for the region of
+λ where a perturbative calculation of βλ is reliable, this
+beta function exhibits evidence for a zero at some (pos-
+itive) value of the quartic coupling. This would be an
+ultraviolet fixed point (UVFP) of the renormalization
+group, i.e., as µ → ∞, λ(µ) would approach this value
+(from below). In previous work we have investigated this
+question up to the five-loop order for the O(N) λ|⃗φ|4
+theory in [18] and up to the six-loop order for the real
+λφ4 theory in [19] and the O(N) λ|⃗φ|4 theory in [20],
+finding evidence against such a UVFP. In the present
+paper, using the results of [15], we extend our analysis to
+the seven-loop level. Our analysis in [20] covered a large
+range of specific N values and also included an argument
+for the absence of a UV zero in the (rescaled) n-loop beta
+function at large N (see Eqs. (3.12)-(3.13) in [20]). Thus,
+it will suffice to focus on the N = 1 theory here.
+In view of this previous evidence against a UV zero in
+βλ and associated UVFP in the O(N) λ|⃗φ|4 theory, it is
+worthwhile to mention one case where an IR-free quan-
+tum field theory is known to have a UVFP, namely, the
+nonlinear O(N) σ model in d = 2 + ǫ spacetime dimen-
+sions. In this theory, an exact solution was obtained in
+the limit N → ∞ with λ(µ)N = x(µ) a fixed function of
+µ and yielded the beta function
+βx = dx
+dt = ǫx
+�
+1 −
+x
+xUV
+�
+(1.4)
+for small ǫ, where xUV = 2πǫ is a UV fixed point of the
+renormalization group [21]. Since the leading term in βx
+is positive for ǫ > 0, this theory is IR-free. Thus, in this
+nonlinear O(N) σ model in d = 2 + ǫ dimensions, the
+
+2
+coupling x(µ) flows (monotonically) from x = 0 at µ = 0
+to x = xUV as µ → ∞. Note that by making ǫ ≪ 1 one
+can arrange that the UVFP at xUV = 2πǫ occurs at an
+arbitrarily small value of the scaled coupling x.
+This paper is organized as follows. In Section II we re-
+view some relevant background. In Section III we present
+the results of our analysis of the seven-loop beta function.
+Section IV contains a further analysis of this question of
+a UV zero using Pad´e approximants, while Section V dis-
+cusses effects of scheme transformations. Our conclusions
+are given in Section VI.
+II.
+BETA FUNCTION
+The n-loop truncation of (1.3), denoted βa,nℓ, is a poly-
+nomial in a of degree n + 1 having an overall factor of
+a2. We may extract this factor and define a reduced beta
+function
+βa,r =
+βa
+βa,1ℓ
+=
+βa
+b1a2
+= 1 + 1
+b1
+∞
+�
+ℓ=2
+bℓaℓ−1 .
+(2.1)
+The n-loop truncation of βa,r, denoted βa,r,nℓ ≡ Rn, is
+defined by taking the upper limit of the sum in (2.1) to
+be ℓ = n rather than ℓ = ∞. .
+The first two coefficients in the beta function of this
+theory are b1 = 3 and b2 = −17/3 [5]. The coefficients bℓ
+with 3 ≤ ℓ ≤ 7 and the resultant higher-loop beta func-
+tion discussed below, are calculated in the MS scheme.
+The coefficients up to the five-loop level are [4, 5, 9, 12]
+b3 = 145
+8
++ 12ζ3 = 32.5497 ,
+(2.2)
+b4 = −3499
+48
+− 78ζ3 + 18ζ4 − 120ζ5
+= −271.606 ,
+(2.3)
+and
+b5 = 764621
+2304
++ 7965
+16 ζ3 − 1189
+8
+ζ4 + 987ζ5 + 45ζ2
+3
+− 675
+2 ζ6 + 1323ζ7
+= 2848.57 ,
+(2.4)
+where the floating-point values are given to the indicated
+accuracy and
+ζs =
+∞
+�
+n=1
+1
+ns
+(2.5)
+is the Riemann zeta function. If s = 2r is even, then
+ζs can be expressed as a rational number times π2r,
+namely ζ2r = (−1)r+1B2r(2π)2r/[2(2r)!], where Bn are
+the Bernoulli numbers; however, we leave these ζ2r in
+their generic form here and below. The six-loop coeffi-
+cient is [13, 14]
+b6 = −18841427
+11520
+− 779603
+240
+ζ3 + 16989
+16
+ζ4 − 63723
+10
+ζ5 − 8678
+5
+ζ2
+3 + 6691
+2
+ζ6 + 162ζ3ζ4 − 63627
+5
+ζ7
+− 4704ζ3ζ5 + 264543
+25
+ζ8 − 51984
+25
+ζ3,5 − 768ζ3
+3 − 46112
+3
+ζ9
+= −34776.13 ,
+(2.6)
+where [22]
+ζ3,5 =
+�
+m>n≥1
+1
+n3m5 .
+(2.7)
+The seven-loop coefficient is considerably more compli-
+cated than b6, and we refer the reader to [15] for the
+analytic expression. The numerical value is
+b7 = 474651.0 .
+(2.8)
+Thus, in summary, the seven-loop beta function of the
+λφ4 theory (calculated in the MS scheme), is
+βa,7ℓ = a2�
+3 − 17
+3 a + 32.5497a2 − 271.606a3
++ 2848.57a4 − 34776.1a5 + 474651a6�
+.
+(2.9)
+III.
+ZEROS OF THE n-LOOP BETA FUNCTION
+UP TO LOOP ORDER n = 7
+In this section we investigate a possible UV zero, de-
+noted aUV,nℓ, of the n-loop beta function, βa,nℓ.
+The
+double zero of βa,nℓ at a = 0 is always present (indepen-
+dent of n); this is an infrared zero and hence will not be
+of interest here.
+
+3
+A necessary condition for there to be robust evidence
+for a UV zero in the beta function of an IR-free the-
+ory is that the values calculated at successive loop orders
+should be close to each other.
+Although the two-loop
+beta function βa,2ℓ does have a UV zero, at aUV,2ℓ =
+9/17 = 0.52941, we found that the three-loop beta func-
+tion βa,3ℓ has no UV zero and, while a UV zero is present
+in βa,4ℓ, it occurs at a considerably smaller value, namely
+aUV,4ℓ = 0.23332. At the five-loop level, βa,5ℓ has no UV
+zero, while at the six-loop level, although βa,6ℓ has a UV
+zero, it occurs at a still smaller value, aUV,6ℓ = 0.16041
+[18, 19].
+Thus, the results of this analysis show that
+the necessary condition that the beta function calculated
+to successively higher loop order should exhibit values
+of aUV,nℓ that are close to each other is not satisfied by
+this theory. At seven-loop order, using βa,7ℓ from [15],
+we find that this function has no physical UV zero. In-
+stead, the zeros are comprised of three complex-conjugate
+pairs, −0.102135±0.079848i, 0.0142348±0.136854i, and
+0.124533 ± 0.0659940i. Summarizing,
+aUV,2ℓ = 0.52941,
+aUV,4ℓ = 0.23332,
+aUV,6ℓ = 0.16041
+no aUV,nℓ for n = 3, 5, 7.
+(3.1)
+The calculations up to seven loops show a pattern,
+namely that for even n = 2, 4, 6, βa,nℓ has a zero, aUV,nℓ,
+but the values for different n are not close to each other,
+while for odd n = 1, 3, 5, 7, βa,nℓ has no UV zero.
+In Fig. 1 we plot the n-loop beta functions for 2 ≤ n ≤
+7 loops. Another way to show this information is via the
+n-loop reduced beta function, βa,r,nℓ = Rn. We plot Rn
+in Fig. 2 for 2 ≤ n ≤ 7. The results discussed above are
+evident in these figures. First, one may inquire how large
+is the interval in a over which the calculations of βa,nℓ to
+the respective n-loop orders are in mutual agreement. As
+one can see from Figs. 1 and 2, the n-loop beta functions
+βa,nℓ with 2 ≤ n ≤ 7 only agree with each other well over
+the small interval of couplings 0 ≤ a <∼ 0.05. As shown
+in Fig. 1, the βa,nℓ with even n = 2, 4, 6 reach maxima
+and then decrease, crossing the (positive) real axis at
+different values listed in Eq. (3.1), while the βa,nℓ with
+odd n increase monotonically with a.
+This seven-loop
+analysis confirms and extends our conclusions in [19, 20]
+at the six-loop level that the zero in the two-loop beta
+function of the λφ4 theory occurs at too large a value of
+a for the perturbative calculation to be reliable.
+IV.
+ANALYSIS WITH PAD´E APPROXIMANTS
+One can gain further insight into the behavior of the
+beta function by the use of Pad´e approximants (PAs).
+We carried out this analysis up to the six-loop level in
+[19, 20], finding no indication of a physical UV zero, and
+here we extend it to the seven-loop level. Since the double
+zero in βa,nℓ at a = 0 is not relevant to the question of a
+UV zero, we use the reduced beta function βa,r,nℓ for this
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+a
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+beta
+FIG. 1:
+Plot of the n-loop β function βa,nℓ as a function of a for
+(i) n = 2 (red, solid), (ii) n = 3 (green, dashed), (iii) n = 4 (blue,
+dotted), (iv) n = 5 (black, dot-dashed), (v) n = 6 (cyan, solid),
+and (vi) n = 7 (brown, solid). At a = 0.16, going from bottom to
+top, the curves are for n = 6, n = 4, n = 2, n = 3, n = 5, n = 7.
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+a
+0.0
+0.5
+1.0
+1.5
+R_n
+FIG. 2: Plot of the ratio Rn of the n-loop beta function βa,nℓ
+divided by βa,1ℓ, as a function of a for (i) n = 2 (red, solid), (ii)
+n = 3 (green, dashed), (iii) n = 4 (blue, dotted), (iv) n = 5 (black,
+dot-dashed), (v) n = 6 (cyan, solid), and (vi) n = 7 (brown, solid).
+At a = 0.16, going from bottom to top, the curves are for n = 6,
+n = 4, n = 2, n = 3, n = 5, and n = 7.
+Pad´e analysis. The [p, q] Pad´e approximant to βa,r,nℓ is
+the rational function [23]
+[p, q]βa,r,nℓ =
+1 + �p
+j=1 rjaj
+1 + �q
+k=1 sk ak
+(4.1)
+with p + q = n − 1, where the coefficients rj and sj are
+independent of a. At seven-loop order, we can calculate
+
+4
+the Pad´e approximants [p, q]βa,r,7ℓ with [p, q] taking on
+the values [6,0], [5,1], [4,2], [3,3], [2,4], [1,5], and [0,6].
+Since the loop order is understood, we write [p, q]βa,r,7ℓ ≡
+[p, q] for brevity of notation. The PA [6,0] is equivalent
+to βa,r,7ℓ itself, which we have already analyzed, and the
+PA [0,6] has no zeros, so we focus here on the remaining
+five Pad´e approximants.
+We list our results for these Pad´e approximants to
+βa,r,7ℓ below:
+[5, 1] = 1 + 11.760a − 14.931a2 + 57.552a3 − 286.17a4 + 1367.8a5
+1 + 13.649a
+,
+(4.2)
+[4, 2] = 1 + 20.541a + 75.687a2 − 49.670a3 + 81.973a4
+1 + 22.430a + 107.21a2
+,
+(4.3)
+[3, 3] = 1 + 25.073a + 152.81a2 + 155.99a3
+1 + 26.962a + 192.89a2 + 318.33a3 ,
+(4.4)
+[2, 4] =
+1 + 22.314a + 103.55a2
+1 + 24.203a + 138.42a2 + 89.390a3 − 91.252a4 ,
+(4.5)
+[1, 5] =
+1 + 14.023a
+1 + 15.912a + 19.205a2 − 45.828a3 + 196.10a4 − 910.03a5 .
+(4.6)
+We recall some necessary requirements for a zero of a
+[p, q] Pad´e approximant to be physically relevant. These
+include the requirement that this zero should occur on
+the positive real axis in the complex a plane and the
+requirement that this zero of the PA should be closer to
+the origin a = 0 than any pole on the real positive a-axis,
+since otherwise the pole would dominate the IR to UV
+flow starting at the origin. If a Pad´e approximant were
+to exhibit such a zero, then one would proceed to inquire
+how close it is to any of the aUV,nℓ in Eq. (3.1). However,
+we find that none of these Pad´e approximants (4.2)-(4.6)
+has a zero on the positive real a axis.
+Explicitly, the
+[5,1] PA has two complex-conjugate pairs of zeros at a =
+−0.12719±0.26046i and a = 0.26922±0.20930i, together
+with a real zero at a = −0.074837.
+This real zero is
+part of a nearly coincident pole-zero pair, with the pole
+of the [5,1] PA being located at a = −0.073267. The
+appearance of a nearly coincident pole-zero pair close to
+a point a0 in a [p, q] Pad´e approximant is typically an
+indication that the function that the PA is fitting has
+neither a pole nor a zero in the local neighborhood of
+a0, since as the locations of the nearly coincident pole-
+zero pair approach each other, they simply divide out in
+the ratio (4.1). Each of the Pad´e approximants that we
+calculate here has a pole-zero pair. The [4,2] PA has zeros
+at the complex-conjugate pair a = 0.42009 ± 0.96575i,
+together with the real values a = {−0.16929, −0.064970}
+and poles at a = {−0.14481, −0.064414}. The [3,3] PA
+has zeros at a = {−0.78531, −0.13282, −0.061458}, and
+poles at a = {−0.42342, −0.12140, −0.061112}. The
+[2,4] PA has zeros at a = {−0.15193, −0.063563}, and
+poles at a = {−0.69186, −0.13432, −0.063100, 1.8689}.
+Finally, the [1,5] PA has a zero at a = −0.071313 and
+poles at a = {−0.22780, −0.070185, 0.44160, 0.035937±
+0.39287i}. Thus, our analysis with Pad´e approximants of
+the seven-loop beta function yields the same conclusion
+as our analysis of the beta function itself, namely that
+there is no evidence for a stable, reliably perturbatively
+calculable UV zero up to this seven-loop level.
+V.
+EFFECTS OF SCHEME
+TRANSFORMATIONS
+Since the terms in the beta function at loop order n ≥ 3
+are scheme-dependent, it is necessary to assess the effect
+of scheme transformations in an analysis of zeros of a
+higher-loop beta function. A scheme transformation can
+be expressed as a mapping between a and a transformed
+coupling a′,
+a = a′f(a′) ,
+(5.1)
+where f(a′) is the scheme transformation function. Since
+this transformation has no effect in the free theory, one
+has f(0) = 1. We consider f(a′) functions that are ana-
+lytic about a = a′ = 0 and hence can be expanded in the
+form
+f(a′) = 1 +
+smax
+�
+s=1
+ks(a′)s ,
+(5.2)
+
+5
+where the ks are constants and smax may be finite or
+infinite. The beta function in the transformed scheme,
+βa′ = da′/d ln µ, has the expansion
+βa′ = a′
+∞
+�
+ℓ=1
+b′
+ℓ(a′)ℓ .
+(5.3)
+In [24], formulas were derived for the b′
+ℓ in terms of bℓ
+and the ks. In addition to b′
+1 = b1 and b′
+2 = b2, these are
+b′
+3 = b3 + k1b2 + (k2
+1 − k2)b1 ,
+(5.4)
+b′
+4 = b4 + 2k1b3 + k2
+1b2 + (−2k3
+1 + 4k1k2 − 2k3)b1 , (5.5)
+and so forth for higher ℓ. These results are applicable
+to the study of both an IR zero in the beta function of
+an asymptotically free theory and a possible UV zero in
+the beta function of an IR-free theory. They were exten-
+sively applied to assess scheme dependence in higher-loop
+studies of an IR fixed point in asymptotically free non-
+Abelian gauge theories [24–28].
+For the present λφ4 theory, a study of scheme depen-
+dence was carried out in [18]. It was shown that even
+when one shifts to a scheme different from the usual MS
+scheme, the beta function still does not satisfy a requi-
+site condition for a physical UV zero, namely that the
+value of this zero (in a given scheme) should not change
+strongly when it is calculated to successive loop orders.
+This result from [18] also holds in the same way in the
+present seven-loop context.
+VI.
+CONCLUSIONS
+In this paper we have investigated whether the real
+scalar field theory with a λφ4 interaction exhibits evi-
+dence of an ultraviolet zero in the beta function.
+Us-
+ing the seven-loop coefficient b7 from [15], our present
+study extends our previous six-loop study in [19, 20] to
+the seven-loop level. Our work includes a study of the
+seven-loop beta function itself, together with an analysis
+of Pad´e approximants. We conclude that, for the range
+of couplings where the perturbative calculation of this
+beta function may be reliable, it does not exhibit robust
+evidence for an ultraviolet zero.
+Acknowledgments
+I would like to thank Oliver Schnetz for valuable dis-
+cussions on [15]. This research was supported in part by
+the U.S. National Science Foundation Grant NSF-PHY-
+22-15093.
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+

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+Demystify Problem-Dependent Power of Quantum Neural Networks on Multi-Class
+Classification
+Yuxuan Du,1, ∗ Yibo Yang,1 Dacheng Tao,1 and Min-Hsiu Hsieh2
+1JD Explore Academy, Beijing 10010, China
+2Hon Hai (Foxconn) Research Institute, Taipei, Taiwan
+Quantum neural networks (QNNs) have become an important tool for understanding the physical
+world, but their advantages and limitations are not fully understood. Some QNNs with specific en-
+coding methods can be efficiently simulated by classical surrogates, while others with quantum mem-
+ory may perform better than classical classifiers. Here we systematically investigate the problem-
+dependent power of quantum neural classifiers (QCs) on multi-class classification tasks. Through
+the analysis of expected risk, a measure that weighs the training loss and the generalization error of
+a classifier jointly, we identify two key findings: first, the training loss dominates the power rather
+than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the
+double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between
+optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we
+propose a method that uses loss dynamics to probe whether a QC may be more effective than a
+classical classifier on a particular learning task. Numerical results demonstrate the effectiveness of
+our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and
+their limitations over convolutional neural networks on image datasets. Our work sheds light on the
+problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.
+I.
+INTRODUCTION
+The advent of hardware fabrication pushes the bound-
+ary of quantum computing from verifying its superiority
+on artificial tasks [1–3] to conquering realistic problems
+with merits [4–6]. This has led to the emergence of a
+popular paradigm known as quantum neural networks
+(QNNs), which combine variational quantum Ans¨atze
+with classical optimizers [7, 8].
+So far, various QNN-
+based methods have been proposed to address difficult
+problems in areas such as quantum physics [9–12], quan-
+tum information theory [13–16], combinatorial optimiza-
+tion [17–21], and machine learning [22–26]. Among these
+applications, QNNs are often deployed as quantum clas-
+sifiers (QCs) to predict correct labels of the input data
+[27–32], e.g., categorize image objects [33–35], classify
+phases of quantum matters [36–39], and distinguish en-
+tangled states from separable states [40, 41].
+To comprehend the full potential of existing quantum
+classifiers (QCs) and to spur the development of novel
+QCs, huge efforts have been made to unveil the learnabil-
+ity of QCs [42–44]. Prior literature establishes the foun-
+dations of QCs from three primary aspects, i.e., model
+capacity [45–48], trainability [49–51], and generalization
+[52–57]. Nevertheless, the advantages and constraints of
+QCs have rarely been proven [57–62]. Meanwhile, pre-
+vious results cannot rigorously explain the empirical ob-
+servations such that QCs generally outperform classical
+classifiers (CCs) on handcraft or quantum data [44, 63]
+but are inferior to them on realistic problems [64]. As
+a result, the need for QCs to address classical issues re-
+mains highly questionable.
+∗ [email protected]
+A principal criteria in characterizing the power of a
+classifier is the expected risk [65], which weighs the em-
+pirical risk (i.e., training loss) and the generalization er-
+ror (i.e., test loss) jointly. An optimal classifier is one
+which achieves zero expected risk. As shown in Fig. 1(a),
+the success of deep neural classifiers is attributed to their
+double-descent risk curves [66, 67]. This means that as
+the hypothesis space is continually expanded, the ex-
+pected risk of a trained deep neural classifier initially
+decreases, increases, and when it overfits the train set,
+undergoes a second descent. As such, to show the supe-
+riority of QCs over CCs, it demands to distill ubiquitous
+rules that capture the risk curve of diverse QCs in addi-
+tion to conditions where the expected risk of QCs can be
+lower than CCs.
+In this study, we unify a broad class of QCs in
+the same framework and understand their problem-
+dependent ability under the expected risk (see Fig. 1(b)).
+Our analysis reveals two substantial outcomes: (i) train-
+ability dominates QCs’ ability more than generalization
+ability; (ii) QCs undergo a U-shape risk curve instead
+of the double-descent curve for CCs.
+These outcomes
+consolidate and refine previous observations. Concretely,
+the first outcome suggests that the deficiency of QCs on
+classical data stems from their limited ability to fit the
+train set, resulting in a larger training loss compared to
+CCs. The second outcome highlights the distinct learn-
+ing behavior of QCs and CCs. Despite the fact that over-
+parameterization is fundamental to enhance the perfor-
+mance of CCs, it adversely affects the power of QCs.
+In line with the diverse dynamics of the risk curves for
+QCs and CCs, we devise an efficient problem-dependent
+method to recognize potential merits of QCs, as shown
+in Fig. 1(a). Conceptually, for a given learning task, our
+method fits the loss (risk) dynamics of QC and CC under
+the prior (i.e., U-shape versus double descent) and then
+arXiv:2301.01597v1  [quant-ph]  29 Dec 2022
+
+2
+(a)
+(b)
+𝑜(") 𝑜($) 𝑜(%)
+𝜌(&)
+ℇ(𝑥(&))
+𝑥(&)
+𝑜(')
+ℛ
+𝐶-ℛ
+Hypothesis space
+𝑄-ℛ
+𝑄-ℛ!"#
+ℛ
+𝐶-ℛ
+Hypothesis space
+𝑄-ℛ
+𝑄-ℛ!"#
+(c)
+𝜌̅(%), 𝑜(%)
+𝜌̅ ' , 𝑜(')
+𝜌̅ ( , 𝑜(() 𝜌̅ ) , 𝑜())
+optimize
+𝜌̅(%), 𝑜(%)
+𝜌̅ ' , 𝑜(')
+𝜌̅ ( , 𝑜(() 𝜌̅ ) , 𝑜())
+FIG. 1. Risk curve and geometry of the unified QCs. (a) The risk curve of QCs and CCs are highlighted by the solid red
+and blue lines (labeled by ‘Q-R’ and ‘C-R’), respectively. The former yields a ‘U’ shape while the latter yields a double-descent
+tendency. Potential advantages of QCs are dominated by the empirical risk, highlighted by the dashed curve. The shaded
+region refers to the potential merits of QCs. (b) The unified QC consists of two parts, the feature state ρ and the measure
+operator o. This model covers diverse QCs. (c) Geometric relationship between {ρ(i,k)} and o of QCs with (near) zero training
+loss: (i) the feature states associated with train samples belonging to the same class concentrate around their class-feature
+mean, i.e., ¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k) for ∀k ∈ [K]; (ii) the class-feature means are maximally distant with each other, i.e.,
+Tr(¯ρ∗(k)¯ρ∗(k′)) ∼ δk,k′; (iii) the measure operator should align with class-feature means, i.e., Tr(¯ρ∗(k)o∗(k′)) ∼ δk,k′.
+identify the ‘advantage’ regime where the risk of QC is
+lower than CC. Numerical simulations are conducted to
+support our theoretical results.
+On the technical level, we approach the two outcomes
+by separately quantifying the empirical risk and gener-
+alization error of QCs. Specifically, we first prove con-
+ditions of QCs that lead to near-zero empirical risk, the
+geometric interpretation of which is depicted in Fig. 1(c).
+As a byproduct, we elucidate how such conditions are
+inherently linked to quantum state discrimination and
+quantum measurement theory.
+In addition, we prove
+that deep QCs can never reach the vanished empirical
+risk by utilizing the concentration property of quantum
+observables [68, 69]. We next analyze the generalization
+error of QCs by exploiting algorithmic robustness [70].
+The derived bound surpasses prior results because it is
+the first non-vacuous bound in the over-parameterized
+regime.
+By combining the unreachable zero empirical
+risk with the manipulatable generalization error, we ob-
+tain the first outcome. The second outcome is gained by
+integrating the fact that deep QCs are unable to reach
+the vanished empirical risk with the first outcome.
+II.
+MAIN RESULTS
+Expected risk.— Let us first introduce a K-class (K ≥
+2) classification task. Denote the input space as X, the
+label (class) space as Y = {1, · · · , K}, and the train set
+as D = �K
+k=1{(x(i,k), y(i,k))}nk
+i=1 with |D| samples drawn
+i.i.d.
+from an unknown probability distribution D on
+Z = X × Y. In standard scenarios, the number of train
+samples in each class is the same, i.e., n1 = ... = nk ≡ nc
+and |D| := n = Knc.
+The purpose of a classification
+algorithm A is using D to infer a hypothesis (a.k.a., a
+classifier) hAD : X → RK from the hypothesis space H
+to separate train examples from different classes. This
+is equivalent to identifying an optimal hypothesis in H
+minimizing the expected risk R(h) = E(x,y)∼D[ℓ(h(x), y)],
+where ℓ(·, ·) is the per-sample loss and for clarity we spec-
+ify it as the square error with ℓ(a, b) = 1
+2∥a − b∥2
+2 [71].
+Unfortunately, the inaccessible distribution D forbids us
+to assess the expected risk directly. In practice, A al-
+ternatively learns an empirical classifier ˆh ∈ H, as the
+global minimizer of the (regularized) loss function
+L(h, D) = 1
+n
+nc
+�
+i=1
+K
+�
+k=1
+ℓ(h(x(i,k)), y(i,k)) + E(h),
+(1)
+where E(h) is an optional regularizer.
+The foremost role of the risk means that quantum ad-
+vantages can be ascertained if R(ˆhQ) < R(ˆhC), where ˆhQ
+and ˆhC are the empirical QC and CC on D. Unlike con-
+ventions merely focusing on a QC on one specific task,
+the above criteria orients to unearth ubiquitous rules of
+QCs with computational advantages.
+To reconcile the
+intractable issue of R(ˆh) and proceed the following anal-
+ysis, we decomposed it into two measurable terms, i.e.,
+R(ˆh) = RERM(ˆh) + RGene(ˆh),
+(2)
+where RERM(ˆh) = 1
+n
+�n
+i=1
+�K
+k=1 ℓ(ˆh(x(i,k)), y(i,k)) is the
+empirical risk and RGene(ˆh) = R(ˆh)−RERM(ˆh) is the gen-
+eralization error. Based on Eq. (2), detecting advances
+of QCs is translated to deriving under what conditions
+do QCs commit both lower RERM and RGene than CCs.
+To better elucidate our results, let us recall that the
+general form of QC is ˆhQ = arg minhQ∈HQ L(hQ, D),
+
+Learning tasks
+Advantages
+No Advantages介83
+where L is defined in Eq. (1) and HQ is the hypothe-
+sis space. For an N-qubit QC, its hypothesis space is
+HQ =
+��
+hQ(·, U(θ), O(k))
+�
+k=1:K
+���θ ∈ Θ
+�
+,
+(3)
+where [·]k=1:K is a K-dimensional vector, its k-th entry
+hQ(x, U(θ), O(k)) = Tr(O(k)U(θ)σ(x)U(θ)†) for ∀k ∈
+[K] refers to the output (prediction) of quantum cir-
+cuits, σ(x) = UE(x)(|0⟩ ⟨0|)⊗NUE(x)† is the input state
+of x with the encoding circuit UE(·), O = {O(k)}K
+k=1
+is a set of measure operators, and U(θ) is the adopted
+Ansatz with trainable parameters θ living in the pa-
+rameter space Θ. Without loss of generality, we define
+U(θ) = �Nt
+l=1(ul(θ)ue) ∈ U(2N), where ul(θ) ∈ U(2m)
+is the l-th parameterized quantum gate operated with at
+most m qubits (m ≤ N) and ue refers to fixed quan-
+tum gates. Similarly, we define UE(x) = �Ng
+g=1 ug(x) ∈
+U(2N), where ug(x) ∈ U(2m) refers to the g-th quan-
+tum gate operated with at most m qubits, and Ng gates
+contain Nge tunable gates and Ng − Nge fixed gates.
+Due to the diverse constructions of U(θ) and UE(·), it
+is necessary to unify various QCs into the same frame-
+work to obtain the generic results. Notably, the unified
+QC should be agnostic to particular forms of these two
+terms. A feasible way is rewritten hQ(·, U(θ), O(k)) as
+hQ(ρ(i,k), o(k)) := Tr(ρ(i,k)o(k)) ∀k ∈ [K],
+(4)
+where O(k) = I2N−D ⊗o(k) with the nontrivial local oper-
+ator o(k) ∈ C2D×2D, D describes the locality, and ρ(i,k) =
+TrD(U(θ)σ(x(i,k))U(θ)†) corresponds to the state before
+measurements, named as feature state. An intuition of
+the unified QC is shown in Fig. 1(b).
+We are now ready to exploit the unified framework
+to analyze the expected risk of QCs. Let ρ = {ρ(i,k)}
+and o = {o(k)} be two sets collecting all feature states
+and measure operators. The following theorem exhibits
+conditions in which QCs allow a low expected risk, where
+the formal statement and the proof are deferred to SM A.
+Theorem 1 (informal). Following notations in Eqs. (1)-
+(4), when the train data size is nO(KNge log KNg
+ϵδ ) with ϵ
+being the tolerable error, and the optimal sets of ρ∗ and
+o∗ satisfy three conditions: (i) the feature states have
+the vanished variability in the same class; (ii) all feature
+states are equal length and are orthogonal in the varied
+classes; (iii) any feature state is alignment with the mea-
+sure operator in the same class, with probability 1−δ, the
+expected risk of QC tends to be zero, i.e., R(ˆhQ) → 0.
+Conditions (i)-(iii) visualized in Fig. 1(c) sculpt the ge-
+ometric interpretations of ρ∗ and o∗. These properties
+come across the design philosophy of CCs, e.g., linear dis-
+criminant analysis and neural collapse phenomenon ap-
+peared in most deep neural classifiers [71–73]. Moreover,
+these conditions unveil the intrinsic connection between
+optimal QCs and the quantum state discrimination [74],
+since ρ∗ and o∗ should maximize the Helstrom bound
+[75], which explains the ultimate limit of QCs observed
+in [76]. However, as will be explained later (see Corol-
+lary 1 and Lemma 1), under certain scenarios, it is hard
+for QCs to meet these conditions. A typical instance is
+applying QC to learn the image dataset, where the dif-
+ficulty stems from the limited nonlinearity of QC to fit
+the train set, thereby inducing a large empirical risk.
+Conditions (i)-(iii) also imply how the quantum mea-
+surement theory can be used to guide the design of QCs.
+Namely, the mean feature states of each class {¯ρ∗(k)}
+compose the equiangular tight frame (ETF) and Con-
+dition (iii) suggests that the optimal measure operators
+{o∗} also satisfy this ETF [77]. Due to the relation be-
+tween symmetric informationally complete (SIC) mea-
+surements and ETF [78, 79], the optimal measure op-
+erators could be estimated by various SIC construction
+strategies [80].
+Besides, the locality D of {o∗} should
+be carefully selected in QCs in which a small D pre-
+cludes the acquisition of the optimal QCs (i.e., the com-
+plex ETF does not exist when 2D = K [81, 82]), while an
+extremely large D may incur the barren plateaus [83, 84].
+Furthermore, when K is large, Pauli-based measurements
+are preferable than computational basis measurements in
+QCs, since the former allows classical shadow techniques
+to accelerate the training of QCs [85, 86].
+The scaling behavior of n indicates that it is data-
+efficient for QCs to attain a low generalization error,
+where the size of train set only linearly depends on the
+class number K and the number of encoding gates Nge
+(see Lemma 3 for the technical elaboration).
+In other
+words, the generalization error of QCs can be well con-
+trolled by the modest-size train data.
+According to Theorem 1, the challenges in satisfying
+Conditions (i)-(iii) and the well controlled generalization
+error pinpoint that the risk of a QC is mostly dominated
+by its empirical loss rather than its generalization error.
+In this view, the core in devising advanced QCs is tailor-
+ing HQ in Eq. (3) so that ˆhQ has a (near) zero empirical
+risk on D, or equivalently examining whether the em-
+ployed QCs can fulfill Conditions (i)-(iii).
+U-shape risk curve.—The risk curve concerns how
+the expected risk of a classifier behaves with the varied
+hypothesis space. It is desired that as with deep neu-
+ral classifiers, QCs undergo a double-descent risk curve
+in the sense that the over-parameterized QCs consent
+a low expected risk when the trainable parameters Nt
+is much greater than the train data n.
+If so, ‘over-
+parameterization’ could serve as a golden law in designing
+QCs. However, the following corollary refutes the exis-
+tence of the double-descent risk curve for QCs.
+Corollary
+1.
+Following
+notations
+in
+Theorem
+1,
+when {UE(x)|x
+∈
+X} follows the Haar distribu-
+tion, with probability 1 − δ, the empirical QC follows
+| Tr
+�
+σ(x(i,k))σ(x)
+�
+− 1
+2N | ≤
+�
+3
+22Nδ. When {U(θ)|θ ∈ Θ}
+follows the Haar distribution, with probability 1 − δ,
+the empirical QC follows | Tr(ρ(i,k)o(k′)) − Tr(o(k′))
+2D
+| <
+
+4
+�
+Tr(o(k′))2+2 Tr((o(k′))2)
+22Dδ
+.
+The proof is deferred to SM B. The achieved results re-
+veal the caveat of deep QCs. Specifically, when UE(x)
+is deep, two encoded states σ(x(i,k)) and σ(x(i′,k)) from
+the same class tend to be orthogonal, which contradicts
+with Conditions (i) in Theorem 1. Besides, when U(θ)
+is deep, the output of QC concentrates to zero, regard-
+less how o(k′) and ρ(i,k) are selected. This violates Con-
+dition (iii) in Theorem 1.
+Overall, over-parameterized
+QCs encounter the high empirical risk and thus the high
+expected risk, which suggests that QCs experience a U-
+shape risk curve. This observation differs the dynamics
+of QCs from variational quantum Eigensolvers, since the
+latter can benefit from over-parameterization, e.g., better
+trainability and convergence rate [87–90]. Moreover, the
+rule of thumb in QCs’ construction is slimming HQ to
+find the valley region. Intriguingly, tailoring the feature
+states echoes with quantum metric learning and quantum
+self-supervised learning [91–95].
+Probe power of QCs via loss dynamics.—The dis-
+tinct tendency of the risk curves between QCs and CCs
+provides a succinct way to recognize the potential quan-
+tum advantages. As shown in Fig. 1(a), given a specific
+data set, the U-shape risk curve of QCs indicates that its
+advantages mostly appear in the valley region. Precisely,
+if the risk values of QC around the basin are lower than
+those of CC, potential merits may exist; otherwise, QC
+is inferior to CC. The proved learning behavior of QCs,
+accompanied with the tight generalization bound, allows
+us to effectively fit its risk curve according to their loss
+dynamics. Specifically, our method contains three steps.
+First, W tuples of {n, Nt, T} are initialized based on The-
+orem 1 so that the collected risk points of QC span the
+basin area with low generalization error. Second, we ex-
+ecute QC and CC under these W hyper-parameter set-
+tings and fit their loss dynamics to attain the risk curve.
+Last, we compare two risk curves and probe potential
+advantages. See SM F for the implementation details.
+Technical analysis.—Theorem 1 is achieved by ana-
+lyzing when RERM(ˆhQ) and RGene(ˆhQ) are (near) zero.
+In the analysis of RERM(ˆhQ), we first consider the most
+general case in which both ρ and o are tunable, where
+ˆhQ ≡ hQ(ρ∗, o∗) with (ρ∗, o∗) = minρ,o L(ρ, o).
+Lemma 1 (Informal). When the regularizer E is consid-
+ered and (ρ∗, o∗) meets the three conditions in Theorem
+1, the global minimizer leads to RERM(ˆhQ) = C2
+1/2 with
+C1 depending on the hyper-parameters in E.
+The achieved properties of o∗ can be used as a priori to
+simplify QCs. To this end, the following lemma quantifies
+RERM(ˆhQ) when o is predefined and E = 0, where ˆhQ ≡
+hQ(ρ∗, o) with ρ∗ = minρ L(ρ, o).
+Lemma 2 (Informal). When the predefined {o(k)} are
+mutually orthogonal with each other and the conditions
+in Theorem 1 are satisfied, we have RERM(ˆhQ) = 0.
+The proofs of Lemmas 1 and 2 are given in SM C&D.
+We next analyze RGene(ˆhQ). Prior results cannot be
+used to prove Theorem 1, since such bounds polynomially
+scale with the trainable parameters and become vacuous
+in the over-parameterized regime. To remedy this issue,
+we utilize the concept of algorithmic robustness [70].
+Definition 1 (Robustness). A learning algorithm A is
+(R, ν(·))-robust with R ∈ N and ν(·) : Zn → R, if Z can
+be partitioned into R disjoint sets, denoted by {Cr}R
+r=1,
+such that the following holds for all D ⊂ Zn : ∀s =
+(x(i), y(i)) ∈ D, ∀z = (x, y) ∈ Z, ∀r ∈ [R],
+s, z ∈ Cr ⇒ |l(hAD(x(i)), y(i)) − l(hAD(x), y)| ≤ ν(D).
+Concisely, robustness measures how much the loss value
+can be varied with respect to the input space Z. A higher
+robustness of a classifier admits lower R, ν(·), and RGene
+[70]. The following lemma quantifies the upper bound of
+RGene(ˆhQ) whose proof is given in SM E.
+Lemma 3. Suppose the measure operator is bounded by
+C2 with maxk∈[K] ∥o(k)∥ ≤ C2. Define ϵ as the tolerable
+error. Following notations in Definition 1, the empiri-
+cal QC is (K(28Nge/ϵ)4mNge, 4L1KC2ϵ)-robust, and with
+probability 1 − δ we have
+RGene(ˆhQ) ≤ 4L1KC2ϵ + 5ξ(ˆhQ)
+�
+|TD|4mNge ln 56KNge
+ϵδ
+n
+,
+where L1 is the Lipschitz constant of ℓ with respect to
+hQ, ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)),
+and TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+The achieved results convey threefold insights. First, our
+bound does not explicitly depend on the number of train-
+able parameters [96]. This unlocks a new way to under-
+stand the generalization ability of QCs, especially for the
+over-parameterized ones. Next, our bound hints that a
+carefully designed UE can enhance performance of QCs
+[53, 97]. Last, RGene(ˆhQ) → 0 requires n ≫ |TD|4mNge.
+Fortunately, a reasonable value of n is sufficient to war-
+rant this condition, because in general m ≤ 2, Nge ∝ |x|,
+and |TD| is continuously decreased from n to K with re-
+spect to the reduced empirical loss.
+III.
+NUMERICAL SIMULATIONS
+We conduct numerical simulations to exhibit that the
+advantages and limitations of QCs on different classifica-
+tion tasks can be interpreted by the derived risk curve
+and feature states. The omitted construction details and
+results are deferred to SM G.
+We first apply QC to accomplish the binary classifica-
+tion on the parity dataset [98–100]. The number of qubits
+is N = 6 and the hardware-efficient Ansatz is adopted
+to realize U(θ).
+The gradient descent method is used
+as the classical optimizer. Two measure operators are
+
+5
+(a)
+(b)
+FIG. 2. Binary classification on the parity dataset. (a)
+The learning performance of QC when the layer number is
+3.
+The x-axis denotes the epoch numbers.
+Shaded region
+represents variance. The Bloch spheres display the quantum
+feature states at different epochs. (b) The fitted risk curve
+of QC and MLP. The x-axis denotes the number of trainable
+parameters. The label ‘QC-risk’ (‘MLP-risk’) refers to the
+fitted risk curve of QC and MLP. The label ‘QC-res’ (‘MLP-
+res’) refers to the collected results used for fitting the curves.
+o(1) = |0⟩ ⟨0| and o(2) = |1⟩ ⟨1|. The simulation results of
+QC with Nt = 54 are displayed in Fig. 2(a). Particularly,
+the averaged train (test) accuracy steadily grows from
+44.1% to 100% within 22 epochs, and the corresponding
+loss decreases from 0.26 to 4 × 10−5. The dynamics of
+the feature states {ρ(i,t)} with t ∈ {0, 10, 20, 30, 40} vi-
+sualized by Bloch spheres echo with Lemma 2. Besides,
+QC becomes more robust when we continue the training.
+Although the train (test) accuracy reaches the optimum,
+the loss can be further reduced and suggests a lower risk
+warranted by Theorem 1. We further compare the risk
+curve between QC and multilayer Perceptron (MLP) on
+this dataset. We fit their risk curves following the pro-
+posed method to probe potential quantum merits.
+As
+shown in Fig. 2(b), QC clearly outperforms MLP when
+the trainable parameters ranges from 20 to 140 and the
+valley of the risk curve is around Nt = 70 [101].
+We then apply QC to learn the Fashion-MNIST im-
+age dataset with K = 9 [102]. The employed number of
+qubits is N = 10 and the Pauli-based measure operators
+are employed.
+Convolutional neural networks (CNNs)
+are exploited as the reference. For all classifiers, the num-
+ber of epochs is fixed to be T = 50 and the number of
+trainable parameters Nt ranges from 60 to 9000. Each
+setting is repeated with 3 times.
+As shown in Fig. 3,
+when the layer number is 50 with Nt = 1500, both the
+train and test accuracies of QC are about 50%.
+This
+performance is inferior to CNN under the similar setting.
+To explore whether QC has the potential to outperform
+CNN on this dataset, we compare their risk curves. As
+shown in Fig. 3(b), unlike the parity dataset, QC is evi-
+dently inferior to CNN on Fashion-MNIST dataset.
+(a)
+(b)
+FIG. 3. Multi-class classification on the image dataset
+with K = 9. (a) The learning performance of QC when the
+layer number is 50. (b) The fitted risk curve of QC and CNN.
+All labels have the same meaning with those used in Fig. 2.
+IV.
+DISCUSSIONS AND OUTLOOK
+We understand the potential of diverse QCs in terms of
+the expected risk. Our theoretical findings demonstrate
+that the efficacy of QCs is dependent on the problem at
+hand, which explains the empirical evidence of their supe-
+riority on synthetic and quantum datasets, yet inferiority
+on realistic tasks. With the clear difference between the
+risk curve of QCs and deep neural classifiers, we present a
+concise technique to investigate potential quantum ben-
+efits by fitting their loss dynamics.
+Numerical results
+validate our theoretical results and the effectiveness of
+our method.
+There are several interesting future research directions.
+The U-shape curve of QCs poses two open questions.
+First, can contemporary QCs attain quantum benefits
+on certain classical data when only limited data and re-
+stricted computing resources are available?
+Secondly,
+is it necessary to redesign QCs such as nonlinear QCs
+[103, 104] that can also exhibit a double-descent risk
+curve? Besides, the unearthed connection between the
+conditions towards optimal empirical risk and quantum
+state discrimination opens a new research avenue that
+amplifies the potential of QCs on quantum data aided
+by quantum information theory.
+Finally, it is intrigu-
+ing to extend the developed non-vacuous generalization
+error bound of QCs to other scenarios, such as out-of-
+distribution data, in order to identify potential quantum
+advantages.
+ACKNOWLEDGMENTS
+The authors thank Xinbiao Wang for valuable input
+and inspiring discussions.
+
+0.3
+1.0
+Loss
+0.2
+Train
+-0.8
+Loss
+Test
+Acc
+0.1
+0.6
+0.0
+0.4
+0
+10
+20
+30
+4010)
+X[0]
+y
+X[0]
+X[0)
+y
+X[0)
+y
+X
+10.6
+0.4
+QC-risk
+K
+S
+QC-res
+R 
+0.2
+MLP-risk
+MLP-res
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+1601.00
+QC-risk
+0.75
+OC-res
+K
+CNN-risk
+0.50
+R
+CNN-res
+0.25
+0.00
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+90001.00
+0.9
+0.75
+Loss
+SS
+Train
+0.50 0
+9 0.6
+Test
+0.25
+0.3
+0.00
+0
+10
+20
+30
+40
+506
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+
+10
+The organization of the supplementary material (SM) is as follows. In SM A, we present the proof of Theorem 1.
+Then, we provide the proof of Corollary 1 in SM B. Subsequently, we demonstrate the proof of Lemma 1 and Lemma
+2 in SM C and SM D, respectively. Next, in SM E, we exhibit the proof of Lemma 3. In the end, we elucidate the
+details of numerical simulations in SM G.
+SM A: Proof of Theorem 1
+For convenience, let us first recall the settings and notations introduced in the main text. When QCs are applied to
+accomplish the multi-class classification task, the training dataset D contains n examples and the number of examples
+in each class is the same with n = ncK. Moreover, the per-sample loss is specified as the mean square error.
+We next introduce the formal description of Theorem 1. In particular, Theorem 1 is established on Lemma 2, where
+the regularization term is set as zero (i.e., E = 0) and the set of measure operator is predefined, i.e., o spans the
+space C2D×2D and satisfies Tr(o(k)o(k′)) = Bδk,k′ where B ≥ 1 is a constant. The requirements of o aims to preserve
+Condition (iii) in Lemma 1. Note that the focus on these specific settings adopted in Lemma 1 instead of the most
+general settings (i.e., o is tunable and E is nonzero) is motivated by Lemma 1, which promises a lower expected risk.
+Following the above elaboration, the loss function of QC to be minimized can be explicitly written as
+L(ρ) = 1
+2n
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ(i,k)o(k))]k=1:K − y(i,k)�2
+,
+(A1)
+where y(i,k) is the unit basis whose k-th entry is 1 for ∀i ∈ [nc], ∀k ∈ [K]. Denote ρ∗ = minρ L(ρ) and the empirical
+risk of QC as RERM(ˆhQ) with ˆhQ ≡ ˆhQ(ρ∗). The formal statement of Theorem 1 is as follows.
+Theorem (Formal statement of Theorem 1). Following notations in Lemmas 2 and 3, with probability 1 − δ, the
+expected risk of QC tends to be zero, i.e., RERM(ˆhQ) = 0, when the size of train dataset satisfies n ≫ O(KNge log KNg
+ϵδ )
+and the global minimizer ρ∗ in Eq. (A1) satisfies
+(i)¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k); (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = Bδk,k′; (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.
+(A2)
+Proof of Theorem 1. Following Eq. (2) and the results in Lemma 3, with probability 1 − δ, the expected risk of an
+optimal empirical QC is upper bounded by
+R(ˆhQ) ≤ RERM(ˆhQ) + 4L1KC2ϵ + 3ξ(ˆh)
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))
+n
+.
+(A3)
+Then, when ρ∗ satisfies Eq. (A2), Lemma 2 warrants RERM(ˆhQ) = 0, which gives
+R(ˆhQ) ≤ 4L1KC2ϵ + 3ξ(ˆh)
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))
+n
+.
+(A4)
+This bound can be further simplified when the training of QC is perfect. Note that Condition (i) implies |TD| = K,
+since all feature states from the same class collapse to the same point. Meanwhile, since ξ(ˆh) and C2 are bounded,
+and m and ϵ are small constant, we can conclude that when n ≫ O(KNge log(KNg/(ϵδ))), the expected risk can
+approach to zero.
+SM B: Proof of Corollary 1
+The proof leverages the following two lemmas related to the Haar measure and the unitary t-design.
+Lemma 4. Let {Wy}y∈Y ⊂ U(d) form a unitary t-design with t > 1, and let A, B : Hd → Hd be arbitrary linear
+operators. Then
+1
+|Y |
+�
+y∈Y
+Tr[WyAW †
+y B] =
+�
+Haar
+dµ(W) Tr[WyAW †
+y B] = Tr[A] Tr[B]
+d
+.
+(B1)
+
+11
+Lemma 5. Let {Wy}y∈Y ⊂ U(d) form a unitary t-design with t > 1, and let A, B, C, D : Hd → Hd be arbitrary
+linear operators. Then
+1
+|Y |
+�
+y∈Y
+Tr[WyAW †
+y B] Tr[WyCW †
+y D] =
+�
+Haar
+dµ(W) Tr[WyAW †
+y B] Tr[WyCW †
+y D]
+=
+1
+d2 − 1 (Tr[A] Tr[B] Tr[C] Tr[D] + Tr[AC] Tr[BD])
+−
+1
+d(d2 − 1) (Tr[AC] Tr[B] Tr[D] + Tr[A] Tr[C] Tr[BD]) .
+(B2)
+Corollary (Restatement of Corollary 1). Following notations in Lemmas 2 and 3, when the encoding unitary
+{UE(x)|x ∈ X} follows the Haar distribution, with probability 1 − δ, the empirical QC follows | Tr
+�
+σ(x(i,k))σ(x)
+�
+−
+1
+2N | ≤
+�
+3
+22Nδ. When the adopted Ansatz {U(θ)|θ ∈ Θ} follows the Haar distribution, with probability 1 − δ, the
+empirical QC follows | Tr(ρ(i,k)o(k′)) − Tr(o(k′))
+2D
+| <
+�
+Tr(o(k′))2+2 Tr((o(k′))2)
+22Dδ
+.
+Proof of Corollary 1. We complete the proof by separately analyzing the concentration behavior of the encoding
+unitary and the Ans¨atze.
+Concentration of the encoding unitary. Recall that Condition (iii) in Lemma 2 concerns the distance between two
+feature states ρ(i,k) and ρ(i′,k′) for ∀i, i ∈ [nc] and ∀k, k′ ∈ [K]. In this regard, we quantify the distance between the
+encoded state σ(x(i,k)) and σ(x) with x ∼ X when the deep encoding Ansatz UE is employed. In particular, we have
+Ex∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+��
+=Ex∼X
+�
+Tr
+�
+σ(x(i,k))UE(x)(|0⟩ ⟨0|)⊗NUE(x)†��
+=
+�
+Haar
+dµ(U) Tr
+�
+σ(x(i,k))U(|0⟩ ⟨0|)⊗NU
+�
+=Tr(σ(x(i,k))) Tr(|0⟩ ⟨0|)⊗N)
+2N
+= 1
+2N ,
+(B3)
+where the third equality uses Lemma 4. Moreover, the variance of the term Tr(σ(x(i,k))σ(x)) yields
+Varx∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+��
+=Ex∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+�2�
+− Ex∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+��2
+=
+�
+Haar
+dµ(U) Tr
+�
+σ(x(i,k))U(|0⟩ ⟨0|)⊗NU
+�
+Tr
+�
+σ(x(i,k))U(|0⟩ ⟨0|)⊗NU
+�
+−
+1
+22N
+=
+1
+22N − 1
+�
+1 + Tr(σ(x(i,k))2)
+�
+−
+1
+22N(22N − 1)
+�
+Tr(σ(x(i,k))2) + 1
+�
+−
+1
+22N
+≤
+1
+22N−2 −
+1
+22N
+= 3
+22N ,
+(B4)
+where the second equality uses the property that the deep encoding unitary follows the Haar distribution and the
+result in Eq. (B3), the third equality comes from Lemma 4, the inequality adopts Tr(σ2) ≤ 1 and 22N − 1 > 22N−1,
+and the last equality is obtained via simplification.
+Supported by the Chebyshev’s inequality Pr(|X − E[X]| ≥ a) ≤ Var[X]/a2, Eqs. (B3) and (B4) indicate
+Pr
+���� Tr
+�
+σ(x(i,k))σ(x)
+�
+− 1
+2N
+��� ≥ τ
+�
+≤
+3
+22Nτ 2 .
+Equivalently, with probability 1 − δ, we have
+��� Tr
+�
+σ(x(i,k))σ(x)
+�
+− 1
+2N
+��� ≤
+�
+3
+22Nδ .
+(B5)
+
+12
+Concentration of the deep Ansatze. Recall Condition (ii) in Lemma 2. Given a feature state ρ(i,k), for ∀i ∈ [nc] and
+∀k ∈ [K] and a measure operator o(k), the optimal feature state should satisfy
+Tr(ρ∗(i,k)o(k′)) = δk,k′.
+In other words, we should examine the value of Tr(ρ(i,k)o(k′)) when ρ(i,k) is prepared by a deep Ansatze U(θ).
+Specifically, we have
+Eθ∼Θ
+�
+Tr(ρ(i,k)o(k′))
+�
+=Eθ∼Θ
+�
+Tr(U(θ)σ(x(i,k))U(θ)†(o(k′) ⊗ I2N−D)
+�
+=
+�
+Haar
+dµ(U) Tr
+�
+Uσ(x(i,k))U †(o(k′) ⊗ I2N−D)
+�
+=Tr(o(k′))(2N−D)
+2N
+=Tr(o(k′))
+2D
+,
+(B6)
+where the first equality comes from the explicit form of QC in Eq. (4), the second equality uses the fact that U follows
+the Haar distribution, and the last second equality comes from Lemma 4.
+We then quantify the variance of Tr(ρ(i,k)o(k′)), i.e.,
+Varθ∼Θ
+�
+Tr(ρ(i,k)o(k′))
+�
+=Eθ∼Θ
+�
+Tr(ρ(i,k)o(k′))2�
+−
+�
+Eθ∼Θ
+�
+Tr(ρ(i,k)o(k′))
+��2
+=
+�
+Haar
+dµ(U) Tr
+�
+Uσ(x(i,k))U †(o(k′) ⊗ I2N−D)
+�2
+− Tr(o(k′))2
+22D
+=
+1
+22N − 1
+�
+Tr(σ(x(i,k))) Tr(o(k′) ⊗ I2N−D) Tr(σ(x(i,k))) Tr(o(k′) ⊗ I2N−D) + Tr(σ(x(i,k))2) Tr((o(k′) ⊗ I2N−D)2)
+�
+−
+1
+2N(22N − 1)
+�
+Tr(σ(x(i,k))2) Tr(o(k′) ⊗ I2N−D)2 + Tr(σ(x(i,k)))2 Tr((o(k′) ⊗ I2N−D)2)
+�
+− Tr(o(k′))2
+22D
+≤
+1
+22N − 1
+�
+Tr(o(k′) ⊗ I2N−D)2 + Tr((o(k′) ⊗ I2N−D)2)
+�
+− Tr(o(k′))2
+22D
+=
+1
+22N − 1
+�
+Tr(o(k′))222N���2D + Tr((o(k′))2)22N−2D�
+− Tr(o(k′))2
+22D
+≤Tr(o(k′))2 + Tr((o(k′))2)
+22D−1
+− Tr(o(k′))2
+22D
+=Tr(o(k′))2 + 2 Tr((o(k′))2)
+22D
+.
+(B7)
+where the second equality uses the fact that U follows the Haar distribution and Eq. (B6), the the third equality
+comes from Lemma 5, the first inequality arises from dropping some positive terms, the last second equality employs
+Tr(A⊗B) = Tr(A) Tr(B) and (A⊗B)(C⊗D) = (AC)⊗(BD), and the last inequality exploits (22N −1)−1 > (2N−1)−1,
+and the last equalities is obtained via simplification.
+Supported by the Chebyshev’s inequality Pr(|X − E[X]| ≥ a) ≤ Var[X]/a2, Eqs. (B6) and (B7) indicate
+Pr
+���� Tr(ρ(i,k)o(k′)) − E
+�
+Tr(ρ(i,k)o(k′))
+� ��� ≥ τ
+�
+≤ Tr(o(k′))2 + 2 Tr((o(k′))2)
+22Dτ 2
+.
+Equivalently, with probability 1 − δ, we have
+��� Tr(ρ(i,k)o(k′)) − Tr(o(k′))
+2D
+��� <
+�
+Tr(o(k′))2 + 2 Tr((o(k′))2)
+22Dδ
+.
+(B8)
+
+13
+SM C: Proof of Lemma 1
+In this section, we derive the geometric properties of the global optimizer under the unconstraint loss function
+L(ρ, o) in which both ρ and o are tunable and the regularization term is considered. Mathematically, the regularizer
+in Eq. (1) is defined as E = λρ
+2
+�nc
+i=1
+�K
+k=1 ∥ρ(i,k)∥2
+F + λo
+2
+�K
+k=1 ∥o(k)∥2
+F with λρ and λo being hyper-parameters. The
+explicit form of the loss function is
+L(ρ, o) = 1
+2n
+nc
+�
+i=1
+K
+�
+k=1
+��
+Tr(ρ(i,k)o(k))
+�
+k=1:K − y(i,k)�2
++ λρ
+2
+nc
+�
+i=1
+K
+�
+k=1
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F .
+(C1)
+Denote the global optima as (ρ∗, o∗) = minρ,o L(ρ, o) and the empirical QC as ˆhQ ≡ hQ(ρ∗, o∗). The restatement of
+Lemma 1 is as follows.
+Lemma (Formal statement of Lemma 1). Define C1 := K
+�
+ncλoλρ. If 2D ≥ K, C1 ≤ 1, and λo ≤ ncλρ, the global
+minimizer (ρ∗, o∗) of L(ρ, o) in Eq. (C1) satisfies for ∀k, k′ ∈ [K]:
+(i)¯ρ∗(k) := ρ∗(1,k) = · · · = ρ∗(nc,k);
+(ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = (1 − C1)
+�
+λo
+nλρ
+δk,k′;
+(iii)o∗(k) =
+�
+nλρ
+λo
+¯ρ∗(k).
+(C2)
+The corresponding empirical risk is RERM(ˆhQ) = C1.
+Proof of Lemma 1. Conceptually, the global optimizer can be identified by lower bounding L(ρ, o), where the equality
+conditions of ρ amount to the properties of global minimizer. In particular, the lower bound of L(ρ, o) yields
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ(i,k)o(j))]j=1:K − y(i,k)�2
++ λρ
+2
+nc
+�
+i=1
+K
+�
+k=1
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+≥
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
++ λρ
+2
+nc
+�
+i=1
+K
+�
+k=1
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+=
+1
+2Knc
+K
+�
+k=1
+nc
+�
+i=1
+nc
+1
+nc
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
++ λρ
+2
+K
+�
+k=1
+nc
+�
+i=1
+nc
+1
+nc
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+≥ 1
+2K
+K
+�
+k=1
+�
+Tr
+� nc
+�
+i=1
+1
+nc
+ρ(i,k)o(k)
+�
+− 1
+�2
++ λρ
+2
+K
+�
+k=1
+nc
+�����
+nc
+�
+i=1
+1
+nc
+ρ(i,k)
+�����
+2
+F
++ λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F ,
+(C3)
+where the first inequality uses the fact ∥a − b∥2 = �
+i(a(i) − b(i))2 ≥ (a(k) − b(k))2 and the k-th entry of y(i,k) equals
+to 1, and the second inequality comes from the Jensen’s inequality f(E(x)) ≤ E(f(x)). The equality condition of the
+first inequality holds if and only if
+Tr
+�
+ρ(i,k)o(j)�
+= 0, (∀j ∈ [K] \ {k}) ∧ (∀i ∈ [nc]) ;
+and the equality condition of the second inequality holds if and only if
+ρ(1,k) = · · · = ρ(i,k) = · · · = ρ(nc,k), ∀k ∈ [K].
+Denote the mean of the feature state for the k-th class as ¯ρ(k) = �nc
+i=1
+1
+nc ρ(i,k) for ∀k ∈ [K]. The above two equality
+conditions suggest that the global minimizer (ρ∗, o∗) satisfies
+¯ρ∗(k) ≡ ρ∗(1,k) = · · · = ρ∗(nc,k), ∀k ∈ [K]
+Tr(¯ρ∗(k)o∗(j)) = 0, ∀j ∈ [K] \ {k}.
+(C4)
+To thins end, we obtain Conditions (i) in Lemma 1, which describe the geometric properties of ρ∗, i.e.,
+(i)¯ρ∗(k) := ρ∗(1,k) = · · · = ρ∗(nc,k).
+(C5)
+
+14
+The next part of the proof is showing that the global minimizer satisfies Condition (iii). Combining Eqs. (C3) and
+(C4), the lower bound of the loss function in Eq. (C3) follows
+L(ρ, o)
+≥ 1
+2K
+K
+�
+k=1
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2
+K
+�
+k=1
+nc
+���¯ρ(k)���
+2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+= 1
+2K
+K
+�
+k=1
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2 K
+K
+�
+k=1
+1
+K nc
+���¯ρ(k)���
+2
+F + λo
+2 K
+K
+�
+j=1
+1
+K ∥o(j)∥2
+F
+≥1
+2
+� K
+�
+k=1
+1
+K Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2 Knc
+�����
+K
+�
+k=1
+1
+K ¯ρ(k)
+�����
+2
+F
++ λo
+2 K∥
+K
+�
+j=1
+1
+K o(j)∥2
+F ,
+(C6)
+where the second inequality comes from the Jensen’s inequality and the equality condition holds if and only if for
+∀k, k′ ∈ [K],
+Tr
+�
+¯ρ(k)o(k)�
+= Tr
+�
+¯ρ(k′)o(k′)�
+, ∥¯ρ(k)∥F = ∥¯ρ(k′)∥F , ∥o(k)∥F = ∥o(k′)∥F .
+(C7)
+Then, supported by the inequality a + b ≥ 2
+√
+ab, the loss L(ρ, o) can be further lower bounded by
+1
+2
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2 Knc
+���¯ρ(k)���
+2
+F + λo
+2 K∥o(j)∥2
+F
+≥1
+2
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ K
+�
+ncλoλρ
+���¯ρ(k)���
+F ∥o(j)∥F ,
+(C8)
+where the equality condition holds if and only if
+λo∥o(j)∥2
+F = ncλρ
+���¯ρ(k)���
+2
+F , ∀k ∈ [K].
+(C9)
+Note that the requirements C1 ≤ 1 and λo ≤ ncλρ in Lemma 1 imply ∥¯ρ∗(k)∥ ≤ 1 and hence ensure that ¯ρ∗(k) is a
+meaningful quantum state for ∀k ∈ [K].
+Since Tr
+�
+¯ρ(k)o(k)�
+= ∥¯ρ(k)∥∥o(k)∥ cos(∠(ρ(k), o(k))), the lower bound of L(ρ, o) in Eq. (C8) is equivalent to
+1
+2
+�
+∥¯ρ(k)∥∥o(k)∥ cos(∠(ρ(k), o(k))) − 1
+�2
++ C1
+���¯ρ(k)���
+F ∥o(j)∥F .
+Define ∥¯ρ(k)∥∥o(k)∥ = a and ∠(ρ(k), o(k)) = α. The above equation is described by the function f(a, α) = (a cos α −
+1)2/2+C1a and its minimum is C1 −C2
+1/2 when α∗ = 0 and a∗ = 1−C1. The derivation is as follows. Since a > 0 and
+its maxima is unbounded, we first consider the case 0 < a < 1. In this case, the minimum of f(a, α) is C1 −C2
+1/2 with
+α∗ = 0 and a∗ = 1 − C1. Otherwise, when a ≥ 1, the minimum of f(a, α) is C1 with α∗ = arccos(1/a) and a∗ = 1.
+Note that the minimum value of f(a, α) in the second case is always larger than that of the first case. Therefore, the
+minimum of f(a, α) is C1 − C2
+1/2 with α∗ = 0 and a∗ = 1 − C1. Combining the observation that ¯ρ∗(k) and o(k) are in
+the same direction with Eq. (C9), we achieve Condition (iii), i.e.,
+o∗(k) =
+�
+ncλρ
+λo
+¯ρ∗(k).
+The last part is proving Condition (ii). Combining the result ∥¯ρ∗(k)∥∥o∗(k)∥ = 1 − C1 for ∀k ∈ [K] with Eq. (C4)
+and Condition (iii), we immediately obtain condition (ii), i.e.,
+(ii)
+�
+ncλρ
+λo
+∥ρ∗(k)∥∥ρ∗(k′)∥ = (1 − C1)δk,k′ ⇒ Tr(¯ρ∗(k)¯ρ∗(k′)) = (1 − C1)
+�
+λo
+ncλρ
+δk,k′.
+(C10)
+To summarize, given the global optima satisfying the above three conditions, the corresponding empirical risk is
+RERM(ˆhQ) = 1
+2n
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ∗(i,k)o∗(k))]k=1:K − y(i,k)�2
+= C2
+1
+2
+(C11)
+
+15
+SM D: Results related to Lemma 2
+This section is composed of two parts. In SM D 1, we present the proof of Lemma 2. In SM D 2, we explain that
+the requirements in Lemma 2 are mild.
+1.
+Proof of Lemma 2
+Different from Lemma 1, here we focus the setting such that the regularization term is set as E = 0 and the operator
+o is predefined. The explicit form of the loss function L is defined in Eq. (A1). Denote the optimal feature states
+ρ∗ = minρ L(ρ), we quantify the value of RERM(ˆhQ) with ˆhQ ≡ hQ(ρ∗).
+We emphasize that the modifications of E and o allow a lower optimal empirical risk. Recall the results of Lemma
+1. In the most general case, the optimal empirical risk depends on the regularization term, i.e., RERM(ˆhQ) → C2
+1/2.
+The dependance on C1 motivates us to explore the empirical risk of QC when E = 0. Furthermore, Condition (iii)
+in Lemma 1 delivers the crucial properties of the optimal measure operator, i.e., the optimal measure operators are
+orthogonal with each other. Such properties contribute to construct a more effective QCs. Instead of optimizing,
+the measure operator o can be predefined by inheriting the properties proved in Lemma 1, that is, o are required to
+span the space C2D×2D and satisfy Tr(o(k)o(k′)) = Bδk,k′ with B ≥ 1 being a constant. Notably, these requirement
+are mild, covering frequently used measures such as computational basis and Pauli-based measures, as explained in
+SM D 2.
+Lemma (Formal statement of Lemma 2). Suppose that the adopted measure operator o spans the space C2D×2D and
+satisfies Tr(o(k)o(k′)) = Bδk,k′ where B ≥ 1 is a constant. The empirical risk of ˆhQ is RERM(ˆhQ) = 0 when the global
+minimizer ρ∗ satisfies
+(i)¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k); (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = Bδk,k′; (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.
+(D1)
+Proof of Lemma 2. The concept of the proof is analogous to Lemma 1, i.e., the global optimizer is identified by lower
+bounding the loss L(ρ). To this end, the lower bound of L(ρ) yields
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ(i,k)o(j))]j=1:K − y(i,k)�2
+≥
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
+=
+1
+2Knc
+K
+�
+k=1
+nc
+�
+i=1
+nc
+1
+nc
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
+≥ 1
+2K
+K
+�
+k=1
+�
+Tr
+� nc
+�
+i=1
+1
+nc
+ρ(i,k)o(k)
+�
+− 1
+�2
+,
+(D2)
+where the first inequality uses the facts n = Knc, ∥a − b∥2 = �
+i(a(i) − b(i))2 ≥ (a(k) − b(k))2, and only the k-th
+entry of y(i,k) equals to 1, and the second inequality comes from the Jensen’s inequality E(f(x)) ≥ f(E(x)) when f(·)
+is convex. Note that the equality condition of the first inequality holds if and only if
+Tr(ρ(i,k)o(j)) = 0, (∀j ∈ [K] \ {k}) ∧ (∀i ∈ [nc]) ;
+And the equality condition of the second inequality holds if and only if
+ρ(1,k) = · · · = ρ(i,k) = · · · = ρ(nc,k), ∀k ∈ [K].
+Denote the mean of the feature state for the k-th class as ¯ρ(k) = �nc
+i=1
+1
+nc ρ(i,k) for ∀k ∈ [K]. The above two equality
+conditions suggest that the global minimizer yields
+¯ρ∗(k) ≡ ρ∗(1,k) = · · · = ρ∗(nc,k), ∀k ∈ [K]
+(D3)
+Tr(¯ρ∗(k)o(j)) = 0, ∀j ∈ [K] \ {k}.
+(D4)
+
+16
+Combining Eqs. (D2)-(D4), the lower bound of the loss function L(ρ) satisfies
+1
+2K
+K
+�
+k=1
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
+≥ 1
+2
+� K
+�
+k=1
+1
+K Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
+,
+(D5)
+where the inequality comes from the Jensen’s inequality and the equality condition holds if and only if ∀k, k′ ∈ [K],
+Tr
+�
+¯ρ(k)o(k)�
+= Tr
+�
+¯ρ(k′)o(k′)�
+.
+(D6)
+Supported by Eq. (D6), we can further lower bound L(ρ) with
+1
+2
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
+≥ 0,
+(D7)
+where the equality condition is achieved when Tr(¯ρ(k)o(k)) = 1 for ∀k ∈ [K].
+Taken together, the global optimizer ρ∗ should satisfy Condition (i)&(iii) in Lemma 2, where
+(i)¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k);
+(iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.
+(D8)
+We last prove that Condition (iii) and the requirements of o lead to Condition (ii).
+In particular, denote the
+vectorization of ρ∗(k) and o(k) as |ρ∗(k)⟩⟩ and |o(k)⟩⟩, respectively. Condition (iii) can be rewritten as
+��
+¯ρ∗(k), o(k′)��
+= δk,k′.
+(D9)
+Moreover, since the set of measure operators {o(k)} is required to be complete in the space of C2D and Tr(o(k)o(k′)) =
+Bδk,k′ with B ≥ 1 for ∀k, k′ ∈ [K], we have
+�
+k
+���o(k)����
+o(k)��� = BI2D.
+Then, Condition (ii) can be derived as follows, i.e.,
+Tr(ρ∗(k)ρ∗(k′))
+=⟨⟨¯ρ∗(k)|I2D|ρ∗(k′)⟩⟩
+= 1
+B
+��
+¯ρ∗(k)���
+�
+k′′
+|o(k′′)⟩⟩⟨⟨o(k′′)|
+���ρ∗(k′)
+��
+= 1
+B
+��
+¯ρ∗(k)���|o(k)⟩⟩⟨⟨o(k)|
+���ρ∗(k′)��
++
+��
+¯ρ∗(k)���
+�
+k′′̸=k
+|o(k′′)⟩ ⟨o(k′′)|
+���ρ∗(k′)
+��
+= 1
+B δk,k′.
+(D10)
+2.
+Requirement of o used in Lemma 2
+Here we elucidate that the requirements adopted in Lemma 2, i.e., o spans the complex space 2D × 2D and satisfies
+Tr(o(k)o(k′)) = Bδk,k′ with B ≥ 1, are mild. Specifically, the employed measurements in most QNN-based classifiers
+satisfy these requirements, including the computational basis measurements and Pauli measurements.
+Computational basis measurements. In this setting, the local measurement o(k) is set as |k⟩ ⟨k| with |k⟩ being the
+k-th computational basis for ∀k ∈ [K].
+When 2D = K, {|k⟩} spans the whole space of C2D×2D and we have
+Tr(o(k)o(k′)) = (⟨k|k′⟩)2 = δk,k′ with B = 1. The assumptions are satisfied.
+Pauli measurements. Denote the Pauli operation applied to the i-th qubit as P (i)
+a
+with a ∈ {X, Y, Z, I} for ∀i ∈ [D].
+Then, there are in total 4D Pauli strings P = ⊗D
+i=1P (i)
+a
+that form a orthogonal basis for the space C2D×2D. With
+setting 2D = K, each o(k) corresponds to one Pauli string with Tr(o(k)o(k′)) = Kδk,k′ with B = K.
+
+17
+SM E: Proof of Lemma 3
+For elucidating, let us restate Lemma 3 below and introduce the proof sketch before moving on to present the proof
+details.
+Lemma (Formal statement of Lemma 3). Denote L1 as the Lipschitz constant of ℓ in Eq. (1) with respect to h. Given
+a QC defined in Eq. (3), let E be a quantum channel with
+hQ(x, U(θ), O(k)) ≡ Tr(o(k)E(σ(x))), ∀k ∈ [K].
+(E1)
+Suppose the measure operator follows maxk∈[K] ∥o(k)∥ ≤ C2. The explicit form of the encoding unitary follows UE(x) =
+�Ng
+g=1 ug(x) ∈ U(2N) with the g-th quantum gate ug(x) ∈ U(2m) operating with at most m qubits with m ≤ N and Ng
+gates consisting of Nge variational gates and Ng − Nge fixed gates,
+Following above notations and Definition 1, the empirical QC is (K( 28Nge
+ϵ
+)4mNge, 4L1KC2ϵ)-robust and with prob-
+ability 1 − δ, its generalization error yields
+RGene(ˆh) ≤ 4L1KC2ϵ + 3ξ(ˆh)
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))
+n
+,
+where L1 is the Lipschitz constant of ℓ with respect to h, ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z ℓ(ˆh, z), and
+TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+The proof of Lemma 3 is established on the following lemma, which leverages the algorithmic robustness to quantify
+the upper bound of the generalization error.
+Lemma 6 (Theorem 1, [105]). If the learning algorithm A is (R, ν(·))-robust with {Cr}R
+r=1, then for any δ > 0, with
+probability at least 1−δ over an i.i.d drawn of n samples D = {z(i)}n
+i=1 with z(i) = (x(i), y(i)), the returned hypothesis
+ˆh by A on D satisfies
+RGene(ˆh) ≤ ν(D) + ξ(ˆh)
+�
+(
+√
+2 + 1)
+�
+|TD| ln(2R/δ)
+n
++ 2|TD| ln(2R/δ)
+n
+�
+,
+(E2)
+where ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)), and TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+The above result hints that given a hypothesis ˆh, its generalization error is upper bounded by the disjoint sets {Cr}R
+r=1,
+where a lower cardinality R allows a lower generalization error. A natural approach to realize these disjoint partitions
+is covering number [70].
+Definition 2 (Covering number, [65]). Given a metric space (U, ∥ · ∥), the covering number N(U, ϵ, ∥ · ∥) denotes the
+least cardinality of any subset V ⊂ U that covers U at scale ϵ with a norm ∥ · ∥, i.e., supA∈U minB∈V ∥A − B∥ ≤ ϵ.
+In conjunction with Lemma 6 and Definition 2, the analysis of RGene(ˆh) of an N-qubit QC amounts to quantifying
+the covering number of the space of the input quantum states, i.e.,
+XQ =
+�
+UE(x)(|0⟩ ⟨0|)⊗NUE(x)†��x ∈ X
+�
+.
+(E3)
+The following lemma connects the robustness and covering number of XQ of QCs whose proof is provided in Sec. E 1.
+Lemma 7. Following the settings in Eqs. (E1)-(E3), the corresponding QC is (K( 28Nge
+ϵ
+)4mNge, 4L1KC2∥E∥⋄ϵ)-robust.
+We are now ready to prove Lemma 3.
+Proof of Lemma 3. The generalization error bound can be acquired by combining Lemmas 6 and 7, i.e.,
+RGene(ˆh) ≤4L1KC2∥E∥⋄ϵ + ξ(ˆh)
+�
+�(
+√
+2 + 1)
+�
+|TD| ln(2K( 28Nge
+ϵ
+)4mNge/δ)
+n
++ 2|TD| ln(2K( 28Nge
+ϵ
+)4mNge/δ)
+n
+�
+�
+≤4L1KC2∥E∥⋄ϵ + ξ(ˆh)
+�
+3
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ 2|TD|4mNge ln(56KNge/(ϵδ))
+n
+�
+≤4L1KC2ϵ + ξ(ˆh)
+�
+3
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ 2|TD|4mNge ln(56KNge/(ϵδ))
+n
+�
+,
+(E4)
+where ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)), and TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+
+18
+1.
+Proof of Lemma 7
+The proof uses the following lemma to quantify the covering number of XQ whose proof is given in SM E 2.
+Lemma 8. Following the settings in Eq. (E1), the covering number of XQ in Eq. (E3) is
+N(XQ, ϵ, ∥ · ∥F ) ≤
+�28Nge
+ϵ
+�4mNge
+.
+(E5)
+Proof of Lemma 7. When QC is applied to accomplish the K-class classification task, the sample space is Z = XQ ×Y
+with Y = {1, 2, ..., K}. Denote ˜
+XQ as the ϵ-cover set of XQ with the covering number N(XQ, ϵ, ∥ · ∥F ) in Definition 2.
+Supported by the ϵ-cover set ˜
+XQ, the space XQ × {i} can be divided into N(XQ, ϵ, ∥ · ∥F ) sets for ∀i ∈ [K]. In other
+words, we can divide Z into KN(XQ, ϵ, ∥ · ∥F ) sets denoted by {Zi}KN (XQ,ϵ,∥·∥F )
+i=1
+.
+We then utilize the divided sets of Z to connect the robustness with covering number according to Definition 1.
+Given a training example (x(i), y(i)) and a test example (x, y), suppose that the corresponding quantum examples
+(σ(x(i)), y(i)) and (σ(x), y) are in the same set of {Zi}KN (XQ,ϵ,∥·∥F )
+i=1
+. For convenience, we abbreviate σ(x(i)) and σ(x)
+as σ(i) and σ, respectively. Following the definition of covering number, we have
+y(i) = y and ∥σ(i) − σ∥F ≤ 2ϵ.
+(E6)
+Since the encoded state takes the form σ = UE(x)(|0⟩ ⟨0|)⊗NUE(x)†, we have
+rank(σ(i) − σ) ≤ 2.
+(E7)
+Then, in accordance with the definition of robustness, we bound the discrepancy of the loss values for σ(i) and σ, i.e.,
+���l(hQ(σ(i)), y(i)) − l(hQ(σ), y)
+���
+≤L1
+���[Tr(E(σ(i))o(k))]k=1:K − [Tr(E(σ))o(k))]k=1:K
+���
+2
+≤L1K max
+k∈K | Tr(E(σ(i)))o(k)) − Tr(E(σ)o(k))|
+≤L1K max
+k
+���o(k)���
+2 Tr(|E(σ(i) − σ)|)
+≤2L1KC2∥E∥⋄∥σ(i) − σ∥F
+≤4L1KC2∥E∥⋄ϵ,
+(E8)
+where the first inequality uses the Lipschitz property of the loss function with ℓ(a, b) − ℓ(c, d) ≤ L1∥a − c∥2 and the
+form of E in Lemma 7, the second inequality comes from the definition of l2 norm, the third inequality exploits von
+Neumann’s trace inequality | Tr(AB)| ≤ ∥A∥p∥B∥q with 1/p + 1/q = 1 and the linear property of CPTP map with
+E(ρ)−E(σ) = E(ρ−σ), the last second inequality employs maxk
+��o(k)��
+2 ≤ C2, the relation ∥E(ρ−σ)∥1 ≤ ∥E∥⋄∥ρ−σ∥1
+and ∥A∥1 ≤ rank(A)∥A∥F , and the last inequality adopts the result in Eq. (E6).
+The above result exhibits that the learned QC is (KN(XQ, ϵ, ∥ · ∥), 4L1KC2∥E∥⋄ϵ)-robust. In this regard, the proof
+can be completed when the upper bound of the covering number N(XQ, ϵ, ∥ · ∥F ) is known. Supported by Lemma 8,
+we obtain N(XQ, ϵ, ∥ · ∥F ) ≤ ( 28Nge
+ϵ
+)4mNge. Taken together, the learned QC is
+�
+K
+�28Nge
+ϵ
+�4mNge
+, 4L1KC2∥E∥⋄ϵ
+�
+− robust.
+2.
+Proof of Lemma 8
+The derivation of the covering number of XQ in Eq. (E3) uses the following lemma.
+Lemma 9 (Lemma 1, [106]). For 0 < ϵ < 1/10, the ϵ-covering number for the unitary group U(2m) with respect to
+the Frobenius-norm distance in Definition 2 obeys
+� 3
+4ϵ
+�4m
+≤ N(U(2m), ϵ, ∥ · ∥F ) ≤
+�7
+ϵ
+�4m
+.
+(E9)
+
+19
+Proof of of Lemma 8. Recall the input state space is XQ = {UE(x)(|0⟩ ⟨0|)⊗NUE(x)†|x ∈ X}, where the encoding
+unitary UE(x) = �Ng
+g=1 ug(x) ∈ U(2N) consists of Nge variational gates and Ng − Nge fixed gates. To quantify the
+covering number N(XQ, ϵ, ∥·∥F ), we define ˜S as the ϵ-covering set for the unitary group U(2m), ˜
+XQ as the ϵ′-covering
+set of XQ, and define a set
+˜UE :=
+�
+�
+�
+�
+i∈{Nge}
+ui(x)
+�
+j∈{Ng−Nge}
+uj(x)
+���ui(x) ∈ ˜S
+�
+�
+� ,
+(E10)
+where ui(θi) and uj specify to the variational and fixed quantum gates, respectively. Note that for any encoding circuit
+UE(x), we can always find a unitary UE,ϵ(x) ∈ ˜UE where each ug(x) is replaced by the nearest element in the covering
+set ˜S. To this end, following the definition of covering number, the discrepancy between UE(x)(|0⟩ ⟨0|)⊗NUE(x)† ∈ XQ
+and UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)† ∈ ˜
+XQ under the Frobenius norm satisfies
+��UE(x)(|0⟩ ⟨0|)⊗NUE(x)† − UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)†��
+F
+≤2
+��UE(x)(|0⟩ ⟨0|)⊗NUE(x)† − UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)†��
+≤2∥UE(x) − UEϵ(x)∥∥(|0⟩ ⟨0|)⊗N∥
+≤4Ngeϵ,
+(E11)
+where the first inequality uses ∥X∥F ≤ rank(X)∥X∥ and the relation in Eq. (E7), the second inequality comes from
+the Cauchy–Schwarz inequality, and the last inequality follows ∥UE(x) − UE,ϵ(x)∥ ≤ Ngeϵ and ∥(|0⟩ ⟨0|)⊗N∥ = 1. In
+other words, ϵ′ = 2Ngeϵ and ˜
+XQ is a (4Ngeϵ)-covering set for XQ. In conjunction with the observation that there are
+| ˜S|Nge combinations for the gates in ˜UE and the results in Lemma 9, we obtain the cardinality of the set ˜UE is upper
+bounded by | ˜UE| ≤
+� 7
+ϵ
+�4mNge. Accordingly, supported by Eq. (E11), the covering number of XQ satisfies
+N(XQ, 4Ngeϵ, ∥ · ∥F ) ≤
+�7
+ϵ
+�4mNge
+.
+(E12)
+After simplification, we have
+N(XQ, ϵ, ∥ · ∥F ) ≤
+�28Nge
+ϵ
+�4mNge
+.
+(E13)
+SM F: Implementation of the algorithm to probe potential advantages of QCs
+The expected risk is the most principal criteria to quantify the power of a classifier. As a result, to probe whether
+a QC holds potential advantages over a CC on a specific learning task, the simplest way is comparing their risk
+curves. Nevertheless, capturing these two risk curves are difficult, because of many flexible hyper-parameter settings
+to initiate a classifier.
+The developed theories in Theorem 1 and Lemmas 1-3 deliver concrete rules to set up these hyper-parameters and
+thus allow an efficient way to estimate these risk curves. In particular, the derived U-shape curve of QCs indicates
+that the minimum risk of QC locates at the modest size of the hypothesis space HQ. In other words, the number
+of trainable parameters NT should be lower than O(poly(N)), with N being the number of qubits in QC. Moreover,
+Lemma 3 hints that the generalization error of QC can be well suppressed by using the modest number of train
+examples. As such, if the available number of training examples in D is tremendous, we can distill a subset from D
+to better recognize quantum advantages.
+The Pseudo code of the proposed method is presented in Alg. 1. To make a fair comparison, the hyper-parameter
+settings applied to QC and CC, especially for those relating to the computational resources, are required to keep
+to be the same. Specifically, in each comparison, the employed loss function, the train examples n, the number of
+trainable parameters Nt, and the number of epochs T applied to QC and CC should be identical. Note that the
+learning rate, the adopted optimizer, and the batch size can be varied of different classifiers to better estimate the
+empirical hypothesis. To ensure that the collected results of QC span its basin of the risk curve, the employed W
+settings of Nt can be acquired by uniformly interpolating from O(1) to O(poly(N)). The iteration T should ensure
+the convergence of QC. Once the loss values of QC and CC under {n(w), N (w)
+t
+, T (w)}W
+w=1 are obtained, we can apply
+certain fitting algorithms to attain their risk curves.
+
+20
+Algorithm 1: Estimate risk curves of quantum and classical classifiers
+Data: The train dataset D, the test dataset DT est, QC hQ associated with the hypothesis space HQ, CC hC associated
+with the hypothesis space HQ, the loss function L(·, ·).
+Result: The estimated risk curves of QC and CC.
+Initialization: W tuples of hyper-parameter settings {n(w), N (w)
+t
+, T (w)}W
+w=1 with n being train examples, Nt being the
+number of trainable parameters, and T being the number of epochs;
+for w = 1, w ≤ W, w + + do
+Initialize train data as D(w) by distilling n(w) examples from D;
+# Collect loss dynamics of QC ;
+Minimize the loss function L(·, ·) via gradient descent methods to obtain the empirical quantum classifier ¯h(w)
+Q
+∈ HQ
+using D(w) within T (w) epochs and NT trainable parameters;
+Record the loss value L(¯h(w)
+Q , DT est) ;
+# Collect loss dynamics of CC ;
+Minimize the loss function L(·, ·) via gradient descent methods to obtain the empirical classical classifier ¯h(w)
+C
+∈ HC
+using D(w) within T (w) epochs and NT trainable parameters;
+Record the loss value L(¯h(w)
+C , DT est) ;
+end
+Fitting the loss dynamics of {L(¯h(w)
+Q , DT est)}W
+w=1 to obtain the estimated risk curve of QC ;
+Fitting the loss dynamics of {L(¯h(w)
+C , DT est)}W
+w=1 to obtain the estimated risk curve of CC.
+Ul(✓)
+RZ(✓(l,1,1))
+RY (✓(l,1,2))
+RZ(✓(l,1,3))
+RZ(✓(l,2,1))
+RY (✓(l,2,2))
+RZ(✓(l,2,3))
+RZ(✓(l,3,1))
+RY (✓(l,3,2))
+RZ(✓(l,3,3))
+RZ(✓(l,4,1))
+RY (✓(l,4,2))
+RZ(✓(l,4,3))
+×𝐿
+(a)
+(b)
+Class 1:
+Class 2:
+Class 3:
+FIG. G.4.
+Visualization of image dataset and hardware-efficient Ansatz.
+(a) Image instances sampled from the
+Fashion-MNIST dataset. (b) The circuit architecture of the employed Hardware-efficient Ansatz. The label ‘×L’ denotes the
+layer number, which means repeating the gates in the dashed box with L times.
+SM G: Numerical simulation details
+Dataset. The construction of the parity dataset mainly follows from Ref. [98]. Note that this task has also been
+broadly studied in the field of deep learning to show the limits of deep neural classifiers [107, 108]. The constructed
+dataset contains in total 64 examples. Each example corresponds to a bit-string with the length 6, i.e., x ∈ {0, 1}6.
+The label of x is assigned to be 1 if the number of ‘0’ in x is even; otherwise, the label is 0. We split it into train dataset
+and test dataset with the train-test-split ratio being 0.75. The number of train examples in each class is controlled to
+be the same. For each example, its feature dimension is 10. The image dataset is adapted from Ref. [102]. Specifically,
+the data from the first nine classes are preserved and the total number of examples is 180. The train-test-split ratio is
+set as 0.5 to construct the train and test dataset. Each example corresponds to an image with 28 × 28 pixels. In the
+preprocessing stage, we flatten all examples followed by padding and normalization. The processed example yields an
+10-qubit state with x ∈ R210 and ∥x∥2
+2 = 1. Some examples after preprocessing are illustrated in Fig. G.4(a).
+Construction of QCs. The quantum subroutine of QC consists of the encoding circuit UE and the Ansatz U(θ).
+For all learning tasks, the hardware-efficient Ansatz is employed whose mathematical expression is U(θ) = �L
+l Ul(θ).
+The layout of the hardware-efficient Ansatz follows the layer-wise structure and the gate arrangement at each layer
+is the same. For ∀l ∈ [L], Ul(θ) = �N
+i=1(RZ(θ(l,i,1)) RY(θ(l,i,2)) RZ(θ(l,i,3)))Uent with Uent being the entanglement
+layer formed by CNOT gates. Fig. G.4(b) depicts the adopted hardware-efficient Ansatz with L layers.
+The encoding methods for the parity dataset classification and the digit images classification are different. The
+former uses the basis encoding method. Specifically, for a classical example x ∈ Rd, the employed encoding unitary
+
+21
+(a)
+(b)
+FIG. G.5. Geometric properties of the quantum feature states on parity dataset. (a) The averaged performance of
+QC evaluated by M1 defined in Eq. (G1). The label ‘Init-C-k’ with k = 1, 2 refers that the value of M(k)
+1
+at the initialization.
+Similarly, the label ‘Final-C-k’ with k = 1, 2 refers that the value of M(k)
+1
+when the training of QC is completed. (b) The
+averaged performance of QC evaluated by M2 defined in Eq. (G2). The label ‘Init-C-1-2’ (‘Final-C-1-2’) refers that the value
+of M2 before and after training of QC. The label ‘L = a’ in the x-axis stands for that the layer number of hardware-efficient
+Ansatz is a.
+is UE(x) |0⟩⊗d = |x⟩, which maps x to a 2d dimensional quantum state UE(x) |0⟩⊗d. The latter uses the amplitude
+encoding method. Given a normalized image x ∈ R64 with ∥x∥2
+2 = 1, the corresponding unitary encodes it into a
+6-qubit state with UE(x) |0⟩⊗6 = �64
+j=1 xj |j⟩.
+The Pauli-based measure operators are used in learning Fashion-MNIST dataset. Since the preprocessed dataset
+contains 9 classes, there are in total 9 measure operators, i.e., o(1) = X⊗X⊗I⊗8, o(2) = X⊗Y ⊗I⊗8, o(3) = X⊗Z⊗I⊗8,
+o(4) = Y ⊗X ⊗I⊗8, o(5) = Y ⊗Y ⊗I⊗8, o(6) = Y ⊗Z ⊗I⊗8, o(7) = Z ⊗X ⊗I⊗8, o(8) = Z ⊗Y ⊗I⊗8, o(9) = Z ⊗Z ⊗I⊗8.
+Multilayer Perceptron.
+To better justify the capability and performance of QCs, we apply the multilayer
+perceptron (MLP) as the reference [109]. MLP is composed of an input layer, L hidden layers with L ≥ 1, and an
+output layer. The dimension of the input layer is equivalent to the feature dimension of the input. ReLU activations
+are added in the hidden layer to perform nonlinear transformation. In the output layer, the activation function,
+Softmax, is employed. The number of layers L depends on the assigned tuples {n, Nt, T}.
+Convolutional neural network. In the task of image classification, convolutional neural networks (CNNs) is
+employed as the reference [109]. The employed CNN is formed by two convolutional layers and one fully-connected
+layer. ReLU activations and the pooling operation are added in the hidden layer to perform nonlinear transformation.
+The number of channels for the first convolutional layer is fixed to be 8 and the corresponding kernel size is 9 × 9.
+The kernel size of the pooling operation applied to the two convolutional layers is 2×2. The kernel size for the second
+convolutional layer is fixed to be 5×5 but the number of output channels is varied depending on the settings in Alg. 1.
+For the sake of fair comparison, the number of output channels is set as 2, 6, 15, 30, 50, 75, where the corresponding
+number of parameters is 860, 1284, 2238, 3828, 5948, and 8598, respectively.
+Optimizer and other hyper-parameters. The adaptive gradient descent method, named AdaGrad optimizer
+[110], is used to optimize QCs and MLPs. Compared to the vanilla gradient descent method, AdaGrad permits better
+performance, since it adapts the learning rate for each feature depending on the estimated geometry of the problem.
+In the task of parity learning, the initial learning rate is set as η = 0.5 for QC and η = 0.01 for MLP, respectively.
+For both classifiers, the batch size is fixed to be 4. In the task of image classification, the initial learning rate is set
+as η = 0.05 for QC and η = 0.01 for CNN, respectively. The batch size for both classifiers is set as 1.
+Curve fitting method. To capture the risk curve, Alg. 1 requests a curve fitting method. For all experiments,
+we adopt the polynomial fitting to derive the risk curve by using the collected results. The least squares method in
+determining the best fitting functions.
+Source code.
+The source code used in numerical simulations will be available at Github repository https:
+//github.com/yuxuan-du/Problem-dependent-power-of-QNNs.
+
+0.8
+Init-C-1-2
+Fin-C-1-2
+0.6
+0.4
+0.2
+0.0
+L=1L=2L=3L=4L=5L=6L=74.8
+Init-C-1
+Init-C-2
+4.0
+Fin-C-1
+Fin-C-2
+3.2
+Distance
+2.4
+1.6
+0.8
+0.0
+L=1L=2L=3L=4L=5L=6L=722
+(a)
+(b)
+FIG. G.6. Train (test) accuracy versus epoch on parity dataset. (a) Train accuracy and test accuracy of QC with the
+varied layer number. The label ‘L = a’ refers that the layer number used in hardware-efficient Ansatz is a. The solid line and
+the dashed line separately correspond to the train and test accuracies of QC. (b) Train accuracy and test accuracy of MLP
+with the varied number of hidden neurons. The label ‘h = a’ refers that the number of neurons is a. The solid and dashed
+lines have the same meaning with those in QC.
+1.
+Simulation results of the binary classification for the parity dataset
+The feature states before and after training. We explore the geometric properties of feature states when the
+layer number of hardware-efficient Ansatz varies from L = 1 to L = 7. Other settings are identical to those introduced
+in the main text. Condition (i) in Lemma 2 is evaluated by the metric
+M(k)
+1
+=
+nc
+�
+i=1
+∥ρ(i,k) − ¯ρ(k)∥,
+(G1)
+where the number of train examples {ρ(i,k)}nc
+i=1 belonging to the k-th class is nc and ¯ρ(k) refers to their class-feature
+mean. Since parity learning is a binary classification task, Condition (ii) in Lemma 2 is evaluated by
+M2 = Tr(¯ρ(0)¯ρ(1)).
+(G2)
+The geometric properties of the feature states in the measure of M(k)
+1
+and M2 are visualized in Fig. G.5. The left
+panel shows that when L ∈ {2, 3, 4, 5}, both the value of M(1)
+1
+(highlighted by the green color) and M(2)
+1
+(highlighted
+by the pink color) decrease from ∼ 3.2 (epoch t = 0) to ∼ 0.5 (epoch t = 40). These results comply with Condition
+(i) in the sense that the feature states in the same class concentrates to the class-feature mean and leads to the low
+empirical risk. By contrast, when L is too small or too large, the value of M(1)
+1
+changes subtly before and after
+optimization, which is above 3.2. The large deviation of feature states incurs the degrade performance of QC. The
+right panel depicts that when L ∈ {2, 3, 4, 5}, the value of M(2)
+1
+decreases from 0.5 (epoch t = 0) to 0.05 (epoch
+t = 40). This reduction means that the class-feature means are maximally separated and thus ensure a good learning
+performance. On the contrary, when L ∈ {1, 6, 7}, the the value of M(2)
+1
+oscillates around 0.5, which implies that the
+class-feature means ¯ρ(1) and ¯ρ(2) are highly overlapped.
+The learning dynamics of QC and MLP. Fig. G.6 visualizes the learning dynamics of QC and MLP with
+respect to the varied trainable parameters. The left panel indicates that when the layer number is L = 2, 3, 4, both
+train and test accuracies of QC fast converge to 100% with 25 epochs. When L = 1, both train and test accuracies
+oscillate to 50%. When L = 7, the number of train data becomes insufficient and the overfitting phenomenon appears.
+These results accord with the U-shape risk curve of QCs. The right panel shows that when the number of hidden
+neurons ranges from h = 1 to h = 18, the test accuracy of MLP is no higher that 55%. These results reflect the
+incapability of MLP in learning parity dataset compared with QCs.
+2.
+Simulation results of multi-class classification for the Fashion-MNIST images dataset
+The feature states before and after training. Here we discuss the geometric properties of feature states when
+the layer number of hardware-efficient Ansatz varies from L = 2 to L = 150. The metrics M(k)
+1
+and M2 defined in
+
+1.00
+h=1
+h=2
+0.85-
+h=6
+h=10
+0.70
+h = 14
+Acc
+h= 18
+0.55
+0.40
+0
+10
+20
+30
+401.00
+0.85
+0.70
+0.55
+0.40
+0
+10
+20
+30
+4023
+(a)
+(b)
+FIG. G.7. Geometric properties of the quantum feature states on Fashion-MNIST dataset. (a) The averaged
+performance of QC evaluated by M1 defined in Eq. (G1). (b) The averaged performance of QC evaluated by M2 defined in
+Eq. (G2). All labels have the same meaning with those introduced in Fig. G.5.
+(a)
+(b)
+FIG. G.8. Train (test) accuracy versus epoch on Fashion-MNIST dataset. (a) Train accuracy and test accuracy of
+QC with the varied layer number. The labels have the same meaning with those presented in Fig. G.6. (b) Train accuracy
+and test accuracy of CNN with the varied number of trainable parameters. The label ‘h = a’ refers that the number of output
+channels at the second layer is a. The solid and dashed lines have the same meaning with those in QC.
+Eqs. (G1) and (G2) are employed. In the measure of M2, since the performance of QC for any two classes is similar,
+we only study the first two classes for ease of visualization.
+Fig. G.7 depicts the geometric properties of the feature states in the measure of M(k)
+1
+and M2. The left panel
+shows that for all settings with L ∈ {2, 5, 25, 50, 100, 150}, the value M(k)
+1
+at the initial step and the final step is very
+similar and M(k)
+1
+is larger than 0.2 for ∀k ∈ {1, 2, ..., 9}. These results indicate that QC cannot satisfy Condition
+(i) when learning Fashion-MNIST dataset, where the feature states from the same class cannot collapse to a unique
+point. Moreover, when we examine the performance of intra-class, the right panel implies that after training, the
+class-feature means of QC are still highly overlapping. The distance for all settings of L is above 0.3. The inability
+to achieve the optimal training loss shows the the limited power of QC on learning Fashion-MNIST dataset.
+The learning dynamics of QC and CNN. Fig. G.8 depicts the learning dynamics of QC and CNN with the
+varied number of trainable parameters. The left panel indicates that QC achieves the best performance when the layer
+number is L ∈ [25, 100], where the corresponding number of parameters ranges from 750 to 3000. In these settings,
+both train and test accuracies of QC are around 30% after 50 epochs. When L < 25 or L > 100, both train and test
+accuracies oscillate at 15%. These results accord with the U-shape risk curve of QCs. The right panel shows that
+the train and test accuracies of CNN are steadily growing with the increased number of channels. That is, when the
+number of channels at the second layer is not less than 6, both the train and test accuracies are higher than 60%.
+These results indicate that the employed QC does not have potential advantages in learning image dataset compared
+with CNN.
+
+Init-C-1-2
+Fin-C-1-2
+0.8
+Distance
+e
+0.6
+0.4
+0.2
+0.0
+L = 2
+L= 5
+5L=25L=50L=100L=150Init-C-1
+Fin-C-5
+Fin-C-1
+Init-C-6
+1.6
+Init-C-2
+Fin-C-6
+Fin-C-2
+Init-C-7
+Init-C-3
+Fin-C-7
+e 1.2
+Fin-C-3
+Init-C-8
+Distance
+Init-C-4
+Fin-C-8
+Fin-C-4
+Init-C-9
+0.8
+Init-C-5
+Fin-C-9
+0.4
+0.0
+L = 2
+L= 5
+L=25L=50L=100L=1501.00
+0.75
+h=2
+00.50
+h=6
+h= 15
+h = 30
+0.25
+h= 50
+h= 75
+0.00
+0
+10
+20
+30
+40
+500.45
+L=2
+L= 5
+L= 25
+0.30
+L = 50
+L= 100
+Acc
+_= 150
+0.15
+0.00
+0
+10
+20
+30
+40
+50

hdA0T4oBgHgl3EQfIP8h/content/tmp_files/2301.02071v1.pdf.txt

@@ -0,0 +1,1404 @@
+Published as a conference paper at EMNLP 2022
+Towards Table-to-Text Generation with Pretrained Language Model: A
+Table Structure Understanding and Text Deliberating Approach
+Miao Chen§ ∗, Xinjiang Lu¶ �, Tong Xu§, Yanyan Li ¶, Jingbo Zhou¶, Dejing Dou¶, Hui Xiong† ‡
+¶BIL, Baidu Research,
+§University of Science and Technology of China
+†Hong Kong University of Science and Technology (Guangzhou)
+‡Guangzhou HKUST Fok Ying Tung Research Institute
+[email protected], {luxinjiang,liyanyanliyanyan,zhoujingbo}@baidu.com,
+[email protected], [email protected], [email protected]
+Abstract
+Although remarkable progress on the neural
+table-to-text methods has been made, the gen-
+eralization issues hinder the applicability of
+these models due to the limited source tables.
+Large-scale pretrained language models sound
+like a promising solution to tackle such issues.
+However, how to effectively bridge the gap
+between the structured table and the text in-
+put by fully leveraging table information to
+fuel the pretrained model is still not well ex-
+plored. Besides, another challenge of integrat-
+ing the deliberation mechanism into the text-
+to-text pretrained model for solving the table-
+to-text task remains seldom studied. In this
+paper, to implement the table-to-text genera-
+tion with pretrained language model, we pro-
+pose a table structure understanding and text
+deliberating approach, namely TASD. To be
+specific, we devise a three-layered multi-head
+attention network to realize the table-structure-
+aware text generation model with the help of
+the pretrained language model. Furthermore,
+a multi-pass decoder framework is adopted to
+enhance the capability of polishing generated
+text for table descriptions. The empirical stud-
+ies, as well as human evaluation, on two public
+datasets, validate that our approach can gener-
+ate faithful and fluent descriptive texts for dif-
+ferent types of tables.
+1
+Introduction
+The task of learning to generate natural language
+descriptions from non-linguistic input, which is
+referred to as data-to-text, is important for many
+applications, such as weather forecast genera-
+tion (Mei et al., 2016), sports news writing (Wise-
+man et al., 2017), biography writing (Lebret et al.,
+2016), market comments writing (Murakami et al.,
+2017) and automatic question-answering (Li et al.,
+2021b). The input data can be in various forms
+∗ This work was done when the first author was an intern
+at Baidu Research under the supervision of the second author.
+for data-to-text though, here we focus on the text
+generation task that takes the table as input.
+Inspired by neural machine translation models,
+previous studies on table-to-text tasks mainly adopt
+traditional seq2seq methods to generate table de-
+scriptions (Lebret et al., 2016; Wiseman et al.,
+2017; Liu et al., 2018; Gong et al., 2019b; Wang
+et al., 2020; Li et al., 2021a). Despite generating
+text with high fluency, lacking numerous source
+tables leads to lower generalizability of the table-
+to-text model. Recent progress in the pretrained
+language model (Devlin et al., 2019; Radford et al.,
+2019) shows remarkable performance in solving
+natural language processing tasks. The model pre-
+trained on large-scale data possesses rich knowl-
+edge, which inspires us with the potential for solv-
+ing generalization issues of the text generation task.
+To exploit the expressive power of the pretrained
+model for the table-to-text task, it is necessary to
+serialize the input table effectively. Several works
+have put efforts to bridge this gap, such as serial-
+izing the table into a token sequence (Zhang et al.,
+2020; Suadaa et al., 2021; Xing and Wan, 2021),
+or introducing an extra task to control the table
+representation (Gong et al., 2020). However, none
+of these leveraged the table structure information
+effectively. Furthermore, the text-to-text pretrained
+model decodes and generates a sequence in a one-
+pass forward process, which means it cannot per-
+ceive the future words in advance on the target side.
+Recently, the deliberation mechanism (Niehues
+et al., 2016; Geng et al., 2018) implemented by
+the multi-pass decoder is proposed to tackle this
+problem. However, how to adapt this approach for
+text-to-text pretraining, which can be further ap-
+plied to the table-to-text task, is another challenge.
+To this end, we propose a table structure under-
+standing and text deliberating approach, namely
+TASD, to solve the table-to-text task with the pre-
+trained language model enhanced by the deliber-
+ation mechanism. Specifically, we first serialize
+1
+arXiv:2301.02071v1  [cs.CL]  5 Jan 2023
+
+Published as a conference paper at EMNLP 2022
+the table input with customized templates which
+do not acquire the target cells to be labeled. Then,
+we employ the multi-head attention in a hierarchi-
+cal way to learn the table representation that is
+aware of table structure and apply it to guide the
+fine-tuning of the text-to-text pretrained model. Af-
+terward, we adopt the multi-pass decoder to realize
+text deliberation. More specifically, we treat the
+above table-structure-aware fine-tuned model as
+the first-pass decoder and adopt another pretrained
+model as the second-pass decoder to further polish
+the descriptive text. In the second-pass decoding
+phase, the table representation can be conveniently
+leveraged as the “original text” in the text deliber-
+ation mechanism. The main contributions of this
+work can be summarized as follows:
+• We propose a novel table-to-text generation
+approach (i.e., TASD) to assimilating the com-
+plete table information with the help of table
+structure distillation, the pretrained language
+model, and the text deliberation.
+• We devise a table-structure-aware text gen-
+eration model (TASATG) via the hierarchi-
+cal multi-head attention network, which can
+realize the content selection automatically.
+And we develop an effective text deliberation
+method dedicated to the table-to-text task.
+• Extensive experiments conducted on two dif-
+ferent datasets demonstrate that TASD out-
+performs comparable baselines in terms of
+various metrics.
+2
+Related Work
+2.1
+Table-to-Text Generation
+Encouraged by the success of seq2seq methods
+in machine translation and text summarization, re-
+searchers proposed to formulate the input table as a
+sequence of records (Lebret et al., 2016; Wiseman
+et al., 2017), and further improve the performance
+of table-to-text methods based on seq2seq by mod-
+eling table representation (Liu et al., 2018; Gong
+et al., 2019a). Introducing auxiliary tasks to enrich
+the table representation (Tian et al., 2019; Li et al.,
+2021a) is another promising paradigm to address
+the table-to-text problem. Moreover, there have
+been studies focusing on how to disaggregate the
+table-to-text pipeline effectively to generate more
+faithful and fluent text, e.g. leveraging content
+selection and planning (Puduppully et al., 2019;
+Trisedya et al., 2020; Bai et al., 2021), combin-
+ing autoregressive and non-autoregressive meth-
+ods (Wang et al., 2021). In addition, recent Trans-
+formers were also applied to solve the table-to-text
+task (Gong et al., 2019b; Wang et al., 2020; Obeid
+and Hoque, 2020). However, current table-to-text
+methods may fail to tackle the overfitting problem
+aroused by the lack of diversity in small datasets.
+Fine-tuning the model pretrained in a large cor-
+pus and adapting to a specific task is an effective
+approach to tackling the generation issues disturbed
+by small data and large parameters (Radford et al.,
+2019). (Kale and Rastogi, 2020) explored the feasi-
+bility of applying the text-to-text pretrained model
+to the table-to-text task, (Gong et al., 2020) applied
+multi-task learning to solve the table-to-text task
+with pretrained language model, and (Suadaa et al.,
+2021) leveraged pretrained language model for fact
+inference in numerical table contents. However,
+these approaches seldom perceived and integrated
+the complete table information into the fine-tuning
+of the pretrained model. A table-to-text pretrained
+model (Xing and Wan, 2021) was proposed though,
+the large and diversified table corpus is often un-
+available. In addition, recent works on fact verifica-
+tion taking tabular as input (Yin et al., 2020; Dong
+and Smith, 2021) have suggested the effectiveness
+of the table-structure-aware pretrained model.
+2.2
+Text Deliberation
+The encoder-decoder framework has been widely
+applied to neural machine translation, while the
+subsequent words are often invisible on the target
+side when decoding a sequence. To alleviate this, re-
+searchers proposed to decode and refine the output
+sequence in multiple passes, like human cognitive
+behavior when polishing an article. Studies have
+been made on text deliberation, such as the solu-
+tion with two separate stages (i.e., generating and
+polishing) (Niehues et al., 2016), combining two
+separate stages as one framework (Xia et al., 2017),
+and deliberating generated text in multiple passes
+adaptively via reinforcement learning (Geng et al.,
+2018) or customized evaluating architecture (Li
+and Yao, 2021). To the best of our knowledge, we
+are the first to apply the deliberation mechanism to
+the table-to-text problem.
+3
+Preliminaries
+3.1
+Problem Formulation
+Our table-to-text problem takes a table as input,
+and we formulate a table as a sequence of records:
+T = {τ1,1, τ1,2, · · · , τi,j, · · · , τm,n}, where m and n
+denote the number of rows and columns of T, re-
+2
+
+Published as a conference paper at EMNLP 2022
+Figure 1: The framework overview of TASD.
+spectively. Then, we aim to generate a document Y
+containing words Y = y1y2 · · · yl that can describe
+the content of T precisely, where l is the document
+length. Formally, given a table T, the table-to-text
+model is excepted to generate a descriptive docu-
+ment Y in an auto-regressive way
+yi = arg max P(yi | T, y1y2 · · · yi−1; Θ), i = 1, · · · , l
+where Θ is the set of model parameters.
+3.2
+Data
+NumericNLG Dataset. The numericNLG dataset
+was released by (Suadaa et al., 2021).
+In this
+dataset, the tables demonstrate experimental re-
+sults in research papers, thus, most of the table
+contents are numerical values. We use this dataset
+to evaluate the accuracy and smoothness of the
+generated descriptions for the table with numerical
+content. In particular, for each table of numer-
+icNLG, <table_id> acts as the pronoun of the
+table, and <caption> is the descriptive text of the
+table. Moreover, for each cell of a table, there are
+<metric>, (row and column) <header>, and
+<value> as different views of a cell.
+Totto Dataset. The Totto dataset (Parikh et al.,
+2020) is an open-domain table-to-text dataset
+collected from Wikipedia. The table contents
+are mainly in text form. The metadata of
+the
+Totto
+dataset
+includes
+<page_title>,
+<section_title> and <section_text>. In
+detail, each cell of a table has corresponding
+<header> and <value>. Unlike numericNLG,
+textual content in our Totto dataset accounts for
+62.4%, which can evaluate the text generation
+effectiveness for the tables with textual records.
+4
+Methodology
+In this section, we introduce the proposed frame-
+work in detail. As shown in Fig. 1, our framework
+mainly consists of three components, i.e., template-
+based table serialization, table-structure-aware
+fine-tuning, and text deliberation. Specifically, we
+first produce a sequence describing the table con-
+tents with customized templates. The templates we
+adopted do not require the target cells to be labeled.
+Then, to generate informative text, we adopt full ta-
+ble representation learning to guide the description
+generation, such that the outcome text is capable
+of emphasizing and delineating the facts in the ta-
+ble from a macroscopic perspective. Finally, we
+employ and adapt the multi-pass decoder to our
+data-to-text problem, which can further fine-tune
+the generated table description. Technical details
+for all three modules will be introduced separately
+in the following subsections.
+4.1
+Template-based Table Serialization
+To well harness the expressive power of the text-to-
+text pretrained model for the input table, it is nec-
+essary to serialize the raw table first. The template-
+based representation offers us a simple yet effective
+linearization approach to generating descriptive
+texts which can reflect the facts in a table without
+yielding an intractable downstream model.
+In particular, the templates we adopted in this
+work are devised to mention all the available facts
+in the table without knowing the emphasized cells
+in advance, which is different from (Suadaa et al.,
+2021). The template for describing facts consists
+of two parts:
+1. The title or descriptive text that comes with
+the table.
+2. A series of expressions, in which each one
+describes the content of a cell.
+More specifically, for the numericNLG dataset,
+we apply the following template:
+<table_id> shows <caption>. <metric1,1
+> of <header1,1> is <value1,1>, · · · , <me
+trici,j> of <headeri,j> is <valuei,j>, · · · .
+For the Totto dataset, we apply another template:
+As <page_title> <section_title>, <se
+ction_text>. <header1,1> is <value1,1>,
+· · · , <headeri,j> is <valuei, j>, · · · .
+The second part of the template enumerates all the
+cells in the table. This preliminary table represen-
+tation, denoted by TS , covers all the available facts
+in a raw table. Note that, the templates we adopt
+may encounter the content selection problem. In
+table-to-text applications, target cells in the input
+table are often not highlighted and the generated
+table description should emphasize certain cells.
+3
+
+table 2 shows the overall mention
+detection results on the test set of
+ontonotes. prec. of our full model
+prec. is 89.6. rec. of our full model
+rec. is 82.2. f1 of our full model f1 is
+table 2 shows the results on the ontonotes
+85..7.....
+test set. f1 score of our full model is 86.2
+our full model outperforms the state-of-
+the-art by 1.8 points in f1 score.
+table 2 shows the results on the ontonotes
+test set. we can see that our full model
+outperforms the state-of-the-art on all
+metrics.Published as a conference paper at EMNLP 2022
+Figure 2: The architecture of table-structure-aware text
+generation model (i.e., TASATG).
+4.2
+Table-Structure-Aware Text Generation
+A text-to-text pretrained model can take the large-
+scale corpus as input to possess vast knowledge
+and generate texts in an unsupervised way so that
+it has been widely applied to text-generation tasks.
+When handling a specific text generation task, it is
+effective to fine-tune the pretrained model on new
+data. However, for the table-to-text task, some hid-
+den information, like table structure, is most likely
+to be overlooked, though the drafted TS mentions
+all the available facts in the table. Thus, we pro-
+pose to exploit table structure information to guide
+fine-tuning of the text-to-text pretrained model.
+As shown in Fig. 2, we first encode the table
+content in a multi-view fashion. To be specific,
+given a cell τi,j in a table T, it can be viewed from
+different perspectives, such as the value of τi,j, the
+row header of τi,j, and the column header of τi,j,
+etc. Then, we treat the k-th view of τi,j as a to-
+ken sequence which is denoted by x(k)
+i,j . Afterward,
+we pad x(k)
+i,j with placeholders (if necessary) and
+concatenate these token sequences as follows:
+xi,j = x(1)
+i,j ⊛ x(2)
+i,j ⊛ · · · ,
+(1)
+where ⊛ denotes the concatenation operator, and
+the multi-viewed representation of a table T is de-
+noted as X = [x1,1, · · · , xi,j, · · · , xm,n]. Each to-
+ken of x(k)
+i,j can be encoded as a d-dimensional em-
+bedding by looking up the text-to-text pretrained
+model and updated accordingly when fine-tuning
+the pretrained model. In this way, we can obtain
+the semantic representation of table T, which is
+denoted by E(0) ∈ Rm×n×s×d, where s is the length
+of concatenated sequence xi,j.
+To realize TASATG for table-to-text, we pro-
+pose to employ multi-head attention (Vaswani et al.,
+2017) to guide fine-tuning of the text-to-text pre-
+trained model. In particular, we adopt three multi-
+head attention (MHA) layers to interactively extract
+the information in the table in a hierarchical way.
+Specifically, the MHA layer is defined as:
+Qi = QWQ
+i , Ki = KWK
+i , Vi = VWV
+i
+head i = Attention (Qi, Ki, Vi) = softmax
+�QiK⊤
+i
+√
+d
+�
+Vi,
+MHA(Q, K, V) = [ head 1, · · · , head h] WO,
+where Q, K, V represent the query, key and value
+in the attention mechanism, respectively.
+As illustrated in Fig. 2, in the first MHA layer,
+we add a cell text position embedding (E(ctpe) ∈
+Rs×d) to each cell of the aforementioned E(0), and
+feed it to the multi-head attention to implement cell
+text self-attention,
+�
+E0 = E(0) ⊕ E(ctpe),
+E(1) = MHA(�
+E(0), �
+E(0), �
+E(0)),
+E(1) = 1
+s
+s
+�
+i=1
+(E(1)[:, :, i, :]) ,
+(2)
+where ⊕ denotes the element-wise addition opera-
+tion. Consequently, E(1) ∈ Rm×n×d can be deemed
+as an initial aggregated table representation. Next,
+in the second MHA layer, we add a table position
+embedding (E(tpe) ∈ Rm×n×d) to E(1) to implement
+table structure self-attention,
+�
+E(1) = E(1) ⊕ E(tpe),
+E(2) = MHA(�
+E(1), �
+E(1), �
+E(1)).
+(3)
+E(2) ∈ Rm×n×d is the table-structure-aware represen-
+tation. Moreover, in the third MHA layer, we ap-
+ply a multi-head cross-attention to take the hidden
+state of the text-to-text pretrained model (denoted
+by H ∈ Rs×d) as the attention query, such that we
+can focus on the important cells of the table,
+�H = MHA(H, E(2), E(2)) ⊕ H.
+(4)
+This new hidden state �H guided by the table repre-
+sentation will replace the original hidden state H
+in the text-to-text pretrained model to generate the
+probability of the next word.
+Note that, the cross attention weights on differ-
+ent table cells based on the previous words can
+realize the content selection automatically. In ad-
+dition, we implement the text-to-text pretrained
+model with GPT2 (Radford et al., 2019), which
+adopts a decoder-only Transformer architecture.
+4
+
+Text-to-Text Pretrained Model
+Multi-Head Self-Attention
+Multi-Head Self-Attention
+Multi-Head Cross-Attention
+Next Word
+Previous
+Probability
+WordsPublished as a conference paper at EMNLP 2022
+(a) Training.
+(b) First and second fine-tuning of TASATG with vali-
+dation data.
+(c) Testing.
+Figure 3: Training, validation and testing procedures of the proposed TASD approach.
+4.3
+Text Deliberation
+The encoder-decoder framework applied in many
+sequence generation tasks often adopts a one-pass
+process while decoding a sequence. Though effi-
+cient, the one-pass decoder cannot perceive future
+context for further text deliberation. Multi-pass de-
+coder extends the capability of generating more
+refined text by exploring global information in the
+sequence (Niehues et al., 2016; Xia et al., 2017).
+For the text-to-text pretrained model, due to the
+huge amount of parameters of the pretrained lan-
+guage model, it is unwise to directly combine the
+models in different passes. A common solution is
+to concatenate the original serialized table content
+and the text generated in the previous pass to fine-
+tune the pretrained model in the next-pass decoding.
+However, in this way, the length of input text prob-
+ably exceeds the limit of the text-to-text pretrained
+model, and the time complexity is too high.
+To effectively implement the fine-tuning of the
+text-to-text pretrained model in multiple passes,
+as shown in Figs. 3a and 3b, we take the table
+representation as the “original text” and feed the
+text generated in the first-pass fine-tuning plus
+the table representation to the second-pass fine-
+tuning. Note that, as shown in Fig. 3a, we sep-
+arately fine-tune the table-to-text generation task
+and the text-to-text deliberation task with two inde-
+pendent TASATG models, and each of them takes
+a text-to-text pretrained model as the backbone.
+5
+Experiments
+5.1
+Experimental Settings
+Data. We conducted experiments on the aforemen-
+tioned datasets, i.e., numericNLG and Totto. The
+statistics of the numericNLG dataset can be found
+in (Suadaa et al., 2021). Besides, the size of the
+original Totto dataset is 120K, which is much larger
+than the numericNLG dataset. To evaluate differ-
+ent methods for table-to-text with comparable data
+size, for the Totto dataset, we filtered out the tables
+with fewer rows and columns, i.e., #rows < 8 and
+#columns < 8, such that the filtered Totto dataset
+contains 1.8K tables. Then, we randomly selected
+1.2K1 tables to generate the new Totto dataset.
+Evaluation Metrics. We calculated BLEU (from
+gram-1 to gram-4) (Papineni et al., 2002), ROUGE-
+L (Lin, 2004) and METEOR (Denkowski and
+Lavie, 2014) to evaluate the quality of the gen-
+erated text. The BLEU-n with a small value of n
+measures the accuracy of the word level, and the
+BLEU-n with a large n can measure the fluency
+of the sentence. The ROUGE-L measures the re-
+call rate based on the longest common sequence
+between source and target texts. The METEOR is
+based on the harmonic mean of unigram precision
+and recall, with recall weighted higher than preci-
+sion. These metrics are widely used to measure the
+accuracy and fluency of the generated sentence.
+Baselines. We compare TASD with the following
+baselines.
+• Template-based Table Serialization.
+We
+use the template designed for table serial-
+ization as a baseline. Note that, the token
+sequence generated by the template-based
+method is denoted as TS .
+• Pointer Generator (See et al., 2017). This
+is a seq2seq model with the attention and
+copy mechanism. We take TS as input for
+the pointer generator model.
+• TRM. We implemented a simplified version
+of the proposed TASD that omits the pos-
+sessed knowledge in the pretrained language
+model and removes text deliberation for focus-
+ing on table representation modeling, namely
+TRM. In particular, TRM adopts the architec-
+ture of GPT2 but initializes the parameters
+randomly and trains 100 epochs at most for
+fine-tuning. Besides, TRM takes TS plus the
+table structure representation as input and is
+fed with TS in the inference phase.
+1The size of numericNLG data is 1.3K.
+5
+
+2nd Fine-tuned TASATG
+TASATG
+TASATG
+1st Fine-tuned TASATG1st Fine-tuned TASATG
+Candidate Fine-tuned TASATG 1
+Candidate Fine-tuned TASATG 1
+Candidate Fine-tuned TASATG 2
+TASATG
+Candidate Fine-tuned TASATG 2
+TASATG
+Candidate Fine-tuned TASATG 3
+Candidate Fine-tuned TASATG 3
+1st Fine-tuned TASATG
+2nd Fine-tuned TASATG2ndFine-tunedTASATG
+1st Fine-tunedTASATGPublished as a conference paper at EMNLP 2022
+Table 1: Performance comparisons of the automatic evaluation on the numericNLG dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+Template-based Method
+10.28
+5.52
+2.83
+1.14
+11.31
+11.49
+Pointer Generator
+5.10±0.59
+2.71±0.19
+1.16±0.17
+0.56±0.04
+7.82±0.15
+15.21±0.14
+TRM
+14.16±0.97
+6.05±0.50
+2.11±0.13
+0.80±0.12
+9.72±0.94
+12.72±0.80
+Fine-tuned GPT2
+16.13±0.56
+9.02±0.31
+4.68±0.22
+2.20±0.22
+10.14±0.32
+17.48±0.36
+TableGPT
+18.69±0.39
+8.21±0.24
+3.31±0.19
+1.51±0.14
+11.06±0.18
+16.90±0.27
+TASD w/o TAS
+18.20±2.40
+9.74±1.01
+4.38±0.31
+1.98±0.39
+10.64±0.86
+19.29±1.77
+TASD w/o D
+18.02±0.50
+10.06±0.25
+5.20±0.13
+2.47±0.20
+10.99±0.29
+18.57±0.27
+TASD w/o 1st-TAS
+20.07±1.94
+10.35±0.69
+4.67±0.35
+2.05±0.34
+11.52±0.80
+20.10±0.62
+TASD
+21.81±1.13
+11.03±0.11
+4.92±0.22
+2.15±0.39
+11.87±0.40
+20.40±0.80
+Table 2: Performance comparisons of the automatic evaluation on the Totto dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+Template-based Method
+0.84
+0.43
+0.23
+0.09
+4.59
+1.51
+Pointer Generator
+11.34±1.57
+2.05±0.83
+0.45±0.27
+0.35±0.13
+5.38±0.78
+14.46±1.46
+TRM
+10.21±1.79
+3.44±0.88
+1.21±0.48
+0.54±0.25
+9.30±1.16
+11.52±2.03
+Fine-tuned GPT2
+9.53±0.51
+3.65±0.34
+1.18±0.37
+0.40±0.26
+9.89±0.39
+10.69±0.27
+TableGPT
+6.80±0.26
+3.51±0.22
+1.33±0.21
+0.76±0.12
+11.10±0.42
+11.73±0.44
+TASD w/o TAS
+13.70±0.90
+4.44±0.69
+1.28±0.47
+0.65±0.35
+10.79±0.83
+14.47±1.11
+TASD w/o D
+10.03±0.39
+4.42±0.29
+1.64±0.36
+0.71±0.38
+10.29±0.49
+10.67±0.34
+TASD w/o 1st-TAS
+13.90±0.60
+5.07±0.61
+1.68±0.52
+0.79±0.25
+10.98±0.40
+14.88±0.71
+TASD
+14.19±1.08
+5.17±0.38
+1.71±0.32
+0.78±0.21
+11.65±0.71
+14.96±1.10
+• Fine-tuned GPT2 (Radford et al., 2019). We
+take the concatenation of TS and Y as the in-
+put for fine-tuning. In the inference phase,
+we only feed TS to the model to generate Y
+starting after the last token of TS .
+• TableGPT (Gong et al., 2020). TableGPT is
+a state-of-the-art table-to-text method. To im-
+prove the text fidelity and exploit the struc-
+tural information at the same time, TableGPT
+employs a multi-task learning paradigm con-
+sisting of two auxiliary tasks, that is, one task
+reconstructs the table structure from represen-
+tations of GPT2, and the other aligns the tables
+and the information in the generated text.
+Implementation Details. The split settings for
+training, validation and, testing were 1084:136:135
+2 for the numericNLG dataset and 960:120:120
+for the Totto dataset, respectively. Regarding auto-
+matic evaluation, all results of deep models were
+obtained by conducting experiments on a Linux
+machine with Nvidia A100 GPU, and the averaged
+results of 5 runs were reported. Besides, an Adam
+2This setting follows the experiments of (Suadaa et al.,
+2021).
+optimizer was utilized (with an initial learning rate
+of 3e-5) for GPT2 fine-tuning, and the training was
+iterated in 20 epochs at most. A beam search algo-
+rithm was adopted when decoding a sequence and
+the beam width was set to 5 3.
+5.2
+Automatic Evaluation
+The comparisons of automatic evaluation results
+between TASD and other baselines can be found
+in Tables 1 and 2. In general, TASD outperforms
+the baselines for all the metrics on two datasets. In
+particular, compared to the reported best result of
+all the baselines, TASD achieves improvements of
+3.12 for BLEU-1 (18.69 → 21.81), 2.01 for BLEU-
+2 (9.02 → 11.03), 0.24 for BLEU-3 (4.68 → 4.92),
+0.56 for METEOR (11.31 → 11.87), and 2.92 for
+ROUGE-L (17.48 → 20.40) on the numericNLG
+dataset, and 2.85 for BLEU-1 (11.34 → 14.19),
+1.52 for BLEU-2 (3.65 → 5.17), 0.38 for BLEU-3
+(1.33 → 1.71), 0.02 for BLEU-4 (0.76 → 0.78),
+0.55 for METEOR (11.10 → 11.65), and 0.50 for
+ROUGE-L (14.46 → 14.96) on the Totto dataset.
+3Our implementation is available at https://github.
+com/ramber1836/TASD.
+6
+
+Published as a conference paper at EMNLP 2022
+In other words, for different types of source tables,
+TASD generates better descriptive texts w.r.t. accu-
+racy at the word level, recall of the sequence, and
+fluency of sentences.
+Besides, we have the following observations: 1)
+The template-based method performs much bet-
+ter on the numericNLG dataset compared to the
+Totto dataset, since the referenced table descrip-
+tions in numericNLG were collected from scientific
+papers, however, the table summaries in the Totto
+dataset are more diverse. 2) In the Totto dadaset,
+the pointer generator model tends to cover more
+words in descriptive text and generate more fluent
+sentences than the template-based method, as the
+contents in source tables of the Totto dataset are
+mostly linguistic. This can also explain why the
+pointer generator performs worse than the template-
+based method on the numericNLG dataset w.r.t.
+BLEU and METEOR. 3) Fine-tuned GPT2 can
+generate more faithful and fluent text than other
+baselines (refer to Tables 1 and 2) most of the time,
+which validates the effectiveness of the pretrained
+language model. 4) In general, TableGPT performs
+better, and even the best, among all the baselines.
+In the numericNLG dataset, the headers of the
+input tables (a.k.a. the attributes of records for
+TableGPT) are more diverse, which may explain
+why the performance of TableGPT is not promising
+as expected on the numericNLG dataset. 5) TRM
+can generate comparable, or even better descriptive
+text as fined-tuned GPT2, which further suggests
+the effectiveness of table structure understanding.
+5.3
+Ablation Analysis
+Moreover, to verify the effectiveness of different
+modules, we compare TASD with its variants.
+• After generating text with fine-tuned GPT2,
+we fed the generated text concatenated with
+TS to another fine-tuned GPT2 to realize the
+second-pass decoder without table structure
+representation.
+• We implemented TASD without deliberating
+on the outcome text, which means that we
+realized TASATG based on GPT2 in a one-
+pass forward process.
+• TASD w/o 1st-TAS. We removed table struc-
+ture modeling in the first-pass decoding from
+TASD, which was implemented by taking the
+fine-tuned GPT2 as the first-pass decoder and
+the table-structure-aware fine-tuned GPT2 as
+the second-pass decoder.
+As can be seen in Tables 1 and 2, TASD w/o TAS
+performs worse than TASD under all metrics, since
+the table structure modeling can benefit the fine-
+tuning of GPT2. This can also be validated by com-
+paring fine-tuned GPT2 to TASD w/o D. Besides,
+the effectiveness of deliberating text can be proven
+by comparing TASD w/o D to TASD (this can also
+be validated by comparing fine-tuned GPT2 to
+TASD w/o TAS). While text deliberation may harm
+sentence fluency as depicted by the results of these
+methods w.r.t. BLEU-3 & 4 in Table 1. In addition,
+TASD w/o 1st-TAS outperforms TASD w/o TAS under
+all metrics suggesting that taking the table repre-
+sentation as the “original text” in the deliberation
+mechanism is also effective.
+5.4
+Qualitative Analysis
+Figs. 4(a) and (b) show two selected source ta-
+bles and corresponding descriptive texts (i.e., cap-
+tion and section_text) in numericNLG and Totto
+datasets. Fig. 4(c) demonstrates the generated de-
+scriptions by different methods. The text that cor-
+rectly reflects the facts of the source table is in
+green, the erroneous text is in red, and the con-
+fusing text is in blue. We can see that, there are
+many grammatical errors in the text produced by
+the pointer generator. Fine-tuned GPT2 tends to
+repeat phrases and sentences due to the limited
+knowledge about the input table, which can also
+explain why the fine-tuned GPT2 can obtain a false
+high score in BLEU-n as n grows. Thanks to the
+semantic knowledge brought by pretraining, fine-
+tuned GPT2 can generate more natural descriptions,
+in which, however, perplexing factual errors ex-
+ist. Compared to fine-tuned GPT2, the description
+generated by TASD is more relevant to the table
+contents. Since the target cells are not known in
+advance, the generated text may miss the empha-
+sized points described in the reference. The text
+generated by TableGPT is also fluent, though coun-
+terfactual descriptions may exist.
+5.5
+Human Evaluation
+We randomly selected 30 samples from the test set
+in numericNLG and Totto datasets, respectively,
+and invited 10 volunteers to evaluate the quality of
+the outcome text by considering three criteria, i.e.,
+grammar, coherence & concise, and factual per-
+spective (correct and relevant). Each criterion has
+scores of five degrees, ranging from 1 (the worst) to
+5 (the best). The averaged scores were reported in
+Table 3, which show that TASD can generate more
+7
+
+Published as a conference paper at EMNLP 2022
+Figure 4: Two examples of the generated table descriptions.
+Table 3: Result of Human Evaluation
+Dataset
+Method
+Grammar Coherence
+& Concise
+Factual per-
+spective
+numericNLG
+Pointer Genera-
+tor
+3.16±0.99
+2.73±1.20
+1.54±0.69
+Fine-tuned
+GPT2
+3.42±0.56
+3.11±0.58
+2.51±0.45
+TASD w/o D
+3.72±0.61
+3.48±0.55
+2.82±0.45
+TASD
+4.17±0.72
+3.98±0.64
+3.15±0.73
+Totto
+Pointer Genera-
+tor
+2.03±0.71
+1.89±0.82
+1.56±0.55
+Fine-tuned
+GPT2
+2.60±0.55
+2.36±0.64
+1.85±0.46
+TASD w/o D
+2.63±0.52
+2.46±0.60
+1.89±0.46
+TASD
+3.4±0.66
+3.18±0.70
+2.25±0.69
+readable and coherent texts, and describe more
+correct facts. Moreover, the pretrained models con-
+sistently achieve better scores than the pointer gen-
+erator on grammar and coherence because of the
+expressive power learned from the large-scale cor-
+pus. In the Totto dataset, the improvement of the
+table structure modeling is smaller than that of the
+polishing mechanism, which is consistent with the
+automatic evaluation results in Table 2.
+6
+Discussion
+In our work, we devised a two-pass decoder frame-
+work dedicated to the table-to-text task with the
+help of the table-structure-aware text generation
+model (i.e., TASATG). However, the effectiveness
+of the text deliberation for the table-to-text task
+should be further explored and integrated into the
+table-structure-aware modeling in a more harmonic
+Figure 5:
+Table reconstruction for table-structure-
+aware modeling enhancement.
+manner. To discuss the limitation of the text de-
+liberation of TASD, we additionally developed a
+table content reconstruction loss and integrate it
+into TASD in a multi-task learning fashion.
+Specifically, given the table-structure-aware em-
+bedding E(2) generated with Eq. (3), we randomly
+mask certain cells of the input table and yield a
+partially corrupted embedding of the input table,
+denoted by �
+E(2). Then, a two-layer MLP (i.e., multi-
+layer perceptron) is adopted to restore the table-
+structure-aware embedding. Afterward, an MSE
+(i.e., mean square error) loss is adopted to mea-
+sure the effectiveness of table reconstruction and
+further integrated into the TASD framework in the
+multi-task learning paradigm. The process of table
+reconstruction is demonstrated in Fig. 5.
+We carried out a series of experiments to evalu-
+ate the performance of TASD w/ and w/o the help
+of table reconstruction loss (i.e., TRLoss) on nu-
+mericNLG and Totto datasets in terms of BLEU-n
+(1 to 4), METEOR, and ROUGE-L. The results can
+be found in Tables 4 and 5.
+8
+
+our model also outperforms transformer
+our model.
+compares.
+transformer significantly outperforms transformer
+the effectiveness of transformer
+the comparison of our model with transformer on
+caption
+frenchenglish translation task.
+the size of our
+model is slightly larger than the size of the transformer model
+the evaluation metrics of our model and the transformer. our
+model achieves the best results on all the metrics
+suvarnabhumi is the busiest airport in thailand
+international airport international airport
+ international airport international airport...
+2018, airport is airport is airport...
+busiest
+after
+suvarnabhumi airport
+is the third-busiest airport in thailand after
+suvarnabhumi airport
+section_titleTASATG
+GPT2 Fine-Tuning Loss
+Table Reconstruction Loss
+MLPPublished as a conference paper at EMNLP 2022
+Table 4: The performances of TASD w/ and w/o the table reconstruction on the numericNLG dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+TASD w/o D
+18.02±0.50
+10.06±0.25
+5.20±0.13
+2.47±0.20
+10.99±0.29
+18.57±0.27
+TASD w/o D w/ TRLoss
+20.56±0.25
+11.57±0.21
+5.90±0.23
+2.98±0.17
+12.00±0.48
+20.50±0.39
+TASD w/ TRLoss
+19.29±0.38
+10.12±0.24
+5.32±0.25
+2.62±0.22
+12.18±0.90
+18.95±0.69
+TASD w/ TRLoss in 1st pass
+18.23±0.68
+9.39±0.52
+4.64±0.26
+2.36±0.24
+11.51±0.78
+18.13±0.45
+TASD w/ TRLoss in 2nd pass
+19.38±2.21
+10.33±1.34
+5.11±0.73
+2.40±0.38
+11.35±0.92
+18.69±1.05
+TASD
+21.81±1.13
+11.03±0.11
+4.92±0.22
+2.15±0.39
+11.87±0.40
+20.40±0.80
+Table 5: The performances of TASD w/ and w/o the table reconstruction on the Totto dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+TASD w/o D
+10.03±0.39
+4.42±0.29
+1.64±0.36
+0.71±0.38
+10.29±0.49
+10.67±0.34
+TASD w/o D w/ TRLoss
+9.94±0.43
+4.35±0.31
+1.63±0.31
+0.75±0.13
+10.37±0.22
+10.62±0.60
+TASD w/ TRLoss
+14.57±0.87
+5.22±0.42
+1.70±0.49
+0.89±0.38
+11.79±0.77
+15.28±0.86
+TASD w/ TRLoss in 1st pass
+14.00±0.82
+5.31±0.27
+1.72±0.25
+0.75±0.13
+11.02±0.77
+14.74±0.51
+TASD w/ TRLoss in 2nd pass
+13.89±0.58
+4.78±0.61
+1.47±0.14
+0.52±0.20
+11.07±0.66
+14.73±0.79
+TASD
+14.19±1.08
+5.17±0.38
+1.71±0.32
+0.78±0.21
+11.65±0.71
+14.96±1.10
+According to the results reported on the nu-
+mericNLG dataset, the TRLoss is helpful in en-
+hancing the capability of table comprehension
+though, the best performance is achieved by TASD
+w/o D w/ TRLoss. It seems that the performance im-
+provement gained by the table comprehension en-
+hancement is sacrificed after the text deliberation
+is adopted. Meanwhile, on the Totto dataset, TASD
+with the table reconstruction (i.e., TASD w/ TRLoss)
+does achieve the best performance in terms of
+BLEU-1, BLEU-2, METEOR, and ROUGE-L,
+though the improvement is not remarkable. The
+contents of the input tables are mainly linguistic
+and the table structures are not too diverse might
+be able to explain the performance improvement
+of TASD w/ TRLoss on the Totto dataset. With the
+above comparisons, we can conclude that, for the
+input tables with diverse structures, the limitation
+of the current text deliberation mechanism cannot
+be neglected if one aims to enhance the capability
+of table comprehension for the table-to-text task.
+Moreover, this also suggests that the generalization
+capability of text deliberation of TASD should be
+improved in the future.
+Limitations. In this work, long tables in the
+Totto dataset are removed since the efficiency and
+performance of TASD on large tables could be low-
+ered. In the future, the capability of handling long
+tables for table-to-text models should be further ex-
+plored. Besides, a large-scale and more exhaustive
+human evaluation is necessary. We plan to recruit
+more volunteers to conduct the human annotation.
+7
+Conclusion
+In this paper, to realize table-to-text with the pre-
+trained language model, we proposed a table struc-
+ture understanding and text deliberating approach,
+namely TASD. The table structure understanding
+was realized by developing a hierarchical multi-
+head attention network, which can benefit the fine-
+tuning of the text-to-text pretrained model. The
+fully represented table information benefits not
+only the pretrained language model but also the
+text deliberation process since the structure infor-
+mation with rich semantics could be fed into the
+second-pass decoding naturally. We carried out ex-
+tensive experiments on two public datasets with
+different table types. Automatic and human-based
+evaluations, as well as qualitative analysis, vali-
+date the effectiveness of our approach to generating
+faithful and fluent table descriptions. In the future,
+we will improve text deliberation by devising a
+unified framework to integrate the multi-pass de-
+coder and refine the descriptive text paying more
+attention to sentence fluency.
+Acknowledgements
+This work is supported in part by Foshan
+HKUST Projects (FSUST21-FYTRI01A, FSUST
+21-FYTRI02A).
+9
+
+Published as a conference paper at EMNLP 2022
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+A
+Human Evaluation Settings
+The criteria adopted in our human-based evaluation
+are (1) Grammar (e.g., is this paragraph grammat-
+ical?), (2) Coherence & Concise (e.g., is this para-
+graph coherent and contextually consistent? does
+it repeat redundant information?), and (3) Factual
+perspective (e.g., are the facts that this paragraph
+describes correct? are these facts related to refer-
+ences and tables?). More specifically, we list the
+detailed justifications on how to score the generated
+text in each criterion as follows.
+Grammar
+• 1 It is more like garbled code than a paragraph.
+• 2 There are many obvious grammatical mis-
+takes.
+• 3 There are a few obvious grammatical mistakes.
+• 4 There are few grammatical mistakes.
+• 5 There are no grammatical mistakes.
+Coherence & Concise
+• 1 The logic of text expression is chaotic and
+nonsense.
+• 2 There are a lot of logical inconsistencies or
+redundant information.
+• 3 There are some logical inconsistencies or re-
+dundant information.
+• 4 There are a few logical inconsistencies or
+redundant information, but it does not affect
+browsing.
+• 5 The logic of the text is smooth without redun-
+dant information.
+Factual Perspective
+• 1 The paragraph does not coincide with the ref-
+erence or table, and it is full of information
+inconsistent with the facts.
+• 2 The paragraph describes the facts incorrectly
+and has a low correlation with reference, but is
+related to the information in the table.
+• 3 The paragraph description is incorrect, but it
+is highly coincident with the reference.
+• 4 The paragraph description is basically correct,
+and the coincidence with the reference is low,
+but it also describes the information in the table.
+• 5 The paragraph description is correct and
+highly coincident with the reference.
+B
+Illustrative Examples of Generated
+Descriptions
+We additionally selected another two examples of
+the generated table descriptions from the numeric-
+NLG and Totto datasets, respectively. The results
+are shown in Figs. 6 and 7. From these four ex-
+amples, we can see that TASD can generate more
+accurate and fluent descriptive texts. While incor-
+rect descriptions can be found in the outcome texts
+generated by different models for cases D and F,
+which suggests that generating faithful descriptions
+for open-domain tables is much more challenging
+and requires more powerful and, thus larger, pre-
+trained language models.
+C
+Extra Implementation Details
+The learning rate of GPT2 was searched from {3e−
+4, 3e − 5, 3e − 6}. In the evaluation of discussing
+the limitation of text deliberation (see Section 6),
+a trade-off parameter for balancing the GPT2 fine-
+tuning loss and the TRLoss was adopted, then the
+trade-off parameter was searched from {1e−1, 5e−
+2, 1e − 2, 5e − 3, 1e − 3}, and 1e-2 was selected for
+the reported performance. Besides, the reported
+results in Tables 4 and 5 were averaged in 3 runs.
+11
+
+Published as a conference paper at EMNLP 2022
+Figure 6: Generated table descriptions on cases C and D.
+Figure 7: Generated table descriptions on cases E and F.
+12
+
+robusttc-fsl,
+et al.（85.8 et al., 2018)
+matching and prototypical networks
+relation networks
+the graph networks
+arsc mean acc dataset
+ snorkel network providing the best performance
+our model outperforms the other baselines on mean acc
+it leads to a significant improvement in mean accuracy for the compared models
+on the other hand, the absence of semantic representation (a) leads to a significant
+caption
+improvement in mean accuracy
+our proposed model achieves competitive accuracy, comparable to
+the state-of-the-art models from yu et al. (2018).
+ steve gregory
+ steve gregory
+seven interceptions seven
+13 games
+35 tackles
+sacks
+tour touchdowns
+ steve gregory
+357 tackles seven interceptions three sacks
+forced fumble.0 yards per game
+steve gregory ended his career with 357 tackles, seven
+page_title
+interceptions, three sacks and 1 forced fumble.0 yards per game ... (repeat)
+section title
+steve gregory
+357 tackles seven sacks
+seven interceptionsconsistency score
+the w/o results
+consistency and novelty scores
+w/o adversarial learning results in a full model
+consistency score of 3.84 and a novelty score of 3.24
+baseline
+adversarial adaptation
+results in a much better consistency than the w/o adaptation
+caption
+combined with the w/o Istm, the coherence improves significantly
+consistency, novelty, diversity,
+and coherence
+same consistency
+dynamic strategy leads to a higher novelty
+110 years
+the district index performed last 57 district index ohio 67 district ( death ) until
+district used holder index ( years district)
+alice sanders (12 may 1897 - 7 november 2007)
+survivors
+alice sanders (12 mav 1897 - 7 november 2007)
+alice sargent
+over 40 years
+page_title
+breast
+and breast breast
+section title

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+Task-sequencing Simulator: Integrated Machine Learning to Execution
+Simulation for Robot Manipulation
+Kazuhiro Sasabuchi1, Daichi Saito1, Atsushi Kanehira1, Naoki Wake1, Jun Takamatsu1, Katsushi Ikeuchi1
+Abstract— A task-sequencing simulator in robotics manip-
+ulation to integrate simulation-for-learning and simulation-
+for-execution is introduced. Unlike existing machine-learning
+simulation where a non-decomposed simulation is used to
+simulate a training scenario, the task-sequencing simulator runs
+a composed simulation using building blocks. This way, the
+simulation-for-learning is structured similarly to a multi-step
+simulation-for-execution. To compose both learning and execu-
+tion scenarios, a unified trainable-and-composable description
+of blocks called a concept model is proposed and used. Using the
+simulator design and concept models, a reusable simulator for
+learning different tasks, a common-ground system for learning-
+to-execution, simulation-to-real is achieved and shown.
+I. INTRODUCTION
+Simulators are important in robotics. Compared to the
+real world, simulators can run endlessly, safely, remove
+stochasticity, and provide ground truth data. One direction
+for using simulators in robotics is to help check collisions
+and safe executions before real robot executions. Another
+direction is using simulators as a tool for machine learning.
+Simulators that suit either one of the above directions are
+available today, however, it is desirable to have a simulator
+which fulfills both purposes to better integrate learning and
+execution, especially in the multi-step manipulation domain.
+In multi-step manipulation, a series of tasks which occur
+sequentially must be simulated. An example of a sequential
+series of tasks is bringing an object, which is composed
+of tasks: grasp, pick, bring. An execution simulator must
+be able to trigger these different tasks and connect them
+into a sequenced simulation. When the simulator is used
+for checking robot executions, the simulation is a matter of
+combining programmed or trained building blocks (e.g., run
+“grasp” then “pick” then “bring”).
+In contrast, machine learning simulators often ignore the
+task-sequence composition and are structured to train a
+specific problem or benchmark (e.g., a non-decomposed
+“pick-and-place” simulation)[1][2]. This structural differ-
+ence causes a gap between simulation-for-execution and
+simulation-for-learning. The learned results become specific
+to the trained scenario and contradicts with the simulation-
+for-execution where scenarios are non-fixed.
+Instead, a machine learning simulator could be designed
+similar to an execution simulator. The “pick-and-place”
+scenario can be decomposed into a sequenced simulation
+of “grasp then pick then bring then place then release.”
+The difference compared to the execution simulator is that
+1All
+authors
+are
+with
+Microsoft,
+Redmond,
+WA,
+USA
+[email protected]
+Physics
+Kinematics
+Rendering
+Post-process
+Task-sequencing simulator
+Concept Interface
+Environment Engine Pipeline
+CM Grasp
+state
+action
+selected task 
+at step t
+demonstrated 
+execution sequence
+configured 
+training sequence
+CM Open
+CM Release
+training 
+algorithms
+Fig. 1.
+The proposed task-sequencing simulator which enables scenario
+composition for both learning and execution in robotics manipulation.
+some of these blocks are “under-training” and are updated
+as data is collected. Once the update has finished, the trained
+block can be combined and reused for a different scenario,
+thus, the simulation-for-learning can directly transition to the
+simulation-for-execution. In addition, such design enables
+learning new manipulation skills on top of programmed or
+prior-trained building blocks. For example, a “grasp” could
+be trained using the sequence “grasp then pick,” where
+“pick” is a programmed task to provide a supervised signal
+(i.e., teach that the “grasp” was successful if the “pick” was
+successful).
+This article introduces a task-sequencing simulator struc-
+ture which enables integrated learning-to-execution simula-
+tion. At its core, the simulator uses a unified block design
+called the “concept model,” which is proposed within this
+article and defines the necessary descriptions for training
+a task, collecting trained tasks, and running the tasks to
+compose a sequence.
+The rest of the article is outlined as below: Section II
+provides a background on existing robotic simulators. Sec-
+tion III explains the overall simulator structure for achieving
+machine learning to robot execution simulation. Section IV
+explains the concept model core component of the simulator
+and Section V provides some detailed example implementa-
+tions of the model. Section VI shows the capabilities of the
+arXiv:2301.01382v1  [cs.RO]  3 Jan 2023
+
+C16simulator in machine-learning-to-robot-execution followed
+by conclusions in Section VII.
+II. BACKGROUND
+While there are many existing simulators for robotics,
+existing simulators may not achieve the integrated learning-
+to-execution multi-step manipulation purpose for one of
+the following reasons: 1) the simulator targets a different
+domain other than manipulation, 2) the simulator can be
+used for manipulation but misses a capability in simulation-
+for-learning, 3) the simulator can be used for manipulation
+but misses a capability in simulation-for-execution, 4) the
+simulator can be used for manipulation and both learning-
+and-execution purposes but not specifically for learning-to-
+execution purposes.
+Popular
+robotics
+simulators[3][4]
+include
+Gazebo[5],
+MuJoCo[6], CoppeliaSim[7], CARLA[8], AirSim[9], and
+Webots[10]. Gazebo has its advantage in its capability to
+simulate executions using ROS integrated sensors and ac-
+tuators but is not the best choice when it comes to data
+collection and machine learning due to its slow simulation
+performance and inconsistency in physics simulation. Thus,
+Gazebo falls into the second category. Engines like MuJoCo
+on the other hand, are suitable for stable physics simulation
+in machine learning but miss some robotics simulation
+capabilities such as inverse-kinematics and visual feedback
+(realistic rendering). The focus is on physics simulation
+rather than an integrated simulator for robot executions,
+therefore falls into the third category. CARLA and Airsim
+mainly target automobiles such as drones and cars therefore
+miss some important features such as kinematics required for
+manipulation and falls into the first category.
+The CoppeliaSim is an integrated simulator with a kine-
+matics and physics engine, and the PyRep toolkit[11] can
+be used with the simulator for machine learning. WeBots
+is also an integrated simulator and frameworks such as
+Deepbots[12] help the simulator to be used for machine
+learning. The machine learning features of these simulators
+are external features that have been developed within the
+community. While it is possible to use these simulators
+for both execution and learning purposes, they have not
+been designed for integrated learning-to-execution but rather
+using for one-or-the-other purpose. That is, these simulators
+are not designed to connect learning-and-execution, rather,
+learning and execution are separate use cases where one uses
+a community provided wrapper for machine learning, and the
+other uses the integrated features to simulate a robotic system
+execution.
+Compared to the existing simulators, the task-sequencing
+simulator was designed to connect simulation-for-learning
+and simulation-for-execution The simulator uses a con-
+cept model which enables composition of pre-trained, pro-
+grammed, or trained tasks, which is a powerful feature for
+going from machine learning to real robot execution. (e.g.,
+such as plugging-in to machine learning platforms but then
+connecting to execute on ROS). More importantly, tied-
+integration allows features such as training using pre-sequent
+and post-sequent task executions, but also collecting reusable
+execution modules through training.
+III. TASK-SEQUENCING SIMULATOR OVERVIEW
+The task-sequencing simulator has two layers: the Concept
+Interface for “action decision” and the Environment Engine
+Pipeline for “state observing” (Figure 1). However, unlike
+a typical learning simulator, where a specific problem has
+a non-decomposable structure and the action decision is
+a single policy being updated as data is collected for the
+problem, the task-sequencing simulator adds an abstraction
+to this action decision so that the problem is composed of
+a sequence of tasks (i.e., switches between a collection of
+tasks, where each task runs an individual policy), This way,
+a learned task policy can become part of a collection of
+policies for execution once the training has finished. Further
+details of each layer are described below.
+A. concept interface
+At each simulation time-step, a robot decides the next
+action depending on the current state of the world. This
+decision is referred to as a policy. When the relation be-
+tween the state, action, and next state (system dynamics) is
+completely known, this policy can be directly programmed.
+When the system dynamics are unknown, either the learning
+of the policy or system dynamics is required through data
+collection. Data collection is efficient if collected only for
+the unknown dynamics and if known dynamics are directly
+computed. Therefore, it is preferable to break down a robot’s
+execution to a series of tasks, where each task executes its
+own policy optimal for the system dynamics the task is cover-
+ing. In addition, breaking down a robot’s execution increases
+the reusability of each task policy and allows composing
+different execution scenarios from the task building blocks.
+The Concept Interface layer chooses and switches between
+the tasks for a training or execution scenario assuming
+(1) the series of tasks to simulate is known (ways for
+knowing are shown in the experiments), and (2) each task
+policy can indicate when the task has been completed (i.e.,
+has a learned or programmed completion signal). During
+simulation-for-execution, the Concept Interface layer chooses
+the according task in a sequence and switches to the next
+task once the current policy returns a task completion signal.
+The simulation-for-learning can be conducted in a similar
+way, except, the policy’s output of the completion signal is
+evaluated (e.g., by the success of a subsequent task). This
+interchangeable structure enables an integrated learning-to-
+execution simulation. To learn the termination signal under
+an arbitrary training sequence, the tasks share a unified
+design called a “concept model,” which is further explained
+in the later sections.
+B. environment engine pipeline
+At a time-step control scale, a robot behaves under a
+sense-plan-act with a physical embodiment. Thus, simulation
+in robotics requires three important engines: kinematics,
+physics (contact dynamics), and rendering. The role of
+
+the kinematics engine is to do the simulation between the
+robot’s action plan and the movement of the actual body.
+The role of the physics engine is to do the simulation
+between the robot’s body and the environment. The role of
+the rendering engine is to do the simulation between the
+environment and the robot’s sensing. In more technical terms,
+the kinematics engine solves the mapping between cartesian
+space and configuration space, the physics engine solves the
+differential algebraic equation[13] using techniques such as
+velocity-impulse linear complementarity-based time-stepping
+methods[14][15], the rendering engine solves the rendering
+equation[16] about lighting paths for pixel color generation.
+While rendering engines simulate the robot’s sensing by
+generating images, there is a gap between sensing and
+perceiving (extracting meaningful states from the gener-
+ated images). Rather than directly learning the sensing-to-
+planning, sometimes it is more efficient to perceive-before-
+planning and extract visual features. Moreover, in robotics
+it is important to combine both visual and force feedback;
+the visual features help compress the feedback so that the
+vision and force have an aligned state dimension. Thus, the
+proposed simulator adds a fourth “post-process engine” in
+conjunction with the rendering engine.
+These different-role engines are triggered in an ordered
+pipeline to calculate the current state of the simulation world.
+Often, simulators package specific engines to produce a
+single simulation world, but in general, these engines could
+run separately and combine/orchestrate multiple simulation
+worlds to produce better simulation. For example, ROS
+MoveIt could be used for accurate inverse kinematics sim-
+ulation, PyBullet for reproducible physics simulation, and
+the Unreal Engine for photo-realistic ray-traced rendering.
+Combining different engines is possible as long as each
+engine is able to load the same models of the robot/objects
+and is able to share the robot and object states among each
+engine (which can be done using TCP connections etc.).
+In addition, instead of using simulation engines, it is also
+possible to connect “real” engines which replace simulated
+physics with the real robot’s torque sensors and simulated
+rendering with real images from the robot’s camera.
+IV. CONCEPT MODELS
+A single task block operates under some system dynamics
+and achieves a goal state from an initial state. Thus, the de-
+tails of a task can be described using the actors of the system,
+an initial state, a goal (end) state, and the parameters of the
+system dynamics (Figure 2). This kind of task description
+can be referred to as a “task model”[17]. This description
+is enough for executing a task if the system dynamics are
+completely known. The initial state is usually the end state
+of the previous task. However, when the dynamics are not
+fully known, learning is required, and during learning, the
+initial state must be randomized.
+Instead of fully describing the task, a task can be described
+using actor configurations, an initial state, a necessary goal
+state that is described from observable system states, a
+sufficient goal state that is described from non-observable
+Task Model
+initial state
+Action
+Programmed policy
+Actors
+obtained before execution
+Parameters
+completely known and obtained 
+before execution
+defined goal state
+Fig. 2.
+An illustration of a task model.
+Concept Model
+initial state
+Action
+Policy
+Actors
+a set of configurated settings
+Parameters
+partially known and randomized at 
+training, estimated at execution
+necessary goal state
+sufficient goal state
+Fig. 3.
+An illustration of a concept model.
+system states, and the partially known parameters of the
+system dynamics (Figure 3). This kind of task description
+will be referred to as a “concept model” which compared to
+the task model may not be concrete enough for execution, but
+describes the concepts of the task to learn the execution. In
+the special case where the system dynamics are fully known
+and the task is programmable, the descriptions of the concept
+model is identical to the task model.
+By providing a structured description format, the Con-
+cept Interface layer can access these blocks interchangeably
+within a training sequence as the structures are the same and
+only differ in the details.
+A. concept model usage in learning
+The concept model descriptions are used to learn the
+task completion signal as well as the actions to handle the
+unknown system dynamics. The necessary goal state and
+sufficient goal state descriptions are used to evaluate whether
+the task completion signal and actions are appropriate. The
+evaluation is done by minimizing the cost of the current
+states to the goal states. Note that the cost to the necessary
+goal state is evaluated after every action decision, whereas
+the cost to the sufficient goal state is only evaluated once a
+task completion signal is chosen during training.
+The actor configurations describe the possible environ-
+ments (the world including the robot and any manipulating
+target) for training the task. If there are no pre-sequent tasks
+involved for training, then one of the actor configurations is
+used to define the initial state of the tasks. Otherwise, the
+end state of the previous task is the initial state. Unlike the
+initial state, the actor configuration is independent from the
+states of the previous task, thus is configurable and can be
+used for randomizing the states for training.
+
+B. concept model usage in execution
+When the dynamics are fully known, the concept model
+acts the same as the task model. The state of the task changes
+from the initial state using actions from a programmed policy
+based on the system dynamics and actors. The initial state
+of the task during execution is the end state of the previous
+task and the task ends once the goal state is achieved.
+When the dynamics are not fully known, the observable
+system states of the task changes using a learned policy.
+Similar to the programmed case, the initial state is the
+observable states at the end of the previous task. However,
+since part of the goal state is non-observable, the end of the
+task cannot be identified just with the model descriptions.
+Instead, the learned task completion signal from the concept
+model descriptions is used to identify the end of the task.
+V. MODEL IMPLEMENTATIONS
+By following the concept model structure, a task is im-
+plemented in a way that can be trained but then collected
+as a building block for execution. In this article, the screw
+theory[18] based separation of dynamics[17] is used to
+separate a task from another task. That is, once the relation
+between the manipulation target and the robot’s end-effector
+is initialized, a task breaks or maintains a contact state
+between the target and the environment. By classifying the
+inequality equation patterns of contact points, this leads to
+seven pure translation tasks and seven pure rotation tasks.
+Figure 4 shows implementation examples of some of these
+tasks as concept models. Below describes the details of some
+of the examples in the figure. Note that as the task classifica-
+tion only depends on the relation between the end-effector,
+manipulation target, and the environment, the movement of
+the arm (configuration space) can be ignored[19] and the task
+only focuses on the movement of the end-effector (cartesian
+space).
+A. grasping
+The
+grasp
+task
+initiates
+the
+relation
+between
+the
+manipulation-target and the robot’s end-effector. The actors
+are a target object, an environment (e.g., table), and the
+end-effector. The initial state is where the target object is
+attached to the environment but not attached to the end-
+effector (including shape of the finger joints). The goal state
+is where the target object is also attached to the end-effector
+in a way such that enough force is exerted for performing a
+subsequent task. The parameters are the distance between the
+target and end-effector as well as the approaching direction
+of the end-effector.
+When details of the target object are completely known, a
+task model can be defined and programmed from the above
+details. However, in the real world, there is uncertainty in the
+shape of the object and distance to the target, distinguishing
+whether enough force is exerted is intractable due to the
+inaccuracy in the real contact sensors or lack of sensors to
+detect slipping, a grasp-failure due to finger-object collision
+during approach may occur if the policy is not carefully
+designed under the uncertainties of the object properties.
+Grasp Concept Model
+freed OBJ-EEF
+Action
+Policy
+Actors
+end-effector (EEF), environment (ENV), a set 
+of objects (OBJ) with randomized shapes
+Parameters
+estimated approach distance OBJ-EEF,
+approach direction of EEF
+NGS: attached OBJ-EEF
+SGS: next task success
+Open Concept Model
+initial OBJ pose
+Action
+Policy
+Actors
+end-effector (EEF), environment (ENV), a set 
+of objects (OBJ) with randomized parameters
+Parameters
+estimated radius, axis center, axis direction 
+to maintain constraint OBJ-ENV
+NGS: maintain OBJ-ENV
+SGS: desired OBJ pose
+Pick Concept Model
+attached OBJ-ENV
+Action
+Programmed
+Actors
+end-effector (EEF), environment (ENV), target
+object (OBJ)
+Parameters
+detach direction to break constraint OBJ-ENV
+freed OBJ, ENV
+Bring Concept Model
+initial OBJ pose
+Action
+Programmed
+Actors
+end-effector (EEF), target object (OBJ)
+Parameters
+move direction and distance of OBJ
+desired OBJ pose
+Place Concept Model
+freed OBJ, ENV
+Action
+Programmed
+Actors
+end-effector (EEF), environment (ENV), target
+object (OBJ)
+Parameters
+attach direction to create constraint OBJ-ENV
+attached OBJ-ENV
+Release Concept Model
+attached OBJ-EEF
+Action
+Programmed
+Actors
+end-effector (EEF), target object (OBJ)
+Parameters
+release distance OBJ-EEF,
+release direction of EEF
+freed OBJ, EEF
+Pour Concept Model
+initial ENV state
+Action
+Policy
+Actors
+end-effector (EEF), set of environment (ENV) 
+and object (OBJs) with randomized parameters
+Parameters
+estimated ENV (cup) filled state,
+estimated OBJ (pitcher) size,
+estimated axis center, direction of OBJ-ENV
+NGS: maintain OBJ-ENV
+SGS: desired ENV state
+Wipe Concept Model
+initial ENV state
+Action
+Policy
+Actors
+end-effector (EEF), set of environment (ENV) 
+and object (OBJs) with randomized parameters
+Parameters
+estimated ENV (plane) clean state,
+estimated normal axis of OBJ-ENV
+NGS: maintain OBJ-ENV
+SGS: desired ENV state
+Fig. 4. Example concept models of eight different tasks in the screw-theory
+based classification.
+Instead, a set of actor configurations is defined as object
+shapes from a randomized range, a necessary goal state is
+defined as the end-effector to be in contact with the target
+object on an appropriate surface (which can be obtained on
+the real robot with the finger configurations and finger-torque
+sensors with a threshold to determine a binary contacted-or-
+not-contacted state), a sufficient goal state is defined as a
+successful performance of a subsequent task, the estimated
+distance is used for the parameters. The defined goal states
+are used to formulate the reward (cost-to-go) for learning the
+policy.
+Using this concept model, the approaching strategy (ad-
+justed movement around the approaching direction of the
+end-effector) and the enough amount of “closing” of the
+fingers to perform a subsequent task is learned. The learned
+policy chooses the sufficient amount based on the object
+shape which can partially be inferred by the shape of the end-
+effector finger joints once touching the object. The policy
+returns a termination signal once reached the enough amount
+of closing.
+B. door-opening
+The door-opening task is a one degrees-of-freedom pure
+rotation task. The actors are a target object (the door), an
+environment (the hinge), and the end-effector (attached to the
+
+door handle). The initial state is where the end-effector and
+target object are at an attached state. The goal state is where
+the target object has moved to some desired orientation. The
+parameters are the rotation radius, the rotation axis center,
+and the rotation axis direction defined by the target and
+environment.
+When details of the target and environment are completely
+known, a task model can be defined and programmed from
+the above details. However, in the real world, there is
+uncertainty in the environment parameters. Instead, a set of
+actor configurations randomizing the radius, a necessary goal
+state that moves the object along the environment constraint
+at each time-step (which can be obtained on the real robot
+by using a force sensor on the wrist and checking against
+a maximum-stress threshold), a sufficient goal state that
+ensures the door has reached the desired orientation, and
+estimation of the parameters are used to describe the concept
+model.
+By using an end-effector with only force-sensor feedback
+on the wrist, this model enables learning a policy which
+updates parameter estimations at each time-step, and then
+generates a hand motion based on the updated parameters.
+The policy returns a termination signal once inferred that the
+desired orientation has been reached.
+C. bringing
+The bringing task is a six degrees-of-freedom translation
+and rotation task. The actors are the target object and the
+end-effector. The initial state is where the end-effector and
+target object are at an attached state. The goal state is where
+the target object has moved to some relative positioning. The
+parameters are the moving direction and distance.
+Since there are no uncertainties in the target or environ-
+ment, the parameters can be manually specified and the goal
+can be directly specified from the parameters, Thus, the
+concept model is identical to the task model and can be
+programmed.
+VI. EXPERIMENTS
+Experiments were conducted using the concept model
+implementations shown in the previous section, and the
+developed task-sequencing simulator.
+For the learning experiments, the series of tasks to simulate
+were predefined. These experiments were performed to show
+the effectiveness of the simulator and its reusability for
+training different tasks.
+By running the pre-sequent tasks of the task-to-train at
+the start of an episode, and by running the subsequent tasks
+at reward return, the simulator is compatible with common
+reinforcement learning platforms, and utilizes off-the-shelf
+learning algorithms. For the experiments, the simulator was
+connected to the Bonsai platform and used the PPO algo-
+rithm.
+For the execution experiments, the series of tasks were
+obtained from human demonstrations and the actions were
+generated using the learned task policies. These experiments
+were performed to show the effectiveness of the simulator for
+execution simulations (execution of different composed sce-
+narios). Note that the exact same simulator and task blocks
+used for training was used for this experiment, showing
+the simulator’s capability to transition from simulation-for-
+learning to simulation-for-execution.
+For simulation, states were obtained by plugging to the
+environment engine pipeline the PyBullet engine for the
+physics engine role, and the Unreal Engine for rendering. For
+the real robot, states were obtained plugging ROS (connected
+via roslibpy) to the environment engine pipeline. For the arm
+kinematics, the ROS MoveIt package was used.
+A. training
+Figure 5-(A) shows the task-sequencing simulator config-
+urations for running the grasp training. A configured training
+sequence “grasp then pick” is passed to the simulator. The
+“grasp” task is the task-to-train, and a programmed “pick”
+is used as the subsequent task for evaluating the sufficient
+goal state.
+Figure 5-(B) shows the trained grasp results performed
+on a real robot. The concept model was designed with
+the object shape parameters as unknown. Regardless of
+such uncertainty, the learned policy successfully grasps the
+different shaped objects including but not limited to a box,
+a cylindrical cup, an oval rice pack, and a diamond-shaped
+candy box.
+Figure 5-(C) shows a learned grasp for a different robot
+hand, which was trained using the same simulator and
+concept models but with a different actor configuration
+(end-effector) setting. The results show the reusability of
+the simulator for training different robots with different
+mechanics (a hand with multiple fingers and a gripper with
+limited degrees of freedom).
+Figure 5-(D) shows that by changing the configured train-
+ing sequence to “grasp then open,” the door-opening task is
+trained using the same simulator. The “open” task is the task-
+to-train, and the trained “grasp” is reused as the pre-sequent
+task for initiating the relation between the end-effector and
+the target door. Regardless of uncertainty in the rotation
+radius, center, and axis direction, the real robot performed the
+door-opening using the learned policy. Although the policy
+was trained only using simulated data, the policy is directly
+applicable to the real robot as the sufficient goal state does
+not require observability on the real robot and because the
+policy action decisions only rely on the states with very small
+sim-to-real gaps.
+B. execution
+Figure 6-(A) shows the task-sequencing simulator used
+with a demonstrated sequence by a human. Instead of a
+configured sequence as in the previous training experiments,
+the sequence is automatically generated through demonstra-
+tion decomposition using the method described in [20]. The
+same concept models from the training experiments are used
+with the policy-update being disabled (the simulator is not
+connected to any training algorithm and instead uses a fixed
+learned policy without updates).
+
+Task-sequencing simulator
+Concept Interface
+Environment Engine Pipeline
+CM Grasp
+state
+action
+configured 
+training sequence
+CM Pick
+training 
+algorithms
+(A)
+(B)
+(C)
+connect learned policy 
+to the real robot
+train policy on a 
+different robot hand
+CM Grasp
+CM Open
+training sequence 
+for door-opening
+(D)
+configure to a different 
+training scenario
+Fig. 5.
+Results of the task-sequencing simulator when used for learning.
+Task-sequencing simulator
+Concept Interface
+Environment Engine Pipeline
+CM Grasp
+state
+action
+demonstrated execution sequence
+CM Pick
+(A)
+simulation
+CM Bring
+CM Bring
+CM Place
+CM Release
+real
+another 
+scenario
+physics
+rendering
+(B)
+(C)
+(D)
+Fig. 6.
+Results of the task-sequencing simulator when used for execution.
+Figure 6-(B) shows a simulated execution of the demon-
+strated sequence. The first row shows the outputs of the
+physics engine and the second row shows the outputs of
+the rendering engine. As both training and execution run
+on the same system, the learned policy can easily be used
+as a simulation-for-execution. The learned policy is already
+a building block that can be combined with other tasks to
+generate an application such as “pick up a cup from the upper
+shelf and re-place it to the bottom shelf.”
+Figure 6-(C) shows a real robot execution of the demon-
+strated sequence by switching the engines in the environment
+engine pipeline to connect with ROS. This shows how the
+simulator can go from the simulated robot execution to the
+real robot execution by using the same policy connections
+but by changing the engines in which the states are obtained,
+and the actions are performed against. Usually, going from
+simulation to real introduces a sim-to-real gap. However,
+only part of the scenario sequence uses a learned policy and
+due to the careful design of the concept models to divide
+learning observable dynamics (necessary goal states) from
+learning hidden dynamics (sufficient goal states), no such
+gap was encountered.
+Figure 6-(D) shows an execution of a different sequence
+“pick up a cup from the table and throw it in the trash.”
+This scenario uses the same concept models and only differs
+in the demonstrated input, showing how using the simulator
+and concept model descriptions enable reusing the learned
+policies for different execution scenarios. If a policy was
+learned against a full “pick-and-place” scenario, the policy
+would not easily scale to the “pick-and-throw” scenario as
+the problem dynamics are different.
+
+VII. CONCLUSIONS
+This article introduced the task-sequencing simulator
+which bridges simulation-for-learning to simulation-for-
+execution. The simulation scenario for learning is created us-
+ing a sequence of tasks. This way the simulation-for-learning
+has the same structure as the simulation-for-execution. At
+its core, the simulator uses a concept model which enables
+sequencing mixed programmed, trained, and under-training
+building blocks. While the simulator has a large advantage
+in terms of integrated system development, the simulator
+also provides new directions for simulation in execution and
+simulation in learning.
+From an execution perspective, the simulator allows com-
+posing a task-sequence using both programmed and trained
+tasks. Unlike programmed-only sequences, the advantage of
+mixing trained blocks is that, some of the tasks can contain
+uncertainty and the goal state of a task can be described using
+implicit system parameters (the goal state does not have to
+be obtained directly from the real robot). The key is that,
+whether the observed state and selected actions suffice the
+goal state is learned as a termination signal through training.
+From a learning perspective, the simulator and concept
+model design have the following advantages: First, the sim-
+ulation is reusable and easily applicable to slight changes
+in the scenario. A policy for a different end-effector can
+be learned by just changing the actor configurations in the
+concept model. A policy can be optimized for different
+scenarios by just changing the subsequent task in the suffi-
+cient goal state. Second, defining the learning problem using
+the concept model design enables a hierarchical learning-
+structure as well as a structure for reducing sim-to-real gaps.
+Any state parameters that do not have a large gap when
+observed with the real robot is used for defining the necessary
+goal state, whereas any state parameters that have a large gap
+when observed with the real robot is a sufficient goal state
+(implicitly learned in simulation but no need to be observed
+with the real robot). This type of formulation is possible
+as only the parts with uncertainty are being learned instead
+of learning the entire scenario sequence. Following this
+structured formulation has allowed going from simulation
+to real without any extra real-world data collection and
+achieving a reusable policy applicable to different execution
+scenarios.
+ACKNOWLEDGMENT
+The authors thank Brice Chung’s team, Aydan Aksoylar
+and Kartavya Neema for their help in the reward designs and
+training of the concept models used in the experiments.
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+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf,len=349
+page_content='Task-sequencing Simulator: Integrated Machine Learning to Execution  Simulation for Robot Manipulation  Kazuhiro Sasabuchi1, Daichi Saito1, Atsushi Kanehira1, Naoki Wake1, Jun Takamatsu1, Katsushi Ikeuchi1  Abstract— A task-sequencing simulator in robotics manip-  ulation to integrate simulation-for-learning and simulation-  for-execution is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Unlike existing machine-learning  simulation where a non-decomposed simulation is used to  simulate a training scenario, the task-sequencing simulator runs  a composed simulation using building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This way, the  simulation-for-learning is structured similarly to a multi-step  simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' To compose both learning and execu-  tion scenarios, a unified trainable-and-composable description  of blocks called a concept model is proposed and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Using the  simulator design and concept models, a reusable simulator for  learning different tasks, a common-ground system for learning-  to-execution, simulation-to-real is achieved and shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' INTRODUCTION  Simulators are important in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Compared to the  real world, simulators can run endlessly, safely, remove  stochasticity, and provide ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' One direction  for using simulators in robotics is to help check collisions  and safe executions before real robot executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Another  direction is using simulators as a tool for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Simulators that suit either one of the above directions are  available today, however, it is desirable to have a simulator  which fulfills both purposes to better integrate learning and  execution, especially in the multi-step manipulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  In multi-step manipulation, a series of tasks which occur  sequentially must be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' An example of a sequential  series of tasks is bringing an object, which is composed  of tasks: grasp, pick, bring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' An execution simulator must  be able to trigger these different tasks and connect them  into a sequenced simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' When the simulator is used  for checking robot executions, the simulation is a matter of  combining programmed or trained building blocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', run  “grasp” then “pick” then “bring”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  In contrast, machine learning simulators often ignore the  task-sequence composition and are structured to train a  specific problem or benchmark (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', a non-decomposed  “pick-and-place” simulation)[1][2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This structural differ-  ence causes a gap between simulation-for-execution and  simulation-for-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The learned results become specific  to the trained scenario and contradicts with the simulation-  for-execution where scenarios are non-fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead, a machine learning simulator could be designed  similar to an execution simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The “pick-and-place”  scenario can be decomposed into a sequenced simulation  of “grasp then pick then bring then place then release.”  The difference compared to the execution simulator is that  1All  authors  are  with  Microsoft,  Redmond,  WA,  USA  Kazuhiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='Sasabuchi@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='com  Physics  Kinematics  Rendering  Post-process  Task-sequencing simulator  Concept Interface  Environment Engine Pipeline  CM Grasp  state  action  selected task  at step t  demonstrated  execution sequence  configured  training sequence  CM Open  CM Release  training  algorithms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The proposed task-sequencing simulator which enables scenario  composition for both learning and execution in robotics manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  some of these blocks are “under-training” and are updated  as data is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Once the update has finished, the trained  block can be combined and reused for a different scenario,  thus, the simulation-for-learning can directly transition to the  simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In addition, such design enables  learning new manipulation skills on top of programmed or  prior-trained building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For example, a “grasp” could  be trained using the sequence “grasp then pick,” where  “pick” is a programmed task to provide a supervised signal  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', teach that the “grasp” was successful if the “pick” was  successful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  This article introduces a task-sequencing simulator struc-  ture which enables integrated learning-to-execution simula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' At its core, the simulator uses a unified block design  called the “concept model,” which is proposed within this  article and defines the necessary descriptions for training  a task, collecting trained tasks, and running the tasks to  compose a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The rest of the article is outlined as below: Section II  provides a background on existing robotic simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Sec-  tion III explains the overall simulator structure for achieving  machine learning to robot execution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Section IV  explains the concept model core component of the simulator  and Section V provides some detailed example implementa-  tions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Section VI shows the capabilities of the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='01382v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='RO]  3 Jan 2023  C16simulator in machine-learning-to-robot-execution followed  by conclusions in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' BACKGROUND  While there are many existing simulators for robotics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  existing simulators may not achieve the integrated learning-  to-execution multi-step manipulation purpose for one of  the following reasons: 1) the simulator targets a different  domain other than manipulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 2) the simulator can be  used for manipulation but misses a capability in simulation-  for-learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 3) the simulator can be used for manipulation  but misses a capability in simulation-for-execution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 4) the  simulator can be used for manipulation and both learning-  and-execution purposes but not specifically for learning-to-  execution purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Popular  robotics  simulators[3][4]  include  Gazebo[5],  MuJoCo[6], CoppeliaSim[7], CARLA[8], AirSim[9], and  Webots[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Gazebo has its advantage in its capability to  simulate executions using ROS integrated sensors and ac-  tuators but is not the best choice when it comes to data  collection and machine learning due to its slow simulation  performance and inconsistency in physics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus,  Gazebo falls into the second category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Engines like MuJoCo  on the other hand, are suitable for stable physics simulation  in machine learning but miss some robotics simulation  capabilities such as inverse-kinematics and visual feedback  (realistic rendering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The focus is on physics simulation  rather than an integrated simulator for robot executions,  therefore falls into the third category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' CARLA and Airsim  mainly target automobiles such as drones and cars therefore  miss some important features such as kinematics required for  manipulation and falls into the first category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The CoppeliaSim is an integrated simulator with a kine-  matics and physics engine, and the PyRep toolkit[11] can  be used with the simulator for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' WeBots  is also an integrated simulator and frameworks such as  Deepbots[12] help the simulator to be used for machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The machine learning features of these simulators  are external features that have been developed within the  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' While it is possible to use these simulators  for both execution and learning purposes, they have not  been designed for integrated learning-to-execution but rather  using for one-or-the-other purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' That is, these simulators  are not designed to connect learning-and-execution, rather,  learning and execution are separate use cases where one uses  a community provided wrapper for machine learning, and the  other uses the integrated features to simulate a robotic system  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Compared to the existing simulators, the task-sequencing  simulator was designed to connect simulation-for-learning  and simulation-for-execution The simulator uses a con-  cept model which enables composition of pre-trained, pro-  grammed, or trained tasks, which is a powerful feature for  going from machine learning to real robot execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=',  such as plugging-in to machine learning platforms but then  connecting to execute on ROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' More importantly, tied-  integration allows features such as training using pre-sequent  and post-sequent task executions, but also collecting reusable  execution modules through training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' TASK-SEQUENCING SIMULATOR OVERVIEW  The task-sequencing simulator has two layers: the Concept  Interface for “action decision” and the Environment Engine  Pipeline for “state observing” (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, unlike  a typical learning simulator, where a specific problem has  a non-decomposable structure and the action decision is  a single policy being updated as data is collected for the  problem, the task-sequencing simulator adds an abstraction  to this action decision so that the problem is composed of  a sequence of tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', switches between a collection of  tasks, where each task runs an individual policy), This way,  a learned task policy can become part of a collection of  policies for execution once the training has finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Further  details of each layer are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' concept interface  At each simulation time-step, a robot decides the next  action depending on the current state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This  decision is referred to as a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' When the relation be-  tween the state, action, and next state (system dynamics) is  completely known, this policy can be directly programmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When the system dynamics are unknown, either the learning  of the policy or system dynamics is required through data  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Data collection is efficient if collected only for  the unknown dynamics and if known dynamics are directly  computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Therefore, it is preferable to break down a robot’s  execution to a series of tasks, where each task executes its  own policy optimal for the system dynamics the task is cover-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In addition, breaking down a robot’s execution increases  the reusability of each task policy and allows composing  different execution scenarios from the task building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The Concept Interface layer chooses and switches between  the tasks for a training or execution scenario assuming  (1) the series of tasks to simulate is known (ways for  knowing are shown in the experiments), and (2) each task  policy can indicate when the task has been completed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=',  has a learned or programmed completion signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' During  simulation-for-execution, the Concept Interface layer chooses  the according task in a sequence and switches to the next  task once the current policy returns a task completion signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The simulation-for-learning can be conducted in a similar  way, except, the policy’s output of the completion signal is  evaluated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', by the success of a subsequent task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This  interchangeable structure enables an integrated learning-to-  execution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' To learn the termination signal under  an arbitrary training sequence, the tasks share a unified  design called a “concept model,” which is further explained  in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment engine pipeline  At a time-step control scale, a robot behaves under a  sense-plan-act with a physical embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus, simulation  in robotics requires three important engines: kinematics,  physics (contact dynamics), and rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The role of  the kinematics engine is to do the simulation between the  robot’s action plan and the movement of the actual body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The role of the physics engine is to do the simulation  between the robot’s body and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The role of  the rendering engine is to do the simulation between the  environment and the robot’s sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In more technical terms,  the kinematics engine solves the mapping between cartesian  space and configuration space, the physics engine solves the  differential algebraic equation[13] using techniques such as  velocity-impulse linear complementarity-based time-stepping  methods[14][15], the rendering engine solves the rendering  equation[16] about lighting paths for pixel color generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  While rendering engines simulate the robot’s sensing by  generating images, there is a gap between sensing and  perceiving (extracting meaningful states from the gener-  ated images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Rather than directly learning the sensing-to-  planning, sometimes it is more efficient to perceive-before-  planning and extract visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Moreover, in robotics  it is important to combine both visual and force feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  the visual features help compress the feedback so that the  vision and force have an aligned state dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus, the  proposed simulator adds a fourth “post-process engine” in  conjunction with the rendering engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  These different-role engines are triggered in an ordered  pipeline to calculate the current state of the simulation world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Often, simulators package specific engines to produce a  single simulation world, but in general, these engines could  run separately and combine/orchestrate multiple simulation  worlds to produce better simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For example, ROS  MoveIt could be used for accurate inverse kinematics sim-  ulation, PyBullet for reproducible physics simulation, and  the Unreal Engine for photo-realistic ray-traced rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Combining different engines is possible as long as each  engine is able to load the same models of the robot/objects  and is able to share the robot and object states among each  engine (which can be done using TCP connections etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  In addition, instead of using simulation engines, it is also  possible to connect “real” engines which replace simulated  physics with the real robot’s torque sensors and simulated  rendering with real images from the robot’s camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' CONCEPT MODELS  A single task block operates under some system dynamics  and achieves a goal state from an initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus, the de-  tails of a task can be described using the actors of the system,  an initial state, a goal (end) state, and the parameters of the  system dynamics (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This kind of task description  can be referred to as a “task model”[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This description  is enough for executing a task if the system dynamics are  completely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is usually the end state  of the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, when the dynamics are not  fully known, learning is required, and during learning, the  initial state must be randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead of fully describing the task, a task can be described  using actor configurations, an initial state, a necessary goal  state that is described from observable system states, a  sufficient goal state that is described from non-observable  Task Model  initial state  Action  Programmed policy  Actors  obtained before execution  Parameters  completely known and obtained  before execution  defined goal state  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  An illustration of a task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Concept Model  initial state  Action  Policy  Actors  a set of configurated settings  Parameters  partially known and randomized at  training, estimated at execution  necessary goal state  sufficient goal state  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  An illustration of a concept model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  system states, and the partially known parameters of the  system dynamics (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This kind of task description  will be referred to as a “concept model” which compared to  the task model may not be concrete enough for execution, but  describes the concepts of the task to learn the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In  the special case where the system dynamics are fully known  and the task is programmable, the descriptions of the concept  model is identical to the task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  By providing a structured description format, the Con-  cept Interface layer can access these blocks interchangeably  within a training sequence as the structures are the same and  only differ in the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' concept model usage in learning  The concept model descriptions are used to learn the  task completion signal as well as the actions to handle the  unknown system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The necessary goal state and  sufficient goal state descriptions are used to evaluate whether  the task completion signal and actions are appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  evaluation is done by minimizing the cost of the current  states to the goal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Note that the cost to the necessary  goal state is evaluated after every action decision, whereas  the cost to the sufficient goal state is only evaluated once a  task completion signal is chosen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The actor configurations describe the possible environ-  ments (the world including the robot and any manipulating  target) for training the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' If there are no pre-sequent tasks  involved for training, then one of the actor configurations is  used to define the initial state of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Otherwise, the  end state of the previous task is the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Unlike the  initial state, the actor configuration is independent from the  states of the previous task, thus is configurable and can be  used for randomizing the states for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' concept model usage in execution  When the dynamics are fully known, the concept model  acts the same as the task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The state of the task changes  from the initial state using actions from a programmed policy  based on the system dynamics and actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state  of the task during execution is the end state of the previous  task and the task ends once the goal state is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When the dynamics are not fully known, the observable  system states of the task changes using a learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Similar to the programmed case, the initial state is the  observable states at the end of the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However,  since part of the goal state is non-observable, the end of the  task cannot be identified just with the model descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead, the learned task completion signal from the concept  model descriptions is used to identify the end of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' MODEL IMPLEMENTATIONS  By following the concept model structure, a task is im-  plemented in a way that can be trained but then collected  as a building block for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In this article, the screw  theory[18] based separation of dynamics[17] is used to  separate a task from another task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' That is, once the relation  between the manipulation target and the robot’s end-effector  is initialized, a task breaks or maintains a contact state  between the target and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' By classifying the  inequality equation patterns of contact points, this leads to  seven pure translation tasks and seven pure rotation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 4 shows implementation examples of some of these  tasks as concept models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Below describes the details of some  of the examples in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Note that as the task classifica-  tion only depends on the relation between the end-effector,  manipulation target, and the environment, the movement of  the arm (configuration space) can be ignored[19] and the task  only focuses on the movement of the end-effector (cartesian  space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' grasping  The  grasp  task  initiates  the  relation  between  the  manipulation-target and the robot’s end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The actors  are a target object, an environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', table), and the  end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is where the target object is  attached to the environment but not attached to the end-  effector (including shape of the finger joints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The goal state  is where the target object is also attached to the end-effector  in a way such that enough force is exerted for performing a  subsequent task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The parameters are the distance between the  target and end-effector as well as the approaching direction  of the end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When details of the target object are completely known, a  task model can be defined and programmed from the above  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, in the real world, there is uncertainty in the  shape of the object and distance to the target, distinguishing  whether enough force is exerted is intractable due to the  inaccuracy in the real contact sensors or lack of sensors to  detect slipping, a grasp-failure due to finger-object collision  during approach may occur if the policy is not carefully  designed under the uncertainties of the object properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Grasp Concept Model  freed OBJ-EEF  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a set  of objects (OBJ) with randomized shapes  Parameters  estimated approach distance OBJ-EEF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  approach direction of EEF  NGS: attached OBJ-EEF  SGS: next task success  Open Concept Model  initial OBJ pose  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a set  of objects (OBJ) with randomized parameters  Parameters  estimated radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' axis center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' axis direction  to maintain constraint OBJ-ENV  NGS: maintain OBJ-ENV  SGS: desired OBJ pose  Pick Concept Model  attached OBJ-ENV  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target  object (OBJ)  Parameters  detach direction to break constraint OBJ-ENV  freed OBJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' ENV  Bring Concept Model  initial OBJ pose  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target object (OBJ)  Parameters  move direction and distance of OBJ  desired OBJ pose  Place Concept Model  freed OBJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' ENV  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target  object (OBJ)  Parameters  attach direction to create constraint OBJ-ENV  attached OBJ-ENV  Release Concept Model  attached OBJ-EEF  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target object (OBJ)  Parameters  release distance OBJ-EEF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  release direction of EEF  freed OBJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' EEF  Pour Concept Model  initial ENV state  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' set of environment (ENV)  and object (OBJs) with randomized parameters  Parameters  estimated ENV (cup) filled state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  estimated OBJ (pitcher) size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  estimated axis center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' direction of OBJ-ENV  NGS: maintain OBJ-ENV  SGS: desired ENV state  Wipe Concept Model  initial ENV state  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' set of environment (ENV)  and object (OBJs) with randomized parameters  Parameters  estimated ENV (plane) clean state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  estimated normal axis of OBJ-ENV  NGS: maintain OBJ-ENV  SGS: desired ENV state  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Example concept models of eight different tasks in the screw-theory  based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a set of actor configurations is defined as object  shapes from a randomized range,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a necessary goal state is  defined as the end-effector to be in contact with the target  object on an appropriate surface (which can be obtained on  the real robot with the finger configurations and finger-torque  sensors with a threshold to determine a binary contacted-or-  not-contacted state),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a sufficient goal state is defined as a  successful performance of a subsequent task,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' the estimated  distance is used for the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The defined goal states  are used to formulate the reward (cost-to-go) for learning the  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Using this concept model, the approaching strategy (ad-  justed movement around the approaching direction of the  end-effector) and the enough amount of “closing” of the  fingers to perform a subsequent task is learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The learned  policy chooses the sufficient amount based on the object  shape which can partially be inferred by the shape of the end-  effector finger joints once touching the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The policy  returns a termination signal once reached the enough amount  of closing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' door-opening  The door-opening task is a one degrees-of-freedom pure  rotation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The actors are a target object (the door), an  environment (the hinge), and the end-effector (attached to the  door handle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is where the end-effector and  target object are at an attached state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The goal state is where  the target object has moved to some desired orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  parameters are the rotation radius, the rotation axis center,  and the rotation axis direction defined by the target and  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When details of the target and environment are completely  known, a task model can be defined and programmed from  the above details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, in the real world, there is  uncertainty in the environment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Instead, a set of  actor configurations randomizing the radius, a necessary goal  state that moves the object along the environment constraint  at each time-step (which can be obtained on the real robot  by using a force sensor on the wrist and checking against  a maximum-stress threshold), a sufficient goal state that  ensures the door has reached the desired orientation, and  estimation of the parameters are used to describe the concept  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  By using an end-effector with only force-sensor feedback  on the wrist, this model enables learning a policy which  updates parameter estimations at each time-step, and then  generates a hand motion based on the updated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The policy returns a termination signal once inferred that the  desired orientation has been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' bringing  The bringing task is a six degrees-of-freedom translation  and rotation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The actors are the target object and the  end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is where the end-effector and  target object are at an attached state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The goal state is where  the target object has moved to some relative positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  parameters are the moving direction and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Since there are no uncertainties in the target or environ-  ment, the parameters can be manually specified and the goal  can be directly specified from the parameters, Thus, the  concept model is identical to the task model and can be  programmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' EXPERIMENTS  Experiments were conducted using the concept model  implementations shown in the previous section, and the  developed task-sequencing simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  For the learning experiments, the series of tasks to simulate  were predefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' These experiments were performed to show  the effectiveness of the simulator and its reusability for  training different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  By running the pre-sequent tasks of the task-to-train at  the start of an episode, and by running the subsequent tasks  at reward return, the simulator is compatible with common  reinforcement learning platforms, and utilizes off-the-shelf  learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For the experiments, the simulator was  connected to the Bonsai platform and used the PPO algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  For the execution experiments, the series of tasks were  obtained from human demonstrations and the actions were  generated using the learned task policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' These experiments  were performed to show the effectiveness of the simulator for  execution simulations (execution of different composed sce-  narios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Note that the exact same simulator and task blocks  used for training was used for this experiment, showing  the simulator’s capability to transition from simulation-for-  learning to simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  For simulation, states were obtained by plugging to the  environment engine pipeline the PyBullet engine for the  physics engine role, and the Unreal Engine for rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For  the real robot, states were obtained plugging ROS (connected  via roslibpy) to the environment engine pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For the arm  kinematics, the ROS MoveIt package was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' training  Figure 5-(A) shows the task-sequencing simulator config-  urations for running the grasp training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' A configured training  sequence “grasp then pick” is passed to the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  “grasp” task is the task-to-train, and a programmed “pick”  is used as the subsequent task for evaluating the sufficient  goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 5-(B) shows the trained grasp results performed  on a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The concept model was designed with  the object shape parameters as unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Regardless of  such uncertainty, the learned policy successfully grasps the  different shaped objects including but not limited to a box,  a cylindrical cup, an oval rice pack, and a diamond-shaped  candy box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 5-(C) shows a learned grasp for a different robot  hand, which was trained using the same simulator and  concept models but with a different actor configuration  (end-effector) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The results show the reusability of  the simulator for training different robots with different  mechanics (a hand with multiple fingers and a gripper with  limited degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 5-(D) shows that by changing the configured train-  ing sequence to “grasp then open,” the door-opening task is  trained using the same simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The “open” task is the task-  to-train, and the trained “grasp” is reused as the pre-sequent  task for initiating the relation between the end-effector and  the target door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Regardless of uncertainty in the rotation  radius, center, and axis direction, the real robot performed the  door-opening using the learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Although the policy  was trained only using simulated data, the policy is directly  applicable to the real robot as the sufficient goal state does  not require observability on the real robot and because the  policy action decisions only rely on the states with very small  sim-to-real gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' execution  Figure 6-(A) shows the task-sequencing simulator used  with a demonstrated sequence by a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Instead of a  configured sequence as in the previous training experiments,  the sequence is automatically generated through demonstra-  tion decomposition using the method described in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  same concept models from the training experiments are used  with the policy-update being disabled (the simulator is not  connected to any training algorithm and instead uses a fixed  learned policy without updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Task-sequencing simulator  Concept Interface  Environment Engine Pipeline  CM Grasp  state  action  configured  training sequence  CM Pick  training  algorithms  (A)  (B)  (C)  connect learned policy  to the real robot  train policy on a  different robot hand  CM Grasp  CM Open  training sequence  for door-opening  (D)  configure to a different  training scenario  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Results of the task-sequencing simulator when used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Task-sequencing simulator  Concept Interface  Environment Engine Pipeline  CM Grasp  state  action  demonstrated execution sequence  CM Pick  (A)  simulation  CM Bring  CM Bring  CM Place  CM Release  real  another  scenario  physics  rendering  (B)  (C)  (D)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Results of the task-sequencing simulator when used for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 6-(B) shows a simulated execution of the demon-  strated sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The first row shows the outputs of the  physics engine and the second row shows the outputs of  the rendering engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' As both training and execution run  on the same system, the learned policy can easily be used  as a simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The learned policy is already  a building block that can be combined with other tasks to  generate an application such as “pick up a cup from the upper  shelf and re-place it to the bottom shelf.”  Figure 6-(C) shows a real robot execution of the demon-  strated sequence by switching the engines in the environment  engine pipeline to connect with ROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This shows how the  simulator can go from the simulated robot execution to the  real robot execution by using the same policy connections  but by changing the engines in which the states are obtained,  and the actions are performed against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Usually, going from  simulation to real introduces a sim-to-real gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However,  only part of the scenario sequence uses a learned policy and  due to the careful design of the concept models to divide  learning observable dynamics (necessary goal states) from  learning hidden dynamics (sufficient goal states), no such  gap was encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 6-(D) shows an execution of a different sequence  “pick up a cup from the table and throw it in the trash.”  This scenario uses the same concept models and only differs  in the demonstrated input, showing how using the simulator  and concept model descriptions enable reusing the learned  policies for different execution scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' If a policy was  learned against a full “pick-and-place” scenario, the policy  would not easily scale to the “pick-and-throw” scenario as  the problem dynamics are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' CONCLUSIONS  This article introduced the task-sequencing simulator  which bridges simulation-for-learning to simulation-for-  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The simulation scenario for learning is created us-  ing a sequence of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This way the simulation-for-learning  has the same structure as the simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' At  its core, the simulator uses a concept model which enables  sequencing mixed programmed, trained, and under-training  building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' While the simulator has a large advantage  in terms of integrated system development, the simulator  also provides new directions for simulation in execution and  simulation in learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  From an execution perspective, the simulator allows com-  posing a task-sequence using both programmed and trained  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Unlike programmed-only sequences, the advantage of  mixing trained blocks is that, some of the tasks can contain  uncertainty and the goal state of a task can be described using  implicit system parameters (the goal state does not have to  be obtained directly from the real robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The key is that,  whether the observed state and selected actions suffice the  goal state is learned as a termination signal through training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  From a learning perspective, the simulator and concept  model design have the following advantages: First, the sim-  ulation is reusable and easily applicable to slight changes  in the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' A policy for a different end-effector can  be learned by just changing the actor configurations in the  concept model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' A policy can be optimized for different  scenarios by just changing the subsequent task in the suffi-  cient goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Second, defining the learning problem using  the concept model design enables a hierarchical learning-  structure as well as a structure for reducing sim-to-real gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Any state parameters that do not have a large gap when  observed with the real robot is used for defining the necessary  goal state, whereas any state parameters that have a large gap  when observed with the real robot is a sufficient goal state  (implicitly learned in simulation but no need to be observed  with the real robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This type of formulation is possible  as only the parts with uncertainty are being learned instead  of learning the entire scenario sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Following this  structured formulation has allowed going from simulation  to real without any extra real-world data collection and  achieving a reusable policy applicable to different execution  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  ACKNOWLEDGMENT  The authors thank Brice Chung’s team, Aydan Aksoylar  and Kartavya Neema for their help in the reward designs and  training of the concept models used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
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+How to get the most out of Twinned Regression
+Methods
+Sebastian J. Wetzel
+University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
+Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
+Homes Plus Magazine Inc., Waterloo, Ontario N2V 2B1, Canada
+Abstract.
+Twinned regression methods are designed to solve the dual problem to
+the original regression problem, predicting differences between regression targets rather
+then the targets themselves. A solution to the original regression problem can be
+obtained by ensembling predicted differences between the targets of an unknown data
+point and multiple known anchor data points. We explore different aspects of twinned
+regression methods: (1) We decompose different steps in twinned regression algorithms
+and examine their contributions to the final performance, (2) We examine the intrinsic
+ensemble quality, (3) We combine twin neural network regression with k-nearest neighbor
+regression to design a more accurate and efficient regression method, and (4) we develop
+a simplified semi-supervised regression scheme.
+Keywords: Artificial Neural Networks, k-Nearest Neighbors, Random Forests, Regression,
+Semi-Supervised Learning
+arXiv:2301.01383v1  [cs.LG]  3 Jan 2023
+
+How to get the most out of Twinned Regression Methods
+2
+1. Introduction
+Regression is one of the most general and common machine-learning tasks, practitioners
+in many different fields of science and industry rely on methods that help them to make
+the most accurate and reliable predictions on new data points inferred from a limited
+amount of training data. Conventional regression methods aim to infer the mapping
+of input features to one or multiple target variables. Twinned regression methods aim
+to solve the dual problem of predicting the difference between target values, a solution
+to the original regression problem can then be obtained by evaluating the predictions
+between a new unknown data point and multiple anchor data points. This process creates
+an ensemble of predictions from a single trained model, which tends to be more accurate
+than solving the original regression problem directly and opens up semi-supervised
+learning and uncertainty estimates for a very low cost [1, 2, 3]. However, the trade-off
+is the poor scaling with large data sets, since the effective training data set size scales
+quadratically with the size of the original training data set.
+Thus, these methods are beneficial in domains where data is either scarce or costly
+to obtain. This is the case in for example real estate where markets differ from city to city
+and data becomes outdated quickly [4, 5, 6]. Another example stems from calculations
+in chemistry where simulations of chemical systems based on quantum mechanics require
+an enormous amount of computational resources [7, 8].
+This article is meant to be a practitioner’s guide to using twinned regression methods
+that guides the reader through advantages and trade-offs and attempts to answer most
+questions that were left open in the recent years.
+The main question on our mind is why can such a simple trick of solving the dual
+problem yield a more accurate prediction than solving the regression problem directly.
+While we do not fully answer this question, we decomposed twin neural network regression
+(TNNR) into different steps each with the potential to enhance the performance over
+traditional algorithms. These include increased effective data set size obtained though
+pairing training data points, or different ensembles of TNNR predictions. Further, by
+mapping extreme cases of TNNR to k-nearest neighbor (k-NN) regression and normal
+artificial neural networks (ANN) we can observe a distinct performance behaviour of
+twinned regression methods different from traditional regression.
+Further, we are eager to improve the accuracy of twinned regression methods. For
+this purpose, we devise an improvement to TNNR based on the idea of weighting the
+predictions from different anchors. This leads us to combine k-nearest neighbors (kNN)
+with TNNR to an even more accurate regression scheme.
+The semi-supervised regression framework for TNNR invented in [2] is specifically
+tailored to neural network-based regression. It is based on enforcing consistency conditions
+on unknown data points through a modified loss function. At the end of this manuscript
+we examine a way to translate a simplified version of this semi-supervised learning scheme
+to a twinned version of random forests (RF) proposed in [3].
+In some projects it is important to apply neural networks with strong memory
+
+How to get the most out of Twinned Regression Methods
+3
+constraints, this might be the case in small chips or autonomous systems [9, 10]. In
+these cases it would normally be very inefficient to store ensembles of machine learning
+models due to the increased number of parameters. In contrast to that, with twinned
+regression methods, one only needs to store additional anchor data points.
+2. Prior Work
+The pairwise comparison inherent to twinned regression methods is inspired by Siamese
+neural networks which were devised to solve the similarity classification problem as
+it occurs in fingerprint recognition or signature verification [11, 12]. Siamese neural
+networks contain two identical neural networks with shared weights which project a
+pair of inputs into a latent space on which the pairwise similarity is determined by the
+distance. Twinned regression methods also take a pair of inputs to predict the difference
+between the labels [1].
+Twin neural network regression [1] was invented as a regression method that solves
+the dual problem of predicting pairwise differences between the target values of pairs of
+input data points. Independently, the same idea has been developed for random forests
+[3]. This kind of regression framework has been shown to have several advantages: (1) it
+allows for a very efficient generation of ensemble predictions [1, 3]. Typically in methods
+that generate ensembles from training a single machine learning model, the predictions
+are strongly correlated [13, 14] since they can be deformed into each other through small
+perturbations. In twinned regression methods however, ensemble members are separated
+by the distance of the input data points themselves. (2) Twinned regression methods
+tend to be more accurate than the underlying base algorithm on many data sets [1, 3],
+(3) consistency conditions allow for the formulation of uncertainty estimators in addition
+to the ensemble variance [1, 3] and (4) loops containing unlabelled data points can be
+supplied while training, hence turning the method into a semi-supervised regression
+algorithm [2]. Further, (5) the intrinsic uncertainty estimation lends itself for active
+learning [3].
+A central contribution of this article is the combination of twin neural network
+regression and k-NN regression to increase the accuracy of over standard twin neural
+network regression. Similarly, artificial neural networks have been employed in tandem
+with k-NN regression in different contexts before [15, 16, 17].
+3. Reformulation of the Regression Problem
+A regression problem can be formulated as follows: Given a labelled training data
+set of n data points Xtrain = (xtrain
+1
+,...,xtrain
+n
+) with their corresponding target values
+Y train = (ytrain
+1
+,...,ytrain
+m
+), we are tasked to find a function f such that the deviation
+between f(xi) and yi is minimized with respect to a predefined objective function for all
+data points xi on the data manifold. In this work, this function is the root mean square
+error LRMSE =
+√
+∑n
+i=1(f(xi) − yi)2. Unless stated otherwise, all performance measures
+
+How to get the most out of Twinned Regression Methods
+4
+x
+y=f(x)
+x1
+x2
+y2-y1=F(x2,x1)
+F(x1,x2)
+f(x2)
+F(x2,x3)
+F(x3,x1)
+f(x3)
+f(x1)
+Traditional Regression  
+Twinned Regression  
+Figure 1: Dual formulation of a regression problem: A traditional solution to a regression problem
+consists of finding an approximation to the function that maps a data point x to its target value f(x) = y.
+Twinned regression methods solve the dual problem of mapping a pair of inputs x1 and x2 to the
+difference between the target values F(x2,x1) = y2 − y1. The resulting function can then be employed
+as an estimator for the original regression problem y2 = F(x2,x1) + y1 given a labelled anchor point
+(x1,y1). Twinned regression methods must satisfy loop consistency: predictions along each loop sum to
+zero: F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.
+are evaluated on unknown test data (Xtest,Y test).
+Twinned regression methods aim to solve a reformulation of the original regression
+problem which is visualized in Fig. 1. For each pair of data points (xtrain
+i
+,xtrain
+j
+) we
+train a regression model to find a function F to predict the difference
+F(xi,xj) = yi − yj
+.
+(1)
+This function F can be used to construct a solution to the original regression problem
+via ypred
+i
+= F(xi,xj) + yj, where (xj,yj) is an anchor whose target value is known. Every
+training data point xtrain
+j
+∈ Xtrain can be used as such an anchor. A more accurate
+estimate for the solution of the original regression problem is obtained by averaging over
+many differences between a fixed unknown data point and different anchor data points
+ypred
+i
+= 1
+n
+n
+∑
+j=1
+(F(xi,xtrain
+j
+) + ytrain
+j
+)
+= 1
+n
+n
+∑
+j=1
+(1
+2F(xi,xtrain
+j
+) − 1
+2F(xtrain
+j
+,xi) + ytrain
+j
+)
+.
+(2)
+The increase in accuracy is based on averaging out the noise from different anchors and
+the reduction of the variance error via an ensemble of predictions. Previous works [1, 3]
+recommended using the whole training data set as anchors, hence creating an ensemble
+of difference predictions yi − yj which is twice as large as the training set for every single
+prediction of yi.
+A major advantage of the dual formulation is the description via loops containing
+multiple data points as can be seen in Fig. 1. In contrast to traditional regression, the
+
+How to get the most out of Twinned Regression Methods
+5
+results of twinned regression methods need to satisfy consistency conditions, for example
+for each three data points x1,x2,x3, summing up the predictions along a closed loop
+should yield zero: F(x1,x2) + F(x2,x3) + F(x3,x1) = 0. During inference, violations of
+these consistency conditions give rise to uncertainty estimates [1, 3]. Enforcing loop
+consistency on predictions involving unlabelled data points in the training phase is what
+makes twinned regression methods into semi-supervised regression algorithms.
+While neural networks are naturally good learners of linear functions this is not the
+case for other algorithms like random forests. For this reason, [3] proposed to augment
+the input features by their difference (xi,xj) → (xi,xj,xi − xj). One might argue that
+this improvement is similar to common data augmentation, however, this is a feature
+that traditional machine learning algorithms don’t have access to, because it requires
+two different data points.
+4. Notes About Experiments
+All experiments in this article are performed on the data sets outlined in Appendix A.
+Since only neural network based methods scale favorably with the data set size, they use
+the full data sets of which 70% are used for training, 10% as validation set and 20% as
+test set. The details of the neural network architectures can be found in the appendix
+Appendix C. Random forests and especially twinned random forests scale poorly with
+the data set size thus only 100 training data points are chosen from the data sets and
+100 data points comprise the test sets. Random forests do not need validation sets, since
+the hyper=parameters are optimized via 5-fold cross-validation. In section Appendix D
+one can find the details about our random forest implementations. All experiments are
+repeated for 25 random but fixed splits of training, test, and if applicable, validation
+data.
+5. Ensemble Performance
+Twinned regression methods have been shown to produce accurate solutions to regression
+problems [1, 3], comparable to or better than other current state-of-the-art algorithms at
+the cost of scaling poorly towards larger data sets. This naturally leads to the question of
+where the increased performance stems from. The reformulation of a regression problem
+into its dual problem of predicting differences between target values opens up several
+potential reasons for improved accuracy. These include increased effective training set
+size, internal ensembling of predictions (see explanation in Appendix B), or the nature of
+solving a different problem. In the following we examine these reasons at the example of
+TNNR, however, we assume the answers will also be valid for other baseline algorithms.
+Let us start with discussing different kinds of ensembles and their effect on accuracy.
+Fig. 2 contains the results of several experiments examining the performance of different
+ensemble types of ANN regression and TNNR. For each data set, the baseline results are
+the solid blue horizontal line, which represents the test RMSE after applying standard
+
+How to get the most out of Twinned Regression Methods
+6
+Figure 2: Comparison of different ensembles of ANNs and TNNR measured by RMSE vs number of
+ensemble members for the dashed lines. Ensembles can be created either by training several models or
+evaluating TNNR for different anchors. The blue solid line corresponds to training and evaluating one
+TNNR model on all possible training pairs and predicting the results with all possible anchors. The
+dashed blue line varies the number of anchors during the inference phase and converges to the solid
+blue line in the limit of increasing the inference anchors to the full training set. The dashed orange line
+indicates traditional ANN ensembles where multiple ANNs are trained independently. The dashed green
+line corresponds to ensembles of independently trained TNNR models, if the ensemble size is 1, this
+is equivalent to the solid blue line. The dashed red line corresponds to independently trained TNNR
+models each having only one inference anchor per prediction.
+full anchor TNNR and the leftmost point of the orange line which represents the results
+of applying a single ANN, confirming that TNNR almost always yields a lower RMSE
+than ANN regression.
+In order to compare traditional ANNs with TNNR, we observe that after training we
+can map TNNR to an ANN for each anchor, since both ANN regression and TNNR use
+the same internal architecture. For each fixed xj ∈ Xtrain an ANN ˜f is defined through
+˜fj(xi) ∶= F(xi,xj) + yj
+(3)
+The features of xj modify the weights of ˜fj while yj is absorbed by the output neuron
+inside its bias. At that point, the only difference between each ˜fj and an equivalent
+ANN is the procedure with which the weights were optimized.
+
+ANNEnsemblevsTNNanchorensemble
+Bio Conservation
+Boston Housing
+Concrete Strength
+3.7
+5.50
+06'0
+3.6 
+5.25
+0.85
+3.5
+RMSE
+5.00
+3.4
+4.75
+08'0
+EE
+3.2
+4.50
+0.75
+3.1
+4.25
+16
+32
+Energy Efficiency
+RCL Circuit
+Test Function
+0.012
+1.0
+0.020
+0.010
+0.9 
+0.018
+0.008
+0.8
+0.016
+0.006
+0.7
+0.014
+0.004
+16
+32
+16
+32
+16
+32
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0.045
+0.70
+all anchors inference
+0.775
++TNN inference anchors
+0.040
+0.750
+ANN ensemble
+0.65
+0.725
+Ensemble of full anchor TNN
+Ensemble of one anchor TNN
+0.030
+0.700
+0.60
+0.675
+0.025
+0.650
+0.55
+2
+16
+32
+2
+4
+16
+32
+2
+4
+16
+32
+Ensemble Members
+Ensemble Members
+Ensemble MembersHow to get the most out of Twinned Regression Methods
+7
+Table 1: Best estimates for test RMSEs obtained by artificial neural network(ANN) regression compared
+to ensembles of ANN regression. Our confidence on the RMSEs is determined by their standard error.
+We train on 70% of the available data, 10% validation data, 20% test data.
+Single ANN
+32 ANN ensemble
+Gain
+Bio Conservation(BC)
+0.7874±0.012
+0.7438±0.0118
+5.53%
+Boston Housing(BH)
+3.5695±0.1051
+3.3291±0.1137
+6.73%
+Concrete Strength(CS)
+5.3829±0.0964
+4.9842±0.0938
+7.41%
+Energy Efficiency(EE)
+1.0310±0.0313
+0.9649±0.0224
+6.42%
+RCL Circuit(RCL)
+0.0193±0.0002
+0.0157±0.0002
+18.41%
+Test Function(TF)
+0.0076±0.0005
+0.0071±0.0005
+6.35%
+Wheatstone Bridge(WSB)
+0.0443±0.0016
+0.0359±0.0014
+19.1%
+Red Wine Quality(WN)
+0.7681±0.024
+0.6487±0.0068
+15.54%
+Yacht Hydrodynamics(YH)
+0.6723±0.0406
+0.6143±0.0357
+8.63%
+Table 2: Best estimates for test RMSEs obtained by Twin Neural Network Regression (TNNR). We
+measure the improvement between using a single anchor during inference phase and using all anchors.
+Further, the latter is compared to an ensemble of full anchor TNNR.
+1 Anchor TNNR
+All Anchor TNNR
+Gain
+Ensemble of 32 TNNR
+Gain
+BC
+0.9021±0.0131
+0.8140±0.0149
+9.77%
+0.7947±0.0134
+2.37%
+BH
+3.4793±0.1262
+3.2897±0.1293
+5.45%
+3.2232±0.1256
+2.02%
+CS
+4.9602±0.1073
+4.5385±0.1124
+8.5%
+4.2791±0.0995
+5.71%
+EE
+0.7445±0.0224
+0.7071±0.0229
+5.02%
+0.6468±0.0196
+8.53%
+RCL
+0.0212±0.0003
+0.0155±0.0002
+26.6%
+0.0130±0.0001
+16.23%
+TF
+0.0106±0.0004
+0.0053±0.0004
+50.24%
+0.0028±0.0001
+46.41%
+WSB
+0.0309±0.0009
+0.0239±0.0009
+22.81%
+0.0227±0.0008
+5.15%
+WN
+0.7654±0.0059
+0.6985±0.0064
+8.75%
+0.6713±0.0055
+3.89%
+YH
+0.6344±0.0363
+0.5798±0.0362
+8.62%
+0.5723±0.0336
+1.29%
+This gives us access to a framework to directly compare ensembles of ANNs and
+the implicit ensembles generated by TNNR using multiple anchors during inference. We
+examine the results of these both models through the orange and blue dashed lines in
+Fig. 2. While both curves reduce the RMSE as we increase the ensemble size, we come
+to the sobering conclusion that TNNR ensembles and ANN ensembles are not equivalent,
+they neither have a uniform slope nor do they converge to similar RMSEs.
+We have just used a single trained TNNR model for all ensemble members while
+training each ANN model from scratch. What happens if we retrain TNNR for each
+single anchor? The results of these experiments are visualized in the red dashed line.
+Since retraining increases the ensemble diversity the red line is consistently below the blue
+
+How to get the most out of Twinned Regression Methods
+8
+Figure 3: How many different pairings are used for training effects the performance of TNNR measured
+by RMSE. The blue solid line corresponds to training TNNR on all possible training pairs. It is
+compared to training TNNR on a randomly chosen but fixed data set of pairs while still employing all
+training data points as inference anchors. A training set multiplier indicates on how many pairs are
+taken into account compared to the original unpaired training set.
+line. Further, we can see that the performance of these independently trained TNNRs
+increases faster with the number of anchors. If the resources are available one can further
+create an ensemble of different TNNR models each having access to all anchors depicted
+in the green line. On seven out of nine data sets this yields clearly the best performance.
+We note that one-anchor TNNR with retraining (red line) converges towards the green for
+more than 32 ensemble members. This tells us that the full ensemble diversity through
+multiple anchors and multiple models can be captured independently.
+A more quantitative version of the out-performance of TNNR ensembles can be
+seen in Table 1 and Table 2. The magnitude of the % improvement of the combined
+anchor+direct ensembling containing multiple TNNs causes a much larger improvement
+than the ensembling of traditional ANNs.
+6. Effective Training Set Size
+Another improvement over a traditional regression analysis is the increased training set
+size that comes from preparing training sets by pairing each training data point with
+
+TrainingSetMultiplier
+Bio Conservation
+Boston Housing
+Concrete Strength
+0.84
+3.5 
+5.0
+0.82
+RMSE
+3.4
+4.8
+0.80
+EE
+4.6
+0.78
+3.2
+4.4
+4
+8
+16
+2
+4
+8
+16
+32
+1
+2
+8
+16
+32
+Energy Efficiency
+RCL Circuit
+Test Function
+11
+0.0170
+0.0065
+1.0
+0.0165
+0.0060
+0.0160
+0.8
+0.0055
+0.0155
+0.7
+0.0050
+1
+a
+16
+32
+1
+7
+4
+8
+16
+32
+1
+2
+4
+8
+16
+32
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0E0'0
+0.75
+all anchors training
+0.70
+anchor pairs per
+0.70
+0.028
+training data
+MSE
+0.69
+0.65
+0.026
+0.68
+0.60
+0.024
+0.67
+0.55
+2
+4
+8
+16
+32
+2
+4
+8
+16
+32
+1
+2
+4
+8
+16
+32
+IrainingPairsperTrainingData
+Training Pairs per Training Data
+TrainingPairsperTrainingDataHow to get the most out of Twinned Regression Methods
+9
+Table 3: Best estimates for test RMSEs obtained by Nearest Neighbor Twin Neural Network Regression
+(NNTNNR).
+TNNR
+NN inference
+Gain
+NN train+inference
+Gain
+BC
+0.8234±0.0144
+0.8162±0.0155
+0.87%
+0.8133±0.0152
+1.23%
+BH
+3.3104±0.1202
+3.2898±0.1202
+0.62%
+3.3563±0.1161
+-1.39%
+CS
+4.4731±0.1242
+4.4458±0.1091
+0.61%
+4.4290±0.1174
+0.99%
+EE
+0.7156±0.0204
+0.7056±0.0193
+1.4%
+0.6825±0.0216
+4.63%
+RCL
+0.0158±0.0002
+0.0151±0.0003
+3.98%
+0.0140±0.0002
+11.33%
+TF
+0.0050±0.0001
+0.0028±0.0002
+43.65%
+0.0021±0.0003
+57.26%
+WSB
+0.0233±0.0006
+0.0224±0.0009
+4.06%
+0.0236±0.0009
+-1.01%
+WN
+0.6998±0.006
+0.6951±0.0062
+0.68%
+0.6944±0.006
+0.77%
+YH
+0.5977±0.0344
+0.5184±0.0331
+13.26%
+0.5009±0.0333
+16.19%
+every other training data point. This transforms a training set of size n into a pairwise
+training set of size n2. In this section, we measure if the increase in the number of
+pairings leads to an increase in accuracy.
+To address this question we look at several curves in Fig. 3. In this figure, we
+compare the effect of increasing the effective pairwise training data set on the accuracy.
+For this purpose, we define the training set multiplier as the number of pairs that are
+created from the original training set to produce the paired training set. A training
+set multiplier of one means that each training data point is paired with only one other
+randomly chosen (without replacement) but fixed data point (on average this means each
+training data point is used twice). Increasing the training set multiplier to the size of the
+training set converges to the standard formulation of twinned regression methods. We
+can see that in all data sets, except two, increasing the training set multiplier increases
+the performance of TNNR. More precisely, a training set multiplier of ≈ 8 − 16 seems
+already to be enough to reach the accuracy of standard TNNR. It is important to
+note, that on two data sets, namely Bio Conservation(BC) and Red Wine Quality(WN),
+increasing the training set multiplier has the effect of reducing the performance. This
+coincides with other algorithms beating TNNR (Fig. 2,Fig. 5) and is a sign that TNNR
+might not be suitable for such regression tasks. A data scientist using TNNR might do
+a training set multiplier check, if he finds a decreasing accuracy while increasing the
+multiplier, he can reject TNNR as optimal regression algorithm.
+7. Nearest Neighbor TNNR
+In this manuscript we propose a new regression algorithm based on a combination
+of k-nearest neighbor regression and TNNR, which of course could be implemented
+for various baseline twinned regression algorithms. In standard twinned regression
+methods the model learns to predict differences between the targets of two arbitrary
+
+How to get the most out of Twinned Regression Methods
+10
+Figure 4: Effect of nearest neighbor pairing on TNNR measured in terms of RMSE vs the number
+of nearest neighbors used if applicable. The solid blue line marks the performance of the original
+TNNR. The dashed blue line displays the results of TNNR trained on all possible pairs while performing
+inference only using nearest neighbor anchors. The dashed orange line is produced by restricting pairs
+to nearest neighbors for training and inference.
+data points. This model is then employed to create an ensemble prediction via averaging
+the approximations of the differences between the target value of a new data point and
+all anchor data points, see (2). However, not all of these anchor data points might be of
+equal importance for the prediction. That is why in this section we restrict the anchor
+points to the nearest neighbors. For this purpose, we define the notation NN(i,m) as
+the set of m nearest neighbors of a data point xi ∈ X within the training set xi ∈ Xtrain
+to reformulate the prediction:
+ypred
+i
+= 1
+m
+∑
+j∈NN(i,m)
+(F(xi,xtrain
+j
+) + ytrain
+j
+)
+(4)
+While we have defined the prediction using nearest neighbors during the inference
+phase, it is an open question whether it is better to train the model to predict
+differences between target values of generic data points or between neighboring data
+points corresponding to the same number of nearest neighbors in the inference phase. The
+different training versions are compared in Fig. 4 where the baseline is set by standard
+
+Nearest Neighbor Training
+Bio Conservation
+Boston Housing
+Concrete Strength
+4.75
+5.4
+1.05
+4.50
+5.2
+1.00
+4.25
+5.0
+4.00
+4.8
+06'0
+3.75
+0.85
+3.50
+4.6
+3.25
+4.4
+0.80
+2
+4
+8
+16
+37
+64
+1
+4
+8
+16
+64
+8
+16
+32
+64
+Energy Efficiency
+RCL Circuit
+Test Function
+1.6 
+0.008
+0.030
+1.4
+0.007
+0.025
+0.006
+0.005
+1.0
+0.020
+0.004
+0.8
+E00'0
+0.015
+0.002
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64 
+16
+32
+64
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0.06
+all anchor training
+0.85
+1.0
+all training pairs
+0.05
+0.80
+nearest neighbor
+0.8
+training pairs
+0.75
+0.03
+0.6
+0.70
+0.02
+2
+A
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64 
+2
+4
+8
+16
+32
+64
+# Nearest Neighbors
+# Nearest Neighbors
+# Nearest NeighborsHow to get the most out of Twinned Regression Methods
+11
+Figure 5: Comparison of k-NN regression and TNNR with different numbers of nearest neighbor
+training pairs measured by RMSE vs the number of neighbors for the dashed lines. The solid blue
+corresponds to normal TNNR with access to all possible pairs during training and inference. The blue
+dashed line restricts the pairs to be nearest neighbors during inference. The orange dashed line restricts
+the pairs to be nearest neighbors during training and inference. The green dashed line describes k-NN.
+TNNR. We emphasize that both versions of training obey the same principle for selecting
+nearest neighbors during inference. When only using the very nearest neighbor as an
+anchor for inference, we can see that for 7 out of 9 data sets both training versions
+underperform traditional TNNR, while training on all possible pairs performs better than
+just training on neighboring training data points. This picture changes as we increase
+the number of nearest neighbors. On all data sets both versions of nearest neighbor
+TNNR converge to standard TNNR in the limit of increasing the number of neighbors to
+the training set size. In 7 out of 9 data sets there is a sweet spot where nearest neighbor
+TNNR with nearest neighbor training outperforms at around 16 to 64 neighbors, in 3
+out of those data sets nearest neighbor training outperforms by a very large margin
+culminating in reducing the RMSE by ≈ 60% on the TF data set, see Table 3. We note
+that this is the data set with zero noise.
+
+NearestNeighborTrainingvsk-NN
+Bio Conservation
+Boston Housing
+Concrete Strength
+12
+1.0
+10
+RMSE
+0.9
+8
+0.8
+i
+16
+32
+64
+4
+64
+2
+8
+16
+32
+64
+Energy Efficiency
+RCL Circuit
+Test Function
+5
+0.25
+0.10
+4
+0.20
+0.08
+E
+0.15
+0.06
+RM
+0.10 
+0.04
+2
+0.05
+0.02
+0.00
+1
+4
+8
+16
+32
+64
+1
+2
+4
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+10
+0.20
+0.85
+8
+all anchor training
+0.15
+0.80
+all training pairs
+MSE
+nearest neighbor
+0.10
+0.75
+training pairs
+k-NN regression
+0.05
+0.70
+2
+4
+8
+16
+32
+64
+4
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64
+# Nearest Neighbors
+# Nearest Neighbors
+# Nearest NeighborsHow to get the most out of Twinned Regression Methods
+12
+8. TNNR vs k-NN
+A natural question is how nearest-neighbor TNNR(NNTNNR) related to k-NN regression.
+Nearest neighbor TNNR can be related to k-NN regression through setting F(xi,xj) ≡ 0,
+then
+ypred
+i
+= 1
+m
+∑
+j∈NN(i,m)
+⎛
+⎜⎜
+⎝
+F(xi,xtrain
+j
+)
+�����������������������������������������������������������
+0
++ytrain
+j
+⎞
+⎟⎟
+⎠
+= 1
+m
+m
+∑
+j∈NN(i)
+ytrain
+j
+(5)
+Assuming F(xi,xj) would just be a minor contribution to k-NN regression we would see
+a qualitatively similar performance of NNTNNR. In order to test this statement, we
+visualize the behavior of k-NN and NNTNNR in Fig. 5. In this figure we can clearly see,
+that NNTNNR beats k-NN regression by an enormous margin on 7 out of 9 data sets.
+However, there are two data sets, namely BC, WN where k-NN is the winner. Again, we
+note that these data sets are exactly where the expected TNNR mechanism fails Fig. 3
+and it coincides with ANN ensembles outperforming TNNR Fig. 2. Further, the number
+of optimal TNNR anchors is much larger than the optimal number of neighbors in k-NN.
+9. Reformulation Benefits
+After having discussed the impacts of ensembling and the increased effective training
+set size, we have now finally the tools to partially answer the question of whether the
+reformulation to the dual problem itself contributes to the increased accuracy of twinned
+regression methods. We have related TNNR to normal ANNs in (3) and connected
+NNTNNR to k-NN regression in (5). If TNNR would be a glorified form of ANN or k-NN
+regression, the performance of TNNR could be related qualitatively to neural networks
+or k-NN. However, as we can see by comparing with ANNs in Fig. 2 or k-NN Fig. 5, it
+is clear that TNNR has a distinct performance profile that beats ANNs and k-NN on
+the same 7 of 9 data sets and underperforms both on the remaining two data sets. As
+we know from testing the impact of the increased training set size Fig. 3, these are the
+data sets where increasing the data sets has an adverse effect on accuracy, which signals
+that the TNNR mechanism fails while ANN and k-NN continue to perform normally.
+All these facts support the conclusion that the reformulation to a dual problem itself
+tends to have a positive effect on accuracy on most data sets.
+10. Miniaturizing accurate networks
+Often it is required to store fully trained neural networks on hardware that has strict
+memory limitations. This might be on chips that allow for autotuning of quantum
+dots [9] or in self driving vehicles [10]. In these cases it is required to consider the
+trade-off between accuracy and memory requirements. Ensembles of neural networks
+
+How to get the most out of Twinned Regression Methods
+13
+Figure 6: Memory requirements for storing real ensembles in terms of number of parameters stored vs
+the ensemble size. The architecture used corresponds to the architecture that was used to produce the
+results in this manuscript. Different architectures cause quantitatively different plots, but qualitatively
+the behave similarly. The solid lines indicate the memory requirements for storing an ensemble of
+independently trained ANN models for different numbers of features f describing each data instance.
+The dashed lines correspond to ensembles generated by storing one TNNR model and a representative
+number of anchors which can be combined to produce ensembles.
+tend to be more accurate than single neural networks, however, storing them requires
+linearly more memory capacity per each ensemble member. TNNR provides an elegant
+solution to this problem, because in order to store an ensemble of predictions it only
+requires the storage of a single set of trained weights and biases together with one
+anchor data point per ensemble member. Our neural network architectures are chosen
+such that they produce accurate results on all nine considered data sets. Thus, we
+have chosen an architecture with two hidden layers, each with 128 neurons. In Fig. 6
+we visualize the number of parameters that are required to be stored in the case of
+traditional ANN and TNNR ensembles for different feature sizes of the input data. It can
+be seen that in all cases TNNR parameters plus anchors need far less storage capacity
+to achieve a similar ensemble size as ANNs. Of course in practice the optimal neural
+network architecture and its number of parameters varies between problems, meaning
+our quantitative analysis of memory requirements might not generalize to other problems.
+However, the qualitative trend remains the same as long as the feature size is smaller
+than the number of parameters of the model.
+11. Making other models semi-supervised
+In this section we explore a simple framework to train any twinned regression method in
+a semi-supervised manner. The idea is based on the semi-supervised regression method
+devised in [2] for TNNR. As we can see in Fig. 1 the dual formulation requires machine
+learning models to predict differences F(xi,xj) = yi − yh between target values instead of
+the targets f(x) = y, themselves. One advantage of this formulation is that a correct
+
+Number of Parameters to store ensemble
+106
+ANN ensemble f=4
+ANN ensemble f=13
+ parameters
+ANN ensemble f=100
+TNN anchors f=4
+TNN anchors f=13
+103
+TNN anchors f=100
+#
+0
+5
+10
+15
+20
+25 
+30
+Ensemble sizeHow to get the most out of Twinned Regression Methods
+14
+Figure 7: Transductive semi-supervised learning with random forests. Random forests have been
+supplied with loops containing unlabelled data points. The magnitude of the influence the loops have
+on the training process is measured by Λ. Tuning Λ ≈ 1 reduces the RMSE on almost all data sets
+compared to purely supervised learning (solid blue line).
+Table 4: Best estimates for test RMSEs obtained by Semi-supervised random forests.
+Supervised RF
+Semi-Supervised RF
+Improvement
+Bio Conservation
+0.7367±0.0081
+0.7357±0.0078
+0.14%
+Boston Housing
+3.8408±0.1322
+3.8281±0.1356
+0.33%
+Concrete Strength
+5.8813±0.2242
+5.8440±0.2276
+0.63%
+Energy Efficiency
+1.9112±0.0417
+1.8919±0.0411
+1.01%
+RCL Circuit
+0.2934±0.0068
+0.2932±0.0068
+0.05%
+Test Function
+0.0857±0.0022
+0.0855±0.0021
+0.19%
+Wheatstone Bridge
+0.0958±0.0025
+0.0958±0.0022
+0.02%
+Red Wine Quality
+0.6066±0.0083
+0.6045±0.0084
+0.35%
+Yacht Hydrodynamics
+1.1138±0.0383
+1.1091±0.0384
+0.42%
+
+RandomForestSemi-SupervisedLearning
+Bio Conservation
+Boston Housing
+Concrete Strength
+0.742
+3.86
+5.88
+0.740
+3.85
+5.87
+RMSE
+0.738
+5.86
+3.84
+5.85
+0.736
+E8E
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+Energy Efficiency
+RCL Circuit
+Test Function
+1910
+0.302
+0.092
+1.905
+0.300
+MSE
+0.090
+0.298
+0.296
+0.088
+1.895
+0.294
+0.086
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0.09725
+0.608
+115
+TNN
+0.09700
+-+ all training pairs
+0.09675
+0.607
+1.14
+ISE
+0.09650
+0.606
+1.13
+0.09625
+112
+0.09600
+0.605
+0.09575
+1.11
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+Loop Consistency Weight A
+Loop Consistency Weight A
+Loop Consistency Weight ^How to get the most out of Twinned Regression Methods
+15
+solution would satisfy loop consistency F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.
+Hence, we propose the following algorithm that is applicable to all twinned regression
+methods: At first we train the regression model on the labelled training data. This model
+is then used to predict the differences between targets along loops randomly sampled
+from an unlabelled data set. For each loop an adjustment a defined by
+a = F(x1,x2) + F(x2,x3) + F(x3,x1)
+(6)
+is then used to propose a label yij = F(xi,xj)−Λ×a for each combination of xi,xj within
+the loops. Here, Λ is the loop weight hyper-parameter. The unlabelled data set together
+with the proposed labels is then added to the labelled training set, on which the model
+is retrained. The algorithm is further depicted in Fig. 1.
+We apply this idea to the pairwise/twinned random forest regression proposed in [3],
+which was originally aimed at solving regression problems on small data sets in chemistry.
+Since random forests don’t scale as well with large data sets, we restrict our data sets to
+100 training and 100 test data points. The details of the training process are outlined in
+section Appendix D.
+Before applying the semi-supervised learning strategy, we convince ourselves that
+twinned random forest regression is suitable for the test bed consisting of the nine data
+sets (Appendix A) used in this paper. The corresponding results can be seen in Table D1.
+Twinned random forests perform equally well, or slightly worse, compared to traditional
+random forests on three data sets (BC,EE,WN). It moderately outperforms on four data
+sets (BH,CS,RCL,YH) and it massively outperforms by cutting the RMSE by more than
+35% on two data sets (TF,WSB).
+After having convinced ourselves of the superior performance of twinned random
+forests, we apply our semi-supervised learning framework in a transductive manner.
+Transductive means that the test data is used as unlabelled training data. This is in
+contrast to inductive semi-supervised learning where the unlabelled training data would
+be kept separate from the final test data. The final results are depicted in Fig. 7 for
+various choices of the loop weight Λ. We can clearly see, that the optimal choice of
+Λ ≈ 1 leads to a reduction of RMSE on six out of nine data sets. However, the relative
+improvement from semi-supervised learning is very small, as shown in Table 4 and most of
+the time less than 1%. If we compare these results with other semi-supervised regression
+algorithms on the same data sets from [2], one can observe that this improvement is
+significantly less than semi-supervised TNNR and slightly less than co-training with
+neural networks.
+12. Negative Results
+Let us briefly discuss in this section different ideas that we tried during our experiments
+but did not lead to a consistent improvement of twinned regression methods.
+While exploring the ideal weighting of anchor during the inference phase of twinned
+regression methods, a straightforward idea was to try incorporating the intrinsic
+
+How to get the most out of Twinned Regression Methods
+16
+Figure 8: Training and inference time comparison in seconds between different versions of TNNR and
+ANN regression. The solid orange line indicates the time for training and inference in the case of ANNs.
+The solid blue line is the same for original TNNR. The blue dotted line indicates restricting the possible
+training/inference pairs to nearest neighbors, the x-axis corresponds to the number of neighbors.
+uncertainty metrics [1]. These include the ensemble standard deviation and the violation
+of loop consistency. Anchors with a lower uncertainty metric should be weighted higher
+than anchors with high uncertainty metrics. While we observed some benefit, we could
+not consistently show that this process improved the accuracy in a statistically significant
+manner. We believe that other uncertainty metrics that unrelated to the intrinsic
+consistency metrics might be better suited as for example in Gaussian processes.
+Our initial plan when devising a strategy to adopt the semi-supervised learning
+framework from [2] to other algorithms was based on an iterative algorithm. After
+training the underlying twinned regression algorithm, the model would predict labels
+on unknown data points. Randomly sampling loops containing these data points would
+allow us to check for loop consistency. The unknown data points would then be added
+to the training data set with a label that corresponds to the original prediction slightly
+modified in the direction which satisfies the loop condition. The idea was to iterative
+refine the labels by repeating this process. However, it turned out that many times this
+process would either not converge for Λ > 1/3, or eventually converge to sub-optimal
+solutions, worse than the initial supervised version.
+Combining k-NN regression with TNNR was also aimed at reducing the
+
+Training+InferenceTimeComparison
+Bio Conservation
+Boston Housing
+Concrete Strength
+1750
+seconds
+2000
+1500
+6000
+1500
+1250
+1000
+4000
+Time
+1000
+750
+500
+2000
+500
+2
+4
+8
+16
+32
+64
+i
+2
+8
+16
+32
+64
+i
+4
+8
+16
+32
+64
+Energy Efficiency
+RCL Circuit
+Test Function
+6000
+25000
+3500
+seconds
+5000
+20000
+3000
+4000
+2500
+15000
+3000
+2000
+Time
+10000
+2000
+1500
+5000
+1000
+1000
+16
+32
+64
+2
+4
+8
+1
+:
+4
+8
+16
+32
+64
+1
+2
+4
+8
+16
+32
+64
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+800
+5000
+seconds
+1200
+600
+4000
+1000
+ANN
+3000
+800
+TNN
+400
+nearest neighbor
+2000
+600
+training data
+200
+1000
+400
+200
+2
+4
+8
+16
+32
+t9
+1
+2
+4
+8
+16
+32
+64
+1
+2
+4
+8
+16
+32
+64
+# Nearest Neighbors
+# Nearest Neighbors
+# Nearest NeighborsHow to get the most out of Twinned Regression Methods
+17
+computational time. As explored in [1], the training time of twinned regression methods
+scales poorly towards larger data sets, mostly caused by the increase in the effective
+data set size through pairing of data points. While for many baseline algorithms a
+clear relationship between data set size and training time, for neural networks it is less
+known. As neural networks training time scales very favorably with training set size
+we focused on TNNR to test the training time improvement from only using nearest
+neighbor paring during training phase. In Fig. 8 we can see that there is a tendency for a
+reduced computational cost on most data sets. However, the training time scaling is too
+minor and inconsistent to use it as a sole justification to use NNTNNR over traditional
+TNNR.
+13. Conclusion
+Twinned regression is a simple and versatile framework to improve performance through
+intrinsic ensembling and semi-supervised learning on small to medium-sized data sets.
+In this article, we have answered several questions about the nature of the twinned
+regression framework. Further, we devised several improvements to further improve the
+already state-of-the-art performance of twinned regression methods.
+We compared the ensemble behaviour of traditional ANNs and TNNR. For this
+purpose we mapped single anchor TNNR to an equivalent ANN model during inference
+phase. By visualizing the results of these examinations in Fig. 2 we can see that the
+performance an ensemble of single anchor TNNs converges towards an ensemble of full
+anchor TNNs at around 32 ensemble members. Increasing the number of anchors would
+not yield any additional gain. This suggests that the intrinsic ensemble diversity of
+TNNR is a subset of the diversity that can be achieved by retraining the network for
+each anchor. The combined anchor+direct ensemling containing multiple TNNs causes a
+much larger improvement than the ensembling of traditional ANNs as can be seen in
+Table 1 and Table 2. This explains one element of the outperformance of TNNR over
+ANN regression.
+Further, we examined what effect the increased training set size through pairing
+data points has on the TNN accuracy, Fig. 3. Generally, more pairings per training data
+point reduced the RMSE. However, to our surprise not all data sets benefited from this
+pairing, on two data sets (BC,WN) TNNR had the lowest RMSE if only one pairing
+per training data point was allowed. By comparing this malfunction to results in Fig. 2
+and Fig. 5, we can see that it occurs in exactly the data sets where k-NN regression
+and ANN ensembles outperform TNNR, signaling a breakdown of the performance
+increasing factors of TNNR. By looking at the properties of the data sets it seems like
+twinned regression methods perform best on continuous data sets where the label can be
+approximated through a deterministic function.
+We also pointed at another advantage of TNNR in the case where it is impossible to
+store a large number of parameters, but one wants to retain the advantages of ensembles.
+TNNR provides the possibility to generate an ensemble of predictions just by storing
+
+How to get the most out of Twinned Regression Methods
+18
+one model and some anchors, which is usually significantly smaller than storing multiple
+ANN models, see Fig. 6.
+While exploring the ideal weighting of anchors during the inference phase of TNNR,
+we found that nearest-neighbor predictions tend to yield the most accurate results.
+This lead us to develop nearest-neighbor TNNR (NNTNNR) which is a combination
+of the k-nearest neighbor algorithm and TNNR. There are two versions of NNTNNR,
+one which respects nearest neighbors during both training and inference and another
+version that only restricts nearest neighbors during the inference phase. Restricting
+to nearest-neighbor training tends to yield slightly better results Fig. 4. Both versions
+outperform standard TNNR especially on low noise data sets, see Table 3. It is important
+to note that NNTNNR is not just a minor improvement to k-NN regression since it has
+a very different performance profile when it comes to varying the number of anchors, or
+nearest neighbors, respectively, as can be seen in Fig. 5.
+We devised a semi-supervised regression framework based on enforcing loop
+consistency that can be applied to any twinned regression algorithm, but we tested it for
+random forests. This method yielded a clearly visible improvement over their supervised
+counterparts as can be seen in Fig. 7. However, the magnitude of the reduction of the
+RMSE is relatively small and almost always less than 1%, see Table 4. Comparing these
+results with other semi-supervised regression algorithms on the same data sets from [2],
+we can see that this improvement is significantly less than semi-supervised TNNR and
+slightly less than co-training with neural networks.
+The code supporting this publication is available at [18].
+14. Acknowledgements
+Let us thank Zurab Jashi for his help with the random forest code. This work was
+supported by Mitacs and Homes Plus Magazine Inc. through the Mitacs Accelerate
+program. We also acknowledge Compute Canada for computational resources. We thank
+the National Research Council of Canada for their partnership with Perimeter on the
+PIQuIL. Research at Perimeter Institute is supported in part by the Government of
+Canada through the Department of Innovation, Science and Economic Development
+Canada and by the Province of Ontario through the Ministry of Economic Development,
+Job Creation and Trade.
+
+How to get the most out of Twinned Regression Methods
+19
+Appendix A. Data sets
+Table A1: Data sets
+Name
+Key
+Size
+Features
+Type
+Bio Concentration
+BC
+779
+14
+Discrete, Continuous
+Boston Housing
+BH
+506
+13
+Discrete, Continuous
+Concrete Strength
+CS
+1030
+8
+Continuous
+Energy Efficiency
+EF
+768
+8
+Discrete, Continuous
+RCL Circuit Current
+RCL
+4000
+6
+Continuous
+Test Function
+TF
+1000
+2
+Continuous
+Red Wine Quality
+WN
+1599
+11
+Discrete, Continuous
+Wheatstone Bridge Voltage
+WSB
+200
+4
+Continuous
+Yacht Hydrodynamics
+YH
+308
+6
+Discrete
+The test function (TF) data set created from the equation
+F(x1,x2) = x3
+1 + x2
+1 − x1 − 1 + x1x2 + sin(x2)
+(A.1)
+and zero noise.
+The output in the RCL circuit current data set (RCL) is the current through an
+RCL circuit, modeled by the equation
+I0 = V0 cos(ωt)/
+√
+R2 + (ωL − 1/(ωC))2
+(A.2)
+with added Gaussian noise of mean 0 and standard deviation 0.1.
+The output of the Wheatstone Bridge voltage (WSB) is the measured voltage given
+by the equation
+V = U(R2/(R1 + R2) − R3/(R2 + R3))
+(A.3)
+with added Gaussian noise of mean 0 and standard deviation 0.1.
+Appendix B. Bias-Variance Tradeoff and Ensembles
+In a regression problem, one is tasked with finding the true labels on yet unlabelled
+data points through the estimation of a function f(x) = y. Given a finite training data
+set D we denote this approximation ˆf(x;D). The expected mean squared error can
+be decomposed by three sources of error, bias error BiasD[ ˆf(x;D)] , variance error
+VarD [ ˆf(x;D)] and intrinsic error of the data set σ.
+MSE = Ex {BiasD[ ˆf(x;D)]2 + VarD [ ˆf(x;D)]} + σ2.
+(B.1)
+
+How to get the most out of Twinned Regression Methods
+20
+If we replace the estimator by an ensemble of two functions ˆf(x;D) = 1/2 ˆfA(x;D) +
+1/2 ˆfB(x;D), each exhibiting the same bias and variance as the original estimator, then
+we can decompose the MSE
+MSE = Ex {BiasD[1/2 ˆfA(x;D) + 1/2 ˆfB(x;D)]2 + VarD [1/2 ˆfA(x;D) + 1/2 ˆfB(x;D)]} + σ2
+(B.2)
+= Ex {BiasD[ ˆf(x;D)]2 + VarD [1/2 ˆfA(x;D)] + VarD [1/2 ˆfB(x;D)]
+(B.3)
++ 2CovD [1/2 ˆfA(x;D),1/2 ˆfB(x;D)]} + σ2
+(B.4)
+= Ex {BiasD[ ˆf(x;D)]2 + 1/2VarD [ ˆfA(x;D)] + 1/2CovD [ ˆfA(x;D), ˆfB(x;D)]} + σ2
+(B.5)
+The more uncorrelated ˆfA(x;D) and ˆfB(x;D) are, the smaller is the ratio between
+variance and covariance. Thus an ensemble consisting of weakly correlated ensemble
+members reduce the MSE by circumventing the bias-variance tradeoff. By induction this
+argument extends to larger ensemble sizes.
+Appendix C. Neural Network Architectures
+Both our traditional neural network regression and twin neural network regression
+methods are build using the same architecture build using the tensorflow library [19].
+They consist of two hidden layers with 128 neurons each and relu activation functions.
+The final layer contains one single neuron without an activation function. We train
+our neural networks using the adadelta optimizer, and use learning rate and early stop
+callbacks that reduce the learning rate by 50% or stop training if the loss stops decreasing.
+For this reason it is enough to set the number of epochs large enough such that the early
+stopping is always triggered, in our cases this is 2000 for ANNs and 10000 for TNNR.
+The batchsizes are in both cases 16.
+Appendix D. Random Forests
+The random forests in this article use the scikit-learn library [20]. They are trained
+on a subset of all data sets: from each data set, we randomly sample 100 training
+data points and 100 test data points. We use five-fold cross-validation to optimize
+the following hyper-parameters of our random forests: ’max depth’ ∈ [4,8,16,32,64],
+’max features’ ∈ [0.33,0.667,1.0], ’min samples leaf’ ∈ [1,2,5], ’min samples split’
+∈ [2,4,8], ’n estimators’ ∈ [100,300,600]. Both, the traditional and the twinned random
+forests choose their optimal hyper-parameters from the same pool. It is important to
+note that for semi-supervised learning the hyper-parameters are only optimized during
+
+How to get the most out of Twinned Regression Methods
+21
+Table D1: Best estimates for test RMSEs obtained by Random Forest compared to Twinned Random
+Forests
+Random Forest
+Twinned Random Forest
+Improvement
+Bio Conservation
+0.7407± 0.0087
+0.7427± 0.0081
+-0.27%
+Boston Housing
+4.0019± 0.1394
+3.8301± 0.1322
+4.29%
+Concrete Strength
+6.3763± 0.2136
+5.8519± 0.2242
+8.22%
+Energy Efficiency
+1.8773± 0.0422
+1.8906± 0.0417
+-0.71%
+RCL Circuit
+0.3168± 0.0071
+0.2958± 0.0068
+6.63%
+Test Function
+0.1402± 0.0044
+0.0874± 0.0023
+37.66%
+Wheatstone Bridge
+0.1461± 0.0039
+0.0942± 0.0025
+35.52%
+Red Wine Quality
+0.5989± 0.0085
+0.6068± 0.0083
+-1.32%
+Yacht Hydrodynamics
+1.1917± 0.029
+1.1117± 0.0383
+6.71%
+the initial supervised learning step, the optimal parameters are then carried forward to
+be used during semi-supervised learning.
+Algorithm 1: Semi-Supervised Learning through Loop Consistency
+Data: Labelled data set DL = (Xtrain,L,Ytrain,L)
+Unlabelled data set DU = (Xtrain,U)
+Input: Loop weight Λ
+Loop number nl =length(DL)/3
+1 create ˜DL = [((xi,xj,xi − xj),yij = yi − yj) for xi ∈ Xtrain,L for xj ∈ Xtrain,L]
+2 initialize machine learning model M
+3 train M on ˜DL
+4 sample nl loops L=[(xi,xj,xk) where xi ∈ Xtrain,L, xj,xk ∈ Xtrain,U]
+5 for (xi,xj,xk) ∈ L do
+6
+predict M(xi,xj),M(xj,xk),M(xk,xi)
+7
+a = M(xi,xj) + M(xj,xk) + M(xk,xi)
+8
+(yij,yjk,yki) = (M(xi,xj) − Λa,M(xj,xk) − Λa,M(xk,xi) − Λa)
+9
+add ((xi,xj,xi − xj),yij),((xj,xk,xj − xk),yjk),((xk,xi,xk − xi),yki) to ˜DL
+10 train M on ˜DL
+Output: Trained Model M
+
+How to get the most out of Twinned Regression Methods
+22
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