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+Triple-stream Deep Metric Learning of Great Ape Behavioural Actions
+Otto Brookes1, Majid Mirmehdi1, Hjalmar K¨uhl2, Tilo Burghardt1
+1Department of Computer Science, University of Bristol, United Kingdom
+2Evolutionary and Anthropocene Ecology, iDiv, Leipzig, Germany
+[email protected], [email protected], [email protected], [email protected]
+Keywords:
+Animal Biometrics, Multi-stream Deep Metric Learning, Animal Behaviour, Great Apes, PanAf-500 Dataset
+Abstract:
+We propose the first metric learning system for the recognition of great ape behavioural actions. Our proposed
+triple stream embedding architecture works on camera trap videos taken directly in the wild and demonstrates
+that the utilisation of an explicit DensePose-C chimpanzee body part segmentation stream effectively com-
+plements traditional RGB appearance and optical flow streams. We evaluate system variants with different
+feature fusion techniques and long-tail recognition approaches. Results and ablations show performance im-
+provements of ∼ 12% in top-1 accuracy over previous results achieved on the PanAf-500 dataset containing
+180,000 manually annotated frames across nine behavioural actions. Furthermore, we provide a qualitative
+analysis of our findings and augment the metric learning system with long-tail recognition techniques show-
+ing that average per class accuracy – critical in the domain – can be improved by ∼ 23% compared to the
+literature on that dataset. Finally, since our embedding spaces are constructed as metric, we provide first data-
+driven visualisations of the great ape behavioural action spaces revealing emerging geometry and topology.
+We hope that the work sparks further interest in this vital application area of computer vision for the benefit of
+endangered great apes. We provide all key source code and network weights alongside this publication.
+positive
+anchor
+fusion
+negative
+ResNet-50
+ResNet-50
+ResNet-50
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+ Metric
+Learning
+d
+d
+d
+d
+LTriplet
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+embedding model
+DensePose-C
+Optical flow
+RGB
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+x0
+ . . . . . .
+x128
+embedding model
+embedding model
+shared weights
+shared weights
+Figure 1: System Overview. Our proposed triple-stream metric learning approach utilises all RGB appearance, optical flow,
+and DensePose-C segmentations of chimps in videos. Exploiting hybrid reciprocal triplet and cross entropy losses, the model
+is then trained to map embeddings representing great ape behavioural actions onto a metric space, where semantically similar
+representations are geometrically close forming natural clusters. This pipeline improves on state-of-the-art classification
+performance and allows for visualisations of the underpinning space of behavioural actions. (best viewed zoomed)
+1
+INTRODUCTION
+As the climate crisis gathers pace, the threat to many
+endangered species grows ever more perilous (Al-
+mond et al., 2022). All species of great apes are, for
+instance, listed as endangered or critically endangered
+according to the IUCN Red List (IUCN, 2022)
+.. . . . . . . . . . . there is urgent need for methods
+Consequently, there is urgent need for methods that
+can help to monitor population status and assess the
+effectiveness of conservation interventions (K¨uhl and
+Burghardt, 2013; Congdon et al., 2022; Tuia et al.,
+2022). This includes the recognition of behaviors and
+variation therein, as an integral part of biological di-
+versity (Dominoni et al., 2020; Carvalho et al., 2022).
+arXiv:2301.02642v1  [cs.CV]  6 Jan 2023
+
+Tripletloss:randomtriplets
+camera_interaction
+climbing down
+climbing_up
+hanging
+running
+sitting
+sitting_on_back
+standing
+walkingTripletloss:randomlyinitialisedweights
+camera_interaction
+climbing_down
+climbing_up
+hanging
+running
+sitting
+sitting_on_back
+standing
+walkingPrevious works have employed deep neural net-
+works which leverage multiple modalities, such as
+RGB, optical flow, and audio (Sakib and Burghardt,
+2020; Bain et al., 2021), for the classification of great
+ape behaviours and actions. However, higher level ab-
+stractions such as pose or body part information have
+remained unexplored for addressing this task. In re-
+sponse, we propose utilising the latter together with
+RGB and optical flow in a triple-stream metric learn-
+ing system (see Fig. 1) for improved classification re-
+sults and domain visualisations relevant to biologists.
+104
+105
+# Samples (log)
+Behavioural action classes
+hanging
+walking
+sitting on back
+standing
+sitting
+climbing up
+camera interaction
+running
+climbing down
+Figure 2: Behavioural Actions in the PanAf-500 Data.
+Examples of each one of the nine behavioural action classes
+(top) and their distribution across the approx. 180k frames
+in the dataset (bottom). Note the imbalance of two orders of
+magnitude in the distribution. (best viewed zoomed)
+Great Ape Activities - This paper will focus on
+great ape activity recognition, where the coarse ac-
+tivity classes used are illustrated in Fig. 2 for the
+utilised PanAf-500 dataset (see Sec. 3).
+Note that
+computer vision would traditionally categorise these
+classes as actions whilst in the biological realm they
+represent behaviour (or aspects thereof) often cap-
+tured in ethograms (Nishida et al., 1999; Zamma and
+Matsusaka, 2015). For clarity, in this paper we will
+refer to these classes as behavioural actions recognis-
+ing historical traditions in both disciplines.
+We will approach the classification task via a deep
+metric learning system (Karaderi et al., 2022) that
+embeds inputs into a latent space and uses geomet-
+ric distances to form distributions that align with the
+semantic similarity captured by the classes (Hermans
+et al., 2017; Musgrave et al., 2020). A major advan-
+tage over standard supervised systems is that sample
+distances in visualisations of the latent space always
+relate to learned similarity and, thus, are more natu-
+rally interpretable by experts. We will also analyse
+the role that additional DensePose-Chimp informa-
+tion (Sanakoyeu et al., 2020) can play in improving
+recognition performance compared to systems that
+utilise RGB and optical flow only. Lastly, as shown
+by Sakib and Burghardt (Sakib and Burghardt, 2020),
+there are significant challenges in correctly classify-
+ing behavioural actions which occur infrequently and
+form the distribution tail (see Fig. 2). To address this,
+we will employ three long-tailed recognition (LTR)
+techniques to improve performance on tail classes; (i)
+logit adjustment (Menon et al., 2020); (ii) class bal-
+anced focal loss (Cui et al., 2019); and (iii) weight
+balancing (Alshammari et al., 2022).
+In summary, our contributions are as follows:
+(i) we implement the first deep metric learning system
+for recognising great ape behavioural actions; (ii) we
+show that utilising explicit pose information has a sig-
+nificant positive effect on recognition performance in
+this domain; and (iii) we establish that existing LTR
+techniques can be applied in a metric learning setting
+to improve performance on tail classes for the prob-
+lem. The proposed approaches improve the state-of-
+the-art performance benchmarks with respect to top-1
+(∼ 85%) and average per class (∼ 65%) accuracy on
+the PanAf-500 dataset.
+2
+RELATED WORK
+Action recognition aims to classify actions observed
+in video (Kalfaoglu et al., 2020; Shaikh and Chai,
+2021). Learning spatio-temporal features character-
+istic for actions (Simonyan and Zisserman, 2014) via
+various deep learning paradigms forms the approach
+of choice in the domain of human action recogni-
+tion (HAR). We will briefly review concepts from this
+field, before discussing specifc relevant great ape be-
+havioural action recognition and LTR methods.
+Human Action Recognition - Although there are
+numerous deep learning approaches to action recog-
+nition (Zhou et al., 2018; Lin et al., 2019; Tran et al.,
+2019; Kalfaoglu et al., 2020; Pan et al., 2019; Majd
+and Safabakhsh, 2020; Sharir et al., 2021; Zhang
+et al., 2021a) this work focuses on multi-stream ar-
+chitectures, which address key aspects of the action
+recognition problem (e.g., spatial and temporal) in-
+
+dependently and explicitly. Feichtenhofer et al. (Fe-
+ichtenhofer et al., 2019) introduced the SlowFast ar-
+chitecture which employs two streams, each operat-
+ing at different frame rates; a slow, low frame-rate
+pathway captures spatial information while the fast,
+high frame-rate pathway captures fine temporal detail.
+Other types of multi-stream networks process differ-
+ent visual modalities. Simonyan (Simonyan and Zis-
+serman, 2014) introduced a two-stream network that
+processes RGB and optical flow to exploit spatial and
+temporal semantics, respectively. Since then, several
+networks that utilise additional modalities, such as
+motion saliency (Zong et al., 2021) and audio (Wang
+et al., 2021), have been introduced. Recently, the in-
+troduction of pose, which is critical for the perception
+of actions (Le et al., 2022), has shown promising re-
+sults in multi-stream architectures (Hong et al., 2019;
+Hayakawa and Dariush, 2020; Duan et al., 2021; Li
+et al., 2022).
+In particular, the DensePose format
+provides an opportunity to exploit fine-grained, seg-
+mentation map-based pose representations for action
+recognition. Hayakawa et al. (Hayakawa and Dar-
+iush, 2020) combine RGB and DensePose estimations
+in a two-stream network and demonstrate strong per-
+formance on egocentric footage of humans. Whilst
+such significant progress has been made in the domain
+of HAR, research into great ape behavioural action
+recognition is still in its infancy and few systems have
+been tested on natural datasets.
+Great Ape Domain -
+To date, two systems
+have attempted automated great ape behavioural ac-
+tion recognition, both are multi-stream architectures.
+The first (Sakib and Burghardt, 2020) is based on the
+two-stream convolutional architecture by Simonyan
+et al. (Simonyan and Zisserman, 2014) and used 3D
+ResNet-18s for feature extraction and LSTM-based
+fusion of RGB and optical flow features. They report
+top-1 accuracy of 73.52% across the nine behavioural
+actions in the PanAf-500 dataset (see Sec. 3) and a
+relatively low average per class accuracy (42.33%),
+highlighting the issue of tail class performance. The
+second, proposed by Bain et al. (Bain et al., 2021),
+is a deep learning system that requires both audio
+and video inputs and detects two specific behaviours;
+buttress drumming and nut cracking. Their system
+utilised a 3D ResNet-18 and a 2D ResNet-18 for ex-
+traction of visual and assisting audio features, respec-
+tively, in different streams. They achieved an aver-
+age precision of 87% for buttress drumming and 85%
+for nut cracking on their unpublished dataset. How-
+ever, the multi-modal method is not applicable to all
+camera trap settings since many older models do not
+provide audio. It cannot be utilised on the PanAf-500
+dataset since many clips there do not contain audio.
+Long-tailed Recognition - Most natural recorded
+data exhibits long-tailed class distributions (Liu et al.,
+2019). This is true of great ape camera-trap footage
+which is dominated by commonly occurring be-
+haviours - even with only the nine classes of the
+PanAf-500 data the distribution shows a clear tail (see
+Fig. 2). Without addressing this issue, models trained
+on such data often exhibit poor performance on rare
+classes.
+Various counter-measures have been pro-
+posed (Verma et al., 2018; Kang et al., 2019; Zhang
+et al., 2021b).
+Class balanced losses assign addi-
+tional weights, typically determined by inverse class
+frequencies, to samples from rare classes and have
+yielded strong results when coupled with techniques
+to reduce per-class redundancy (Cui et al., 2019).
+Similarly, logit adjustment uses class frequencies to
+directly offset output logits in favour of minority
+classes during training (Menon et al., 2020).
+An
+orthogonal approach, based on the observation that
+weight norms for rare classes are smaller in naively
+trained classifiers, is to perform weight balancing (Al-
+shammari et al., 2022).
+These techniques have
+achieved strong results on several LTR benchmarks.
+Before detailing how we use triple-stream metric
+learning with explicit DensePose-Chimp processing
+and LTR extensions for behavioural action recogni-
+tion, we will briefly outline the utilised dataset.
+3
+DATASET
+The Pan-African dataset, gathered by the Pan African
+Programme: ‘The Cultured Chimpanzee’, comprises
+∼ 20,000 videos from footage gathered at 39 study
+sites spanning 15 African countries. Here we utilise
+a 500 video subset, PanAf-500, specifically ground-
+truth labelled for use in computer vision under re-
+producible and comparable benchmarks. It includes
+frame-by-frame annotations for full-body locations of
+great apes and nine behavioural actions (Sakib and
+Burghardt, 2020) across approximately 180k frames
+(see. Fig. 3). Fig. 2 displays the behavioural actions
+classes in focus together with their distribution. We
+utilised the PanAf-500 dataset for all experiments and
+employ the same training and test partitions described
+in (Sakib and Burghardt, 2020).
+4
+METHOD
+The proposed system utilises three visual modali-
+ties as input; RGB, optical flow, and DensePose-C
+estimations (Sanakoyeu et al., 2020), as illustrated
+in Fig. 1). All optical flow images are pre-computed
+using OpenCV’s implementation of the Dual TV L1
+algorithm (Zach et al., 2007). We employ the model
+developed by Sanakoyeu et al. (Sanakoyeu et al.,
+
+Figure 3: Frame-by-frame Ground Truth Annotations.
+Four still frames from PanAf-500 videos with annotations
+of location (green boxes) and behavioural actions (visu-
+alised as text) of the apes in-frame. (best viewed zoomed)
+x0 .... x128
+Concatenation
+Conv3D
+ResNet50
+ResNet50
+ResNet50
+b x 6144 x 5 x 8 x 8
+MaxPool3D
+Feature maps
+2048 x 5 x 8 x 8
+feature extractors
+Triple stream
+AdaptiveAvgPool3D
+Fc1
+b x 2048 x 3 x 5 x 5
+Fc2
+b x 1024
+x0 .... x128
+x0 .... x128
+x0 .... x128
+Multiplication
+L2 norm
+x0 .... x128
+RGB
+Optical flow
+Output
+DensePose-C
+Figure 4: Fusion Head Schematics. A component break-
+down of fusion by element-wise multiplication (left) and
+convolutional fusion (right) as applied for our work to ex-
+plore their impact on performance.
+2020) to generate DensePose-C segmentations de-
+scribing chimpanzee pose. The model predicts dense
+correspondences between image pixels and a 3-D ob-
+ject mesh where each mesh represents a chimpanzee
+body part specified by a selector I and local surface
+coordinates within each mesh indexed by U and V.
+Frame-by-frame application to each of the PanAf-
+500 videos yields DensePose-C estimates expressed
+in IUV coordinates.
+Each of the three input modalities is fed into a 3D
+ResNet-50 (Du Tran et al., 2017) backbone, which
+together act as a feature extractor (see Fig. 1). The
+input tensors into the backbones are 3D since inputs
+are processed in snippets, that is each stream accepts a
+sequence of n consecutive RGB frames, optical flow
+images, or IUV coordinates, respectively. The final
+fully-connected layer outputs an n-dimensional en-
+coding for each stream. These are fused into a single
+embedding using three popular approaches; (i) sim-
+ple averaging across streams; (ii) convolutional fusion
+whereby stream features are concatenated and passed
+to a 3D convolutional layer as a volume; and (iii)
+element-wise multiplication of all three embedding
+vectors followed by L2 normalisation. The latter two
+approaches are illustrated in detail in Fig. 4. A lin-
+ear layer at the end of the fusion head finally outputs
+the unified embedding as logits. Whilst this system
+was trained via metric learning - visually sketched in
+Fig. 1 (right) - a k-NN classifier is used to perform
+inference in the embedding space during evaluation.
+Let the parameters of this network fθ(·) be de-
+noted by θ. Furthermore, let fθ(x) = x be the short-
+hand for referring to embeddings. Our metric learn-
+ing objective is, thus, to minimise the distance be-
+tween anchor-positive embedding pairs d(xa,xp) and
+maximise distance between anchor-negative embed-
+ding pairs d(xa,xn), where d represents a Euclidean.
+Instead of using standard triplet loss (Hermans et al.,
+2017) LTL, we use an improved version (Andrew
+et al., 2021), where the model is optimised via a hy-
+brid reciprocal triplet and softmax cross-entropy loss:
+LRC = LCE +λ LRT.
+(1)
+It is assembled from two components balanced by
+λ = 0.1 as given in (Andrew et al., 2021). The two
+components themselves are evaluated as:
+LRT = d(xa,xp)+
+1
+d(xa,xn)
+(2)
+LCE = −log
+�
+exy
+∑C
+i=1 exi
+�
+,
+(3)
+where C denotes the total number of classes and y are
+the class labels.
+In order to extend this system into the LTR do-
+main we substitute the softmax cross-entropy term
+for losses calculated using; (i) cross-entropy soft-
+max with logit adjustment (Menon et al., 2020) LLA;
+(ii) class-balanced focal loss (Cui et al., 2019) LCB;
+and (iii) class-balanced focal loss with weight balanc-
+ing (Alshammari et al., 2022). The first two losses are
+evaluated as follows:
+LLA = −log
+� exy +τ · log πy
+∑C
+i=1 exi+τ · log πi
+�
+,
+(4)
+LCB = − 1−β
+1−βny
+C
+∑
+i=1
+(1− pi)γ log(pi),
+(5)
+
+Bushnestandingcomera_interaction
+camara
+interactionwhere π represents the class priors (i.e., class frequen-
+cies in the training set) and temperature factor τ = 1,
+β = 0.99 is the re-weighting hyper-parameter, n is the
+total number of samples, y are the classes, γ = 1 is the
+focal loss hyper-parameter and pi = σ(xi). Balancing
+the network weights θ is performed via a MaxNorm
+constraint ∥θl,i∥2
+2 ≤ δ2,∀i given in (Alshammari et al.,
+2022) imposed on each class filter i in the last layer l
+of the network where δ is the L2-norm ball radius. We
+will reference a LCB-based optimisation where weight
+balancing is performed via LWB.
+Methodologically, this described architecture ap-
+proaches the learning of behavioural great ape actions
+via five key capabilities: 1) utilisation of multiple rel-
+evant input modalities across an entire video snippet;
+2) effective streamed content encoding; 3) fusion into
+a single embedding space; 4) metric space optimisa-
+tion so that distances naturally reflect semantic sim-
+ilarity; and 5) taking into account class imbalances
+common to the domain content.
+5
+EXPERIMENTS
+5.1
+General Training Setup
+We train our architecture via SGD optimisation using
+batch size 32 and learning rate 10−4. Feature extrac-
+tor backbones are initialised with Kinetics-400 (Kay
+et al., 2017) pre-trained weights and training runs are
+distributed over 8 Tesla V100 GPUs for 100 epochs.
+5.2
+Baselines and Stream Ablations
+As shown in Tab. 1, we first establish performance
+benchmarks for one and two stream baseline archi-
+tectures of our system (rows 2–5) against the cur-
+rent state-of-the-art (row 1), which uses a ResNet-18
+backbone with focal loss LFL, SGD, and LSTM-based
+frame fusion (Sakib and Burghardt, 2020). As ex-
+pected, we confirmed that - using identical setups and
+losses - adding an optical flow stream is beneficial
+in the great ape domain mirroring HAR results (see
+rows 2 vs 4, and 3 vs 5). Additionally, models trained
+using LRC consistently outperformed standard triplet
+loss LRC scenarios (see rows 2 vs 3, and 4 vs 5). Fi-
+nally, a dual-stream version of our proposed architec-
+ture trained with LRC outperforms the state-of-the-art
+by a small margin (see rows 1 vs 5).
+5.3
+Triple-Stream Recognition
+As given in Tab. 1 rows 6–8, our proposed triple-
+stream architecture significantly outperforms all base-
+lines with regards to top-1 accuracy, achieving up to
+85.86%. Thus, explicit DensePose-C information ap-
+pears a useful information source for boosting be-
+havioural action recognition in great apes. However,
+Table 1: Behavioural Action Recognition Benchmarks.
+Top-1 and average per-class (C-Avg) accuracy performance
+on the PanAf-500 dataset for the current state-of-the-
+art (row 1), single and dual-stream baselines (rows 2–5),
+and our triple-stream networks (rows 6–8) for different fu-
+sion methodologies and losses tested.
+Models/Streams
+Fusion
+Loss
+Top-1
+C-Avg
+Sakib et al. 2020
+1
+RGB+OF
+LSTM
+LFL
+73.52%
+42.33%
+Up to Dual-Stream
+2
+RGB only
+None
+LTL
+55.50%
+32.67%
+3
+RGB only
+None
+LRC
+74.24%
+55.76%
+4
+RGB+OF
+Avg
+LTL
+62.90%
+39.10%
+5
+RGB+OF
+Avg
+LRC
+75.02%
+61.97%
+Triple-Stream (Ours)
+6
+RGB+OF+DP
+Avg
+LRC
+81.71%
+46.61%
+7
+RGB+OF+DP
+Conv
+LRC
+82.04%
+56.31%
+8
+RGB+OF+DP
+Elem
+LRC
+85.86%
+50.50%
+without LTR techniques all our triple-stream models
+are significantly outperformed by a dual-stream set-
+ting (row 5) with regards to average per-class accu-
+racy. This reduction is caused by significantly poorer
+performance on minority classes (see Sec. 5.4).
+Since the learned behavioural action embeddings
+are constructed as metric from the outset, they can
+be visualised meaningfully – we note that such data-
+driven visualisations are novel in the primatology do-
+main. Fig. 5 depicts such learned spaces for our data
+and architecture where, independent of stream cardi-
+nality, embeddings cluster the training data cleanly.
+This is of course expected given above 99% top-1
+training accuracy in all settings. Yet, behavioural ac-
+tions of great apes are highly intricate as well as vari-
+able and, even with approx. 144,000 training frames
+used, the model clearly shows signs of overfitting. As
+a result, test set embeddings exhibit significant cluster
+overlap. Sample groups representing sitting, standing,
+and walking, for instance, blend into one another. In
+addition to overfitting, this also highlights the transi-
+tional nature of these often temporarily adjacent and
+smoothly changing actions. Thus, future temporally
+transitional ground truth labelling may be needed to
+represent behavioural great ape action in the PanAf-
+500 dataset more authentically.
+5.4
+Fusing Streams
+When looking at the impact of information fusion
+methods on performance in more detail, we find that
+benchmarks vary significantly (see Tab. 1 rows 6–8)
+when we test averaging, element-wise multiplication,
+and convolutional fusion, as described in Sec. 4. Re-
+sults show that convolution and element-wise mul-
+tiplication improve performance slightly across both
+metrics when compared with averaging: top-1 accu-
+
+camera interaction
+climbing up
+climbing down
+hanging
+running
+sitting
+sitting on back
+standing
+walking
+Behavioural actions
+Single Stream (RGB)
+Kinetics pretrained (no training)
+Training
+Training
+Test
+Dual Stream (RGB+OF)
+Triple Stream (AllThree)
+Figure 5: Visualisations of Great Ape Behavioural Action Spaces. A 2D t-SNE (Wattenberg et al., 2016) visualisation of
+the 128-dimensional training (top-right) and test (bottom-right) embeddings produced by the single, dual and three-stream
+network with convolutional fusion. We can see that training set embeddings from all classes are clustered cleanly. In contrast,
+test set embeddings show significant overlap and only embeddings from majority classes form distinct clusters. This is
+consistent with the high top-1 accuracy and relatively low average per-class accuracy reported in Tab. 1
+racy improves by 0.33% and 4.1%, respectively (see
+rows 6–8). However, the most significant gains are
+observed with respect to average per class accuracy
+which increases by 3.44% for element-wise multipli-
+cation and 9.7% for convolutional fusion. Learnable
+parameters in the convolution method clearly help
+blending information even when only fewer samples
+are available for training. Building on this improve-
+ment, we will next investigate the impact of LTR
+methods in order to benefit tail class performance.
+5.5
+Long-tail Recognition
+When grouping behavioural actions into head (cov-
+ering sitting, standing, and walking) and remain-
+ing tail classes based on frequency in the data (see
+Fig. 2), a significant performance gap becomes appar-
+ent even when using the so far best C-Avg performing
+model (see Tab. 2 row 1). Employing LTR techniques
+can, however, reduce this gap and improve average
+per-class accuracy further as quantified across rows
+2–4 in Tab. 2). Fig. 6 shows t-SNE visualisations of
+the three LTR triple-stream approaches when trained
+with convolutional feature fusion. Particularly for the
+class-balanced approaches and weight-balancing se-
+tups (two rightmost), tail class clusters appear more
+clearly separated and class overlap is generally re-
+duced. Thus, for the great ape domain underrepre-
+sented classes are indeed an effective source of infor-
+mation for improving action separability in general.
+6
+CONCLUSION
+In this work we introduced the first deep metric learn-
+ing system for great ape behavioural action recogni-
+tion. We demonstrated that the proposed triple-stream
+architecture can provide leading state-of-the-art per-
+formance when tested on the PanAf-500 camera trap
+dataset covering 180,000 annotated frames across 500
+videos taken in the wild. We demonstrated that the ad-
+dition of a DensePose-C chimpanzee pose estimation
+stream into the embedding architecture is highly ef-
+fective and leads to system performance of 85.86%
+top-1 accuracy on the data.
+We also showed that
+adding LTR techniques that address poor tail class
+performance to the system can improve the average
+per-class accuracy to 65.66% on the dataset. Despite
+these improvements we note that both larger anno-
+tated datasets to counteract overfitting as well as more
+temporally blended forms of annotation (e.g. action
+transition annotations) would benefit the authenticity
+of data-driven great ape behavioural representations.
+We hope that the research presented here sparks fur-
+ther interest in this vital application area for the bene-
+fit of endangered species such as great apes.
+ACKNOWLEDGEMENTS
+We thank the Pan African Programme:
+‘The Cultured
+Chimpanzee’ team and its collaborators for allowing the use
+of their data for this paper. We thank Amelie Pettrich, An-
+tonio Buzharevski, Eva Martinez Garcia, Ivana Kirchmair,
+
+climbing up
+hanging
+running
+sitting
+sitting on back
+standing
+walking
+Logit adjustment
+No LTR augmentation
+Weight balanced
+CB (+focal loss)
+climbing down
+camera interaction
+Test
+Figure 6: Long-tail Test Embeddings. A 2D t-SNE visualisation of the 128-dimensional test embeddings produced by the
+three-stream network with convolutional fusion alone (leftmost) and augmented with each LTR technique; (i) logit adjustment
+(ii) CB (+focal loss) and (iii) weight balancing. All LTR-augmented methods improve clustering of embeddings belonging to
+tail classes. They appear more clearly separated and exhibit less overlap when compared with the non-LTR method.
+Sebastian Sch¨utte, Linda Gerlach and Fabina Haas. We also
+thank management and support staff across all sites; specif-
+ically Yasmin Moebius, Geoffrey Muhanguzi, Martha Rob-
+bins, Henk Eshuis, Sergio Marrocoli and John Hart. Thanks
+to the team at https://www.chimpandsee.org particularly
+Briana Harder, Anja Landsmann, Laura K. Lynn, Zuzana
+Mach´aˇckov´a, Heidi Pfund, Kristeena Sigler and Jane Wid-
+ness. The work that allowed for the collection of the dataset
+was funded by the Max Planck Society, Max Planck Society
+Innovation Fund, and Heinz L. Krekeler. In this respect we
+would like to thank: Ministre des Eaux et Forˆets, Minist`ere
+de l’Enseignement sup´erieur et de la Recherche scientifique
+in Cˆote d’Ivoire; Institut Congolais pour la Conservation de
+la Nature, Minist`ere de la Recherche Scientifique in Demo-
+cratic Republic of Congo; Forestry Development Authority
+in Liberia; Direction Des Eaux Et Forˆets, Chasses Et Con-
+servation Des Sols in Senegal; Makerere University Biolog-
+ical Field Station, Uganda National Council for Science and
+Technology, Uganda Wildlife Authority, National Forestry
+Authority in Uganda; National Institute for Forestry De-
+velopment and Protected Area Management, Ministry of
+Agriculture and Forests, Ministry of Fisheries and Environ-
+ment in Equatorial Guinea. This work was supported by the
+UKRI CDT in Interactive AI under grant EP/S022937/1.
+Table 2:
+LTR-enabled Behavioural Action Recogni-
+tion Benchmarks.
+Average per-class accuracy for our
+triple-stream network with convolutional fusion for best
+performing non-LTR method (row1), and three LTR ap-
+proaches (rows 2–4) targetting poor tail class performance.
+Method/Loss
+C-Avg
+Head
+Tail
+Non-LTR Triple-Stream
+1
+LRC
+56.31
+80.57
+44.78
+LTR Triple-Stream
+2
+LLA
+61.76
+83.22
+50.7
+3
+LCB
+63.56
+77.60
+55.95
+4
+LWB
+65.66
+82.55
+56.26
+REFERENCES
+Almond, R., Grooten, M., Juffe Bignoli, D., and Petersen,
+T. (2022).
+Wwf (2022) living planet report 2022 -
+building a nature-positive society. 1
+Alshammari, S., Wang, Y.-X., Ramanan, D., and Kong, S.
+(2022). Long-tailed recognition via weight balancing.
+In CVPR, pages 6897–6907. 2, 3, 4, 5
+Andrew, W., Gao, J., Mullan, S., Campbell, N., Dowsey,
+A. W., and Burghardt, T. (2021). Visual identification
+of individual holstein-friesian cattle via deep metric
+learning. Computers and Electronics in Agriculture,
+185:106133. 4
+Bain, M., Nagrani, A., Schofield, D., Berdugo, S., Bessa, J.,
+Owen, J., Hockings, K. J., Matsuzawa, T., Hayashi,
+M., Biro, D., et al. (2021). Automated audiovisual
+behavior recognition in wild primates.
+Science ad-
+vances, 7(46):eabi4883. 2, 3
+Carvalho, S., Wessling, E. G., Abwe, E. E., Almeida-
+Warren, K., Arandjelovic, M., Boesch, C., Danquah,
+E., Diallo, M. S., Hobaiter, C., Hockings, K., et al.
+(2022). Using nonhuman culture in conservation re-
+quires careful and concerted action. Conservation Let-
+ters, 15(2):e12860. 1
+Congdon, J., Hosseini, M., Gading, E., Masousi, M.,
+Franke, M., and MacDonald, S. (2022). The future
+of artificial intelligence in monitoring animal identifi-
+cation, health, and behaviour. 1
+Cui, Y., Jia, M., Lin, T.-Y., Song, Y., and Belongie, S.
+(2019). Class-balanced loss based on effective num-
+ber of samples. In CVPR, pages 9268–9277. 2, 3,
+4
+Dominoni, D. M., Halfwerk, W., Baird, E., Buxton, R. T.,
+Fern´andez-Juricic, E., Fristrup, K. M., McKenna,
+M. F., Mennitt, D. J., Perkin, E. K., Seymoure, B. M.,
+et al. (2020). Why conservation biology can benefit
+from sensory ecology. Nature Ecology & Evolution,
+4(4):502–511. 1
+Du Tran, H. W., Torresani, L., Ray, J., Lecun, Y., and Paluri,
+M. (2017). A closer look at spatiotemporal convolu-
+tions for action recognition.(2017). OK. 4
+Duan, M., Qiu, H., Zhang, Z., and Wu, Y. (2021). Ntu-
+
+densepose: A new benchmark for dense pose action
+recognition. In Big Data, pages 3170–3175. IEEE. 3
+Feichtenhofer, C., Fan, H., Malik, J., and He, K. (2019).
+Slowfast networks for video recognition.
+In ICCV,
+pages 6202–6211. 3
+Hayakawa, J. and Dariush, B. (2020).
+Recognition and
+3d localization of pedestrian actions from monocular
+video. In ITSC, pages 1–7. IEEE. 3
+Hermans, A., Beyer, L., and Leibe, B. (2017). In defense
+of the triplet loss for person re-identification. arXiv
+preprint arXiv:1703.07737. 2, 4
+Hong, J., Cho, B., Hong, Y. W., and Byun, H. (2019).
+Contextual action cues from camera sensor for multi-
+stream action recognition. Sensors, 19(6):1382. 3
+IUCN (2022). Iucn red list of threatened species version
+2022.1. 1
+Kalfaoglu, M. E., Kalkan, S., and Alatan, A. A. (2020).
+Late temporal modeling in 3d cnn architectures with
+bert for action recognition. In ECCV, pages 731–747.
+Springer. 2
+Kang, B., Xie, S., Rohrbach, M., Yan, Z., Gordo, A., Feng,
+J., and Kalantidis, Y. (2019). Decoupling representa-
+tion and classifier for long-tailed recognition. arXiv
+preprint arXiv:1910.09217. 3
+Karaderi, T., Burghardt, T., Hsiang, A. Y., Ramaer, J., and
+Schmidt, D. N. (2022). Visual microfossil identifica-
+tion via deep metric learning. In ICPRAI, pages 34–
+46. Springer. 2
+Kay, W., Carreira, J., Simonyan, K., Zhang, B., Hillier, C.,
+Vijayanarasimhan, S., Viola, F., Green, T., Back, T.,
+Natsev, P., et al. (2017). The kinetics human action
+video dataset. arXiv preprint arXiv:1705.06950. 5
+K¨uhl, H. S. and Burghardt, T. (2013).
+Animal biomet-
+rics: quantifying and detecting phenotypic appear-
+ance. TREE, 28(7):432–441. 1
+Le, V.-T., Tran-Trung, K., and Hoang, V. T. (2022). A com-
+prehensive review of recent deep learning techniques
+for human activity recognition. Computational Intel-
+ligence and Neuroscience, 2022. 3
+Li, Y., Lu, Z., Xiong, X., and Huang, J. (2022). Perf-net:
+Pose empowered rgb-flow net. In WACV, pages 513–
+522. 3
+Lin, J., Gan, C., and Han, S. (2019). Tsm: Temporal shift
+module for efficient video understanding. In ICCV,
+pages 7083–7093. 2
+Liu, Z., Miao, Z., Zhan, X., Wang, J., Gong, B., and Yu,
+S. X. (2019). Large-scale long-tailed recognition in
+an open world. In CVPR, pages 2537–2546. 3
+Majd, M. and Safabakhsh, R. (2020). Correlational con-
+volutional lstm for human action recognition. Neuro-
+computing, 396:224–229. 2
+Menon, A. K., Jayasumana, S., Rawat, A. S., Jain, H., Veit,
+A., and Kumar, S. (2020). Long-tail learning via logit
+adjustment. arXiv preprint arXiv:2007.07314. 2, 3, 4
+Musgrave, K., Belongie, S., and Lim, S.-N. (2020). Pytorch
+metric learning. 2
+Nishida, T., Kano, T., Goodall, J., McGrew, W. C., and
+Nakamura, M. (1999).
+Ethogram and ethnography
+of mahale chimpanzees.
+Anthropological Science,
+107(2):141–188. 2
+Pan, Y., Xu, J., Wang, M., Ye, J., Wang, F., Bai, K., and
+Xu, Z. (2019). Compressing recurrent neural networks
+with tensor ring for action recognition. In AAAI, vol-
+ume 33, pages 4683–4690. 2
+Sakib, F. and Burghardt, T. (2020). Visual recognition of
+great ape behaviours in the wild. VAIB. 2, 3, 5
+Sanakoyeu, A., Khalidov, V., McCarthy, M. S., Vedaldi, A.,
+and Neverova, N. (2020). Transferring dense pose to
+proximal animal classes. In CVPR, pages 5233–5242.
+2, 3, 4
+Shaikh, M. B. and Chai, D. (2021). Rgb-d data-based action
+recognition: A review. Sensors, 21(12):4246. 2
+Sharir, G., Noy, A., and Zelnik-Manor, L. (2021). An image
+is worth 16x16 words, what is a video worth? arXiv
+preprint arXiv:2103.13915. 2
+Simonyan, K. and Zisserman, A. (2014). Two-stream con-
+volutional networks for action recognition in videos.
+NeurIPS, 27. 2, 3
+Tran, D., Wang, H., Torresani, L., and Feiszli, M. (2019).
+Video classification with channel-separated convolu-
+tional networks. In ICCV, pages 5552–5561. 2
+Tuia, D., Kellenberger, B., Beery, S., Costelloe, B. R.,
+Zuffi, S., Risse, B., Mathis, A., Mathis, M. W., van
+Langevelde, F., Burghardt, T., et al. (2022). Perspec-
+tives in machine learning for wildlife conservation.
+Nature communications, 13(1):1–15. 1
+Verma, V., Lamb, A., Beckham, C., Najafi, A., Courville,
+A., Mitliagkas, I., and Bengio, Y. (2018). Manifold
+mixup: learning better representations by interpolat-
+ing hidden states. 3
+Wang, L., Yuan, X., Zong, M., Ma, Y., Ji, W., Liu, M.,
+and Wang, R. (2021). Multi-cue based four-stream 3d
+resnets for video-based action recognition. Informa-
+tion Sciences, 575:654–665. 3
+Wattenberg, M., Vi´egas, F., and Johnson, I. (2016). How to
+use t-sne effectively. Distill. 6
+Zach, C., Pock, T., and Bischof, H. (2007). A duality based
+approach for realtime tv-l 1 optical flow. In DAGM,
+pages 214–223. Springer. 3
+Zamma, K. and Matsusaka, T. (2015).
+Ethograms and
+the diversity of behaviors, page 510–518. Cambridge
+University Press. 2
+Zhang, Y., Li, X., Liu, C., Shuai, B., Zhu, Y., Brattoli, B.,
+Chen, H., Marsic, I., and Tighe, J. (2021a).
+Vidtr:
+Video transformer without convolutions.
+In ICCV,
+pages 13577–13587. 2
+Zhang, Y., Wei, X.-S., Zhou, B., and Wu, J. (2021b). Bag
+of tricks for long-tailed visual recognition with deep
+convolutional neural networks. In AAAI, volume 35,
+pages 3447–3455. 3
+Zhou, B., Andonian, A., Oliva, A., and Torralba, A. (2018).
+Temporal relational reasoning in videos. In ECCV,
+pages 803–818. 2
+Zong, M., Wang, R., Chen, X., Chen, Z., and Gong, Y.
+(2021). Motion saliency based multi-stream multiplier
+resnets for action recognition. Image and Vision Com-
+puting, 107:104108. 3
+

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+Elliptic flow measurement of J/ψ in PHENIX Run14
+Au+Au at √sNN = 200 GeV ∗
+Luis Bichon III (for the PHENIX collaboration,
+https://doi.org/10.5281/zenodo.7430208)
+Department of Physics and Astronomy, Vanderbilt University, Nashville, TN
+37235 USA
+We obtain the first measurement of J/ψ elliptic flow at RHIC energies
+in forward rapidity using data from the PHENIX detector and applying
+an event plane method. The dataset used contains 19 billion events from
+the PHENIX experiment’s Run 14 Au + Au dataset at √sNN = 200 GeV.
+PHENIX has measured a J/ψ v2 in a centrality range of 10 − 60% that is
+consistent with zero. Taken together with results from LHC the measure-
+ment of v2, which is consistent with zero may indicate that J/ψ production
+by coalescence is not significant at forward rapidity at RHIC energy.
+1. Introduction
+The QGP has been found to exhibit a nearly perfect fluid behavior [1].
+This behavior manifests itself as strong correlations between particles pro-
+duced in nuclear collisions. Presently, the detailed interactions of the heavy
+quarks in the QGP medium are under investigation and, because heavy fla-
+vor quarks will have relatively larger masses, they may not be thermalized
+and flow with the medium.
+The production of J/ψ in p+p collisions is
+theoretically well understood because they are produced in hard scattering
+processes. This feature in addition to their production in hard scattering
+events in the initial stages of the collision make them ideal probes for testing
+the properties of the QGP medium. However, in nucleus+nucleus collisions
+some of the produced J/ψ mesons may be dissolved by the QGP, which may
+create anisotropies in the observed J/ψ azimuthal distributions due to the
+different path length in the medium. Additionally, a similar signal may be
+created if the J/ψ thermalizes inside the medium and follows the pressure
+gradients as lighter particles do, or the J/ψ may dissociate, and the charm
+∗ Presented at the 29th International Conference on Ultrarelativistic Nucleus-Nucleus
+Collisions (Quark Matter 2022)
+(1)
+arXiv:2301.04186v1  [nucl-ex]  10 Jan 2023
+
+2
+QM˙Proceedings˙Bichon
+printed on January 12, 2023
+quarks could equilibrate which could lead to J/ψ regeneration. We present
+a preliminary result for J/ψ v2 using the PHENIX Run14 Au+Au dataset
+at √sNN = 200 GeV.
+2. Data Analysis & Methodology
+2.1. Dataset and Detectors
+In this analysis, we use the Run 14 Au+Au Muon Arm dataset at
+√sNN = 200 GeV containing 19 billion events. The dimuon decay channel
+is used to reconstruct candidate J/ψ mesons. The PHENIX experiment has
+a unique coverage at forward rapidity with muon identification. This in ad-
+dition to the large dataset of Au+Au collisions collected in 2014 allows for
+a statistically improved measurement of J/ψ elliptic flow at RHIC energies.
+The key detector in this analysis is the Forward Silicon Vertex Detector
+(FVTX). With the FVTX, an increase in precision vertexing capabilities
+was added to the muon spectrometers, enabling the rejection of muons from
+the decay of relatively long-lived particles, the rejection of muons from the
+decays of relatively long-lived particles, and an additional way of determin-
+ing the event plane [2].
+2.2. Combinatorial Background Subtraction
+To obtain a pure signal for the J/ψ from dimuon mass distributions we
+employ event-mixing as the standard method of removing the background
+dimuons.
+For this event-mixing method, the background is constructed
+from dimuon pairs of opposite sign, but the single muons come from differ-
+ent events. Mixed event dimuon pairs are only formed if two events have a
+centrality closer than 5%, a Z vertex closer than 0.75 cm and a event plane
+angle closer than π/20 rad.
+Using events instead of individual dimuons
+allows us to increase the likelihood that we are using combinatorial back-
+ground dimuons. A normalization factor must be applied for the background
+which can be obtained by using the ratio of like-sign pairs from the same
+event to like-sign pairs from mixed events. The signal is then obtained by
+the subtraction of the normalized background from the foreground.
+2.3. Fitting the Dimuon Mass Distribution
+In the fitting of the mass distributions, we assume the shape of the
+J/ψ signal to be a Crystal Ball function, and given the statistical precision
+of the dataset, we also apply the same shape to the Ψ(2S) to avoid their
+inclusion in the higher mass J/ψ region. The parameters of the Crystal Ball
+function are obtained using J/ψ embedded Monte Carlo simulation data.
+We produce simulated mass distributions for low/high pT and South/North
+
+QM˙Proceedings˙Bichon
+printed on January 12, 2023
+3
+arm rapidities, fitting the distributions allowing for the function to have
+free (α, n, ¯x, and σ) parameters. The J/ψ count for each distribution is
+obtained by the integral of the J/ψ crystal ball function in the fit (see Figure
+1).
+Fig. 1. Mass distributions using mixed-event subtraction for the unweighted “stan-
+dard” set. These are binned by pT in each column, and rapidity+∆φ angle for
+each row. The green/dashed curve is a Crystal Ball fitted to the J/ψ peak, the
+blue/dashed-dot curve is a Crystal Ball fitted to the ψ(2S) peak, the red/dotted
+curve is an exponential fitted to the remaining background after subtraction, and
+the black/solid curve is the total fit.
+
+0 <Pr [GeV/c] ≤ 1
+1 <Pr [GeV/c] ≤ 2
+2 <Pr [GeV/c] ≤ 3
+3 <pr [GeV/c] ≤ 5
+300E
+160
+250
+元-4
+x = 3.125 ± 0.003 GeV/c2
+x=3.125+0.003GeV/c2
+x = 3.118 ± 0.004 GeV/c2
+80
+140
+x =3.118 ± 0.004 GeV/c2
+VI
+ = 0.135 ± 0.003 GeV/c2
+250
+ = 0.135 ± 0.003 GeV/c2
+ = 0.160 ± 0.002 GeV/c2
+ = 0.160 ± 0.002 GeV/c2
+200
+α= 1.19
+α= 1.19
+120
+α = 1.00
+α = 1.00
+n = 3.94
+n= 3.94
+n = 4.06
+n = 4.06
+200
+60
+150
+100
+>
+Nj/±= 1422±70
+Nj/ =1683±74
+150
+South Arm - 0 
+100
+80
+PH米ENIX
+PH米ENIX
+40
+100
+PH米ENIX
+PH米ENIX
+60
+preliminary
+preliminary
+preliminary
+preliminary
+50
+40
+20+
+20
+2
+2.5
+3
+3.5
+4.5
+2
+2.5
+3
+3.5
+4.5
+2
+2.5
+3
+3.5
+4.5
+2.5
+3
+3.5
+4.5
+Mass [GeV/c2]
+Mass [GeV/c2
+Mass [GeV/c2]
+Mass [GeV/c2]
+300日
+250
+80
+π-2
+T+T
+x= 3.125 ±0.003 GeV/c2
+250E
+ = 3.125 ± 0.003 GeV/c2
+x=3.118 ±0.004 GeV/c2
+x = 3.118 ± 0.004 GeV/c2
+VI
+ = 0.135 ± 0.003 GeV/c2
+ = 0.135 ± 0.003 GeV/c2
+ = 0.160 ± 0.002 GeV/c2
+ = 0.160 ± 0.002 GeV/c2
+200
+α= 1.19
+α= 1.19
+α = 1.00
+α = 1.00
+v
+n = 3.94
+200
+n = 3.94
+n = 4.06
+60
+100
+n = 4.06
+150
+V
+Nj/ = 1374±66
+150E
+Nj/Φ =1582±72
+N
+π一4
+100
+40
+PH米ENIX
+100E
+PH米ENIX
+PH米ENIX
+PH米ENIX
+South Arm -
+preliminary
+preliminary
+preliminary
+preliminary
+50
+20
+-50
+50田
+50
+2
+2.5
+3.5
+4.5
+2.5
+4.5
+2.5
+4.5
+2.5
+Mass [GeV/c2]
+Mass [GeV/c2]
+Mass [GeV/c2]
+Mass [GeV/c2]
+140F
+80
+允-4
+120
+3.147±0.006GeV/c2
+120
+x=3.147±0.006 GeV/c2
+x = 3.159 ± 0.006 GeV/c2
+50E
+x = 3.159 ± 0.006 GeV/c2
+VI
+ = 0.160 ± 0.002 GeV/c2
+ = 0.160 ± 0.002 GeV/c2
+ = 0.146 ± 0.006 GeV/c2
+ = 0.146 ± 0.006 GeV/c2
+>-
+100
+α = 0.63
+100
+α = 0.63
+60
+α= 1.17
+α = 1.17
+n = 7.60
+n= 7.60
+n = 2.55
+40
+n = 2.55
+80E
+Nj/ =902±63
+[Nj/=1013±63
+30
+60
+60
+PH*ENIX
+North Arm -
+PH米ENIX
+PH米ENIX
+PH米ENIX
+20
+preliminary
+40
+preliminary
+preliminary
+preliminary
+20
+10吨
+-20
+2
+2.5
+3.5
+2.5
+2.5
+3.5
+4.5
+2.5
+Mass [GeV/c2]
+Mass [GeV/c2
+Mass [GeV/c2]
+Mass [GeV/c2]
+160
+150
+100
+40
+π-2
+X= 3.147 ± 0.006 GeV/c2
+x=3.147 ±0.006 GeV/c2
+x=3.159±0.006 GeV/c2
+x= 3.159 ± 0.006 GeV/c2
+VI
+140
+ = 0.160 ± 0.002 GeV/c2
+ = 0.160 ± 0.002 GeV/c2
+ = 0.146 ± 0.006 GeV/c2
+g = 0.146 ± 0.006 GeV/c2
+α = 0.63
+α = 0.63
+80
+α = 1.17
+30
+α = 1.17
+120
+n = 7.60
+100
+n = 7.60
+n = 2.55
+n = 2.55
+>
+100
+60
+"j/ = 893±60
+Nj/=474±35
+π-4
+20
+PH米ENIX
+50
+PH米ENIX
+PH米ENIX
+PH米ENIX
+North Arm -
+60E
+preliminary
+preliminary
+preliminary
+10-
+preliminary
+40
+20
+-20E
+50
+20—
+2
+2.5
+3
+3.5
+4.5
+2.5
+3
+3.5
+4.5
+2
+2.5
+3
+3.5
+4.5
+2.5
+3
+3.5
+Mass [GeV/c2]
+Mass [GeV/c2]
+Mass [GeV/c2]4
+QM˙Proceedings˙Bichon
+printed on January 12, 2023
+2.4. Event Plane Method and Measuring v2
+We are primarily using the In/Out ratio method, which is an event plane
+method [3] that uses the counts of the J/ψ in bins of ∆φ to measure v2.
+The In/Out ratio method splits the distributions into 2 bins of ∆φ one in
+plane with the event plane and the other out of plane. We measure v2 using
+this method by looking at the difference between these bins. If there is no
+preference in either plane, we would observe a flow around zero.
+2.5. Systematic Uncertainties
+The systematic uncertainties are determined by changing various aspects
+of the analysis. As of this time, we have employed changing the primary
+detector of the analysis from the FVTX to the Central Arm Spectrometers
+(CNT), which covers a different pseudorapidity range.
+We have used a
+different method for our combinatorial background subtraction, the like-sign
+method, which constructs the background with dimuon pairs of the same
+sign (µ+µ+ and µ−µ−) that come from the same event. The uncertainty in
+the normalization factor in the event-mixing method was also incorporated
+into the systematic uncertainty. The last systematic uncertainty we consider
+comes from the mass fitting of the dimuon distribution, where the shape
+of the continuum distribution was assumed to be an exponential function,
+and the uncertainty in this assumption can be explored by assuming no
+continuum contribution in the J/ψ mass region.
+3. Results
+Figure 2 shows the pT -dependent J/ψ v2.
+The measurement in this
+analysis for PHENIX Run 14 at forward rapidity in a centrality range of 10
+- 60% is shown in red. The measurement made by STAR at mid-rapidity
+and in a centrality range of 10-40% is shown in black. The ALICE result
+at forward rapidity in a centrality range of 20-40% is shown in blue. Boxes
+surrounding the data points represent systematic uncertainties.
+PHENIX observes a larger suppression of J/ψ yield in forward rapidity
+when compared to mid-rapidity. This is contrary to expectations, because
+effects that dissolve the J/ψ have been determined to be stronger at mid-
+rapidity [4]. To understand this observation we begin by looking into the
+production of c¯c pairs. The majority of c¯c pairs per event in central collisions
+at RHIC are produced at mid-rapidity. At LHC energies, less suppression
+is observed, where many more c¯c pairs per event in central collisions are
+produced [5]. To explain this behavior, theoretical models require a contri-
+bution of coalescence via a recombination mechanism between charm and
+anticharm quarks [6]. It was found that the strength of this coalescence
+
+QM˙Proceedings˙Bichon
+printed on January 12, 2023
+5
+effect increases with the initial number of produced c¯c pairs relative to the
+total number of quarks, increasing with the collisions energy.
+At LHC energies, a nonzero v2 is observed, this is in line with J/ψ
+formed by coalescence in the QGP medium, and carrying the azimuthal
+anisotropy of the system [7]. At RHIC energies, STAR has measured v2 that
+is consistent with zero, but due to limited statistics remains inconclusive [8].
+With coalescence being the dominant mechanism for nonzero J/ψ v2 it
+should follow that systems where fewer c¯c pairs are formed should have a
+smaller azimuthal anisotropy.
+Fig. 2. Plot of pT dependent J/ψ v2. The PHENIX result in light gray/red/circle
+is compared to STAR [8] in black/star and ALICE [7] gray/blue/square.
+From the figure we can see the clear nonzero v2 measured by ALICE.
+Although the ALICE measurement is at a much higher energy, we know
+
+0.3
+Au+Au → J/ + X /Snn = 200 GeV
+PHENIX Run14. 10 - 60%. 1.2 < Iyl < 2.2
+★ STAR, 10 - 40%, lyl < 1 (PRL 111, 052301 (2013))
+0.2
+Pb+Pb → J/Φ + X /Snn = 5.02 TeV
+ALICE, 20 - 40%, 2.5 < lyl < 4.4 (JHEP 10 (2020) 141)
+0.1
+-0.1
+PH米ENIX
+-0.2
+preliminary
+-0.3
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+pT [GeV/c]6
+QM˙Proceedings˙Bichon
+printed on January 12, 2023
+v2 does not scale with energy for J/ψ, so it makes for a good comparison
+that the ALICE result which is clearly nonzero is different from our mea-
+surement. In our measurement, we see a v2 that is clearly consistent with
+zero across all pT bins. The systematic uncertainties were conservatively
+estimated, not taking into account cancellations or correlations of uncer-
+tainties from different sources. Additional data from Run 16 of RHIC will
+be included in the final results, and we expect that both statistical and
+systematic uncertainties will be significantly reduced.
+4. Conclusion and Outlook
+We have presented PHENIX Run 14 pT -dependent J/ψ v2 at forward
+rapidity at √sNN = 200 GeV. PHENIX has measured a J/ψ v2 that is
+consistent with zero. We have determined that the ALICE result, where
+there is clearly nonzero v2, is distinctly different from our measurement,
+and that forward and mid-rapidity results at RHIC are consistent, but the
+uncertainties are still large. In the future, we will incorporate Run 16 data
+in our measurement, essentially doubling the current dataset and reducing
+statistical uncertainties accordingly. We also plan to study open heavy flavor
+v2 to obtain a more complete understanding of the heavy flavor dynamics
+at RHIC.
+REFERENCES
+[1] Ulrich Heinz.
+The strongly coupled quark–gluon plasma created at RHIC.
+Journal of Physics A: Mathematical and Theoretical, 42(21):214003, May 2009.
+[2] C. Aidala et al. The PHENIX forward silicon vertex detector. Nuclear Instru-
+ments and Methods in Physics Research Section A: Accelerators, Spectrometers,
+Detectors and Associated Equipment, 755:44–61, Aug 2014.
+[3] A. M. Poskanzer and S. A. Voloshin. Methods for analyzing anisotropic flow in
+relativistic nuclear collisions. Physical Review C, 58(3):1671–1678, Sep 1998.
+[4] A. Adare et al. J/ψ suppression at forward rapidity in Au+Au collisions at
+√sNN = 200 GeV. Physical Review C, 84:054912, Nov 2011.
+[5] Anton Andronic, Peter Braun-Munzinger, Krzysztof Redlich, and Johanna
+Stachel. Decoding the phase structure of QCD via particle production at high
+energy. Nature, 561(7723):321–330, Sep 2018.
+[6] H. Pereira Da Costa et al. Charmonium production in Pb–Pb collisions with
+ALICE at the LHC. Nuclear Physics A, 956:705–708, Dec 2016.
+[7] S. Acharya et al. J/ψ elliptic and triangular flow in Pb-Pb collisions at √sNN
+= 5.02 TeV. Journal of High Energy Physics, 2020(10), Oct 2020.
+[8] L. Adamczyk et al.
+Measurement of J/ψ Azimuthal Anisotropy in Au+Au
+Collisions at √sNN = 200 GeV. Physical Review Letters, 111(5), Aug 2013.
+

1NE2T4oBgHgl3EQf5Ahd/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf,len=276
+page_content='Elliptic flow measurement of J/ψ in PHENIX Run14  Au+Au at √sNN = 200 GeV ∗  Luis Bichon III (for the PHENIX collaboration,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='7430208)  Department of Physics and Astronomy, Vanderbilt University, Nashville, TN  37235 USA  We obtain the first measurement of J/ψ elliptic flow at RHIC energies  in forward rapidity using data from the PHENIX detector and applying  an event plane method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The dataset used contains 19 billion events from  the PHENIX experiment’s Run 14 Au + Au dataset at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  PHENIX has measured a J/ψ v2 in a centrality range of 10 − 60% that is  consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Taken together with results from LHC the measure-  ment of v2, which is consistent with zero may indicate that J/ψ production  by coalescence is not significant at forward rapidity at RHIC energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Introduction  The QGP has been found to exhibit a nearly perfect fluid behavior [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  This behavior manifests itself as strong correlations between particles pro-  duced in nuclear collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Presently, the detailed interactions of the heavy  quarks in the QGP medium are under investigation and, because heavy fla-  vor quarks will have relatively larger masses, they may not be thermalized  and flow with the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  The production of J/ψ in p+p collisions is  theoretically well understood because they are produced in hard scattering  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' This feature in addition to their production in hard scattering  events in the initial stages of the collision make them ideal probes for testing  the properties of the QGP medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' However, in nucleus+nucleus collisions  some of the produced J/ψ mesons may be dissolved by the QGP, which may  create anisotropies in the observed J/ψ azimuthal distributions due to the  different path length in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Additionally, a similar signal may be  created if the J/ψ thermalizes inside the medium and follows the pressure  gradients as lighter particles do, or the J/ψ may dissociate, and the charm  ∗ Presented at the 29th International Conference on Ultrarelativistic Nucleus-Nucleus  Collisions (Quark Matter 2022)  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='04186v1  [nucl-ex]  10 Jan 2023  2  QM˙Proceedings˙Bichon  printed on January 12, 2023  quarks could equilibrate which could lead to J/ψ regeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' We present  a preliminary result for J/ψ v2 using the PHENIX Run14 Au+Au dataset  at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Data Analysis & Methodology  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Dataset and Detectors  In this analysis, we use the Run 14 Au+Au Muon Arm dataset at  √sNN = 200 GeV containing 19 billion events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The dimuon decay channel  is used to reconstruct candidate J/ψ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The PHENIX experiment has  a unique coverage at forward rapidity with muon identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' This in ad-  dition to the large dataset of Au+Au collisions collected in 2014 allows for  a statistically improved measurement of J/ψ elliptic flow at RHIC energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  The key detector in this analysis is the Forward Silicon Vertex Detector  (FVTX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' With the FVTX, an increase in precision vertexing capabilities  was added to the muon spectrometers, enabling the rejection of muons from  the decay of relatively long-lived particles, the rejection of muons from the  decays of relatively long-lived particles, and an additional way of determin-  ing the event plane [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Combinatorial Background Subtraction  To obtain a pure signal for the J/ψ from dimuon mass distributions we  employ event-mixing as the standard method of removing the background  dimuons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  For this event-mixing method, the background is constructed  from dimuon pairs of opposite sign, but the single muons come from differ-  ent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Mixed event dimuon pairs are only formed if two events have a  centrality closer than 5%, a Z vertex closer than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='75 cm and a event plane  angle closer than π/20 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  Using events instead of individual dimuons  allows us to increase the likelihood that we are using combinatorial back-  ground dimuons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' A normalization factor must be applied for the background  which can be obtained by using the ratio of like-sign pairs from the same  event to like-sign pairs from mixed events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The signal is then obtained by  the subtraction of the normalized background from the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Fitting the Dimuon Mass Distribution  In the fitting of the mass distributions, we assume the shape of the  J/ψ signal to be a Crystal Ball function, and given the statistical precision  of the dataset, we also apply the same shape to the Ψ(2S) to avoid their  inclusion in the higher mass J/ψ region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The parameters of the Crystal Ball  function are obtained using J/ψ embedded Monte Carlo simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  We produce simulated mass distributions for low/high pT and South/North  QM˙Proceedings˙Bichon  printed on January 12, 2023  3  arm rapidities, fitting the distributions allowing for the function to have  free (α, n, ¯x, and σ) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The J/ψ count for each distribution is  obtained by the integral of the J/ψ crystal ball function in the fit (see Figure  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Mass distributions using mixed-event subtraction for the unweighted “stan-  dard” set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' These are binned by pT in each column, and rapidity+∆φ angle for  each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The green/dashed curve is a Crystal Ball fitted to the J/ψ peak, the  blue/dashed-dot curve is a Crystal Ball fitted to the ψ(2S) peak, the red/dotted  curve is an exponential fitted to the remaining background after subtraction, and  the black/solid curve is the total fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  0 <Pr [GeV/c] ≤ 1  1 <Pr [GeV/c] ≤ 2  2 <Pr [GeV/c] ≤ 3  3 <pr [GeV/c] ≤ 5  300E  160  250  元-4  x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  x=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='125+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003GeV/c2  x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='118 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='004 GeV/c2  80  140  x =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='118 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='004 GeV/c2  VI  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  250  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  200  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='19  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='19  120  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='00  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='00  n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='94  n= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='94  n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='06  n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='06  200  60  150  100  >  Nj/±= 1422±70  Nj/ =1683±74  150  South Arm - 0  100  80  PH米ENIX  PH米ENIX  40  100  PH米ENIX  PH米ENIX  60  preliminary  preliminary  preliminary  preliminary  50  40  20+  20  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  Mass [GeV/c2]  Mass [GeV/c2  Mass [GeV/c2]  Mass [GeV/c2]  300日  250  80  π-2  T+T  x= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='125 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  250E  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  x=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='118 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='004 GeV/c2  x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='118 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='004 GeV/c2  VI  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='003 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  200  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='19  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='19  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='00  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='00  v  n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='94  200  n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='94  n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='06  60  100  n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='06  150  V  Nj/ = 1374±66  150E  Nj/Φ =1582±72  N  π一4  100  40  PH米ENIX  100E  PH米ENIX  PH米ENIX  PH米ENIX  South Arm -  preliminary  preliminary  preliminary  preliminary  50  20  50  50田  50  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  Mass [GeV/c2]  Mass [GeV/c2]  Mass [GeV/c2]  Mass [GeV/c2]  140F  80  允-4  120  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='147±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006GeV/c2  120  x=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='147±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='159 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  50E  x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='159 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  VI  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='146 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='146 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  >-  100  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='63  100  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='63  60  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='17  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='17  n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='60  n= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='60  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='55  40  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='55  80E  Nj/ =902±63  [Nj/=1013±63  30  60  60  PH*ENIX  North Arm -  PH米ENIX  PH米ENIX  PH米ENIX  20  preliminary  40  preliminary  preliminary  preliminary  20  10吨  20  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  Mass [GeV/c2]  Mass [GeV/c2  Mass [GeV/c2]  Mass [GeV/c2]  160  150  100  40  π-2  X= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='147 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  x=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='147 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  x=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='159±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  x= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='159 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  VI  140  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='002 GeV/c2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='146 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='146 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='006 GeV/c2  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='63  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='63  80  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='17  30  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='17  120  n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='60  100  n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='60  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='55  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='55  >  100  60  "j/ = 893±60  Nj/=474±35  π-4  20  PH米ENIX  50  PH米ENIX  PH米ENIX  PH米ENIX  North Arm -  60E  preliminary  preliminary  preliminary  10-  preliminary  40  20  20E  50  20—  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  Mass [GeV/c2]  Mass [GeV/c2]  Mass [GeV/c2]4  QM˙Proceedings˙Bichon  printed on January 12, 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Event Plane Method and Measuring v2  We are primarily using the In/Out ratio method, which is an event plane  method [3] that uses the counts of the J/ψ in bins of ∆φ to measure v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  The In/Out ratio method splits the distributions into 2 bins of ∆φ one in  plane with the event plane and the other out of plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' We measure v2 using  this method by looking at the difference between these bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' If there is no  preference in either plane, we would observe a flow around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Systematic Uncertainties  The systematic uncertainties are determined by changing various aspects  of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' As of this time, we have employed changing the primary  detector of the analysis from the FVTX to the Central Arm Spectrometers  (CNT), which covers a different pseudorapidity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  We have used a  different method for our combinatorial background subtraction, the like-sign  method, which constructs the background with dimuon pairs of the same  sign (µ+µ+ and µ−µ−) that come from the same event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The uncertainty in  the normalization factor in the event-mixing method was also incorporated  into the systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The last systematic uncertainty we consider  comes from the mass fitting of the dimuon distribution, where the shape  of the continuum distribution was assumed to be an exponential function,  and the uncertainty in this assumption can be explored by assuming no  continuum contribution in the J/ψ mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Results  Figure 2 shows the pT -dependent J/ψ v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  The measurement in this  analysis for PHENIX Run 14 at forward rapidity in a centrality range of 10  60% is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The measurement made by STAR at mid-rapidity  and in a centrality range of 10-40% is shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The ALICE result  at forward rapidity in a centrality range of 20-40% is shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Boxes  surrounding the data points represent systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  PHENIX observes a larger suppression of J/ψ yield in forward rapidity  when compared to mid-rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' This is contrary to expectations, because  effects that dissolve the J/ψ have been determined to be stronger at mid-  rapidity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' To understand this observation we begin by looking into the  production of c¯c pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The majority of c¯c pairs per event in central collisions  at RHIC are produced at mid-rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' At LHC energies, less suppression  is observed, where many more c¯c pairs per event in central collisions are  produced [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' To explain this behavior, theoretical models require a contri-  bution of coalescence via a recombination mechanism between charm and  anticharm quarks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' It was found that the strength of this coalescence  QM˙Proceedings˙Bichon  printed on January 12, 2023  5  effect increases with the initial number of produced c¯c pairs relative to the  total number of quarks, increasing with the collisions energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  At LHC energies, a nonzero v2 is observed, this is in line with J/ψ  formed by coalescence in the QGP medium, and carrying the azimuthal  anisotropy of the system [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' At RHIC energies, STAR has measured v2 that  is consistent with zero, but due to limited statistics remains inconclusive [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  With coalescence being the dominant mechanism for nonzero J/ψ v2 it  should follow that systems where fewer c¯c pairs are formed should have a  smaller azimuthal anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Plot of pT dependent J/ψ v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The PHENIX result in light gray/red/circle  is compared to STAR [8] in black/star and ALICE [7] gray/blue/square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  From the figure we can see the clear nonzero v2 measured by ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  Although the ALICE measurement is at a much higher energy, we know  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='3  Au+Au → J/ + X /Snn = 200 GeV  PHENIX Run14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' 10 - 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='2 < Iyl < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='2  ★ STAR, 10 - 40%, lyl < 1 (PRL 111, 052301 (2013))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='2  Pb+Pb → J/Φ + X /Snn = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='02 TeV  ALICE, 20 - 40%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5 < lyl < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='4 (JHEP 10 (2020) 141)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='1  PH米ENIX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='2  preliminary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='5  5  pT [GeV/c]6  QM˙Proceedings˙Bichon  printed on January 12, 2023  v2 does not scale with energy for J/ψ, so it makes for a good comparison  that the ALICE result which is clearly nonzero is different from our mea-  surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' In our measurement, we see a v2 that is clearly consistent with  zero across all pT bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The systematic uncertainties were conservatively  estimated, not taking into account cancellations or correlations of uncer-  tainties from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Additional data from Run 16 of RHIC will  be included in the final results, and we expect that both statistical and  systematic uncertainties will be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Conclusion and Outlook  We have presented PHENIX Run 14 pT -dependent J/ψ v2 at forward  rapidity at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' PHENIX has measured a J/ψ v2 that is  consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' We have determined that the ALICE result, where  there is clearly nonzero v2, is distinctly different from our measurement,  and that forward and mid-rapidity results at RHIC are consistent, but the  uncertainties are still large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' In the future, we will incorporate Run 16 data  in our measurement, essentially doubling the current dataset and reducing  statistical uncertainties accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' We also plan to study open heavy flavor  v2 to obtain a more complete understanding of the heavy flavor dynamics  at RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  REFERENCES  [1] Ulrich Heinz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  The strongly coupled quark–gluon plasma created at RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  Journal of Physics A: Mathematical and Theoretical, 42(21):214003, May 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Aidala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' The PHENIX forward silicon vertex detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Nuclear Instru-  ments and Methods in Physics Research Section A: Accelerators, Spectrometers,  Detectors and Associated Equipment, 755:44–61, Aug 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Poskanzer and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Voloshin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Methods for analyzing anisotropic flow in  relativistic nuclear collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Physical Review C, 58(3):1671–1678, Sep 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Adare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' J/ψ suppression at forward rapidity in Au+Au collisions at  √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Physical Review C, 84:054912, Nov 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [5] Anton Andronic, Peter Braun-Munzinger, Krzysztof Redlich, and Johanna  Stachel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Decoding the phase structure of QCD via particle production at high  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Nature, 561(7723):321–330, Sep 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Pereira Da Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Charmonium production in Pb–Pb collisions with  ALICE at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Nuclear Physics A, 956:705–708, Dec 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' J/ψ elliptic and triangular flow in Pb-Pb collisions at √sNN  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='02 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Journal of High Energy Physics, 2020(10), Oct 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Adamczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content='  Measurement of J/ψ Azimuthal Anisotropy in Au+Au  Collisions at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}
+page_content=' Physical Review Letters, 111(5), Aug 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'}

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+Inaccessible Neural Language Models Could
+Reinvigorate Linguistic Nativism
+Patrick Perrine
+California Polytechnic State University
+1 Grand Ave, San Luis Obispo, CA, 93410
+[email protected]
+Abstract
+Large Language Models (LLMs) have been mak-
+ing big waves in the machine learning community
+within the past few years. The impressive scalabil-
+ity of LLMs due to the advent of deep learning can
+be seen as a continuation of empiricist lingusitic
+methods, as opposed to rule-based linguistic meth-
+ods that are grounded in a nativist perspective. Cur-
+rent LLMs are generally inaccessible to resource-
+constrained researchers, due to a variety of factors
+including closed source code. This work argues
+that this lack of accessibility could instill a nativist
+bias in researchers new to computational linguis-
+tics, given that new researchers may only have rule-
+based, nativist approaches to study to produce new
+work. Also, given that there are numerous critics
+of deep learning claiming that LLMs and related
+methods may soon lose their relevancy, we spec-
+ulate that such an event could trigger a new wave
+of nativism in the language processing community.
+To prevent such a dramatic shift and placing favor
+in hybrid methods of rules and deep learning, we
+call upon researchers to open source their LLM
+code wherever possible to allow both empircist and
+hybrid approaches to remain accessible.
+1
+Introduction
+Large Language Models (LLMs) have been a pop-
+ular topic of research among the academic com-
+munity (Srivastava et al., 2022). The promise of
+a near-general purpose neural model for a variety
+of language processing tasks is indeed an attrac-
+tive one (Xu et al., 2022). Deep learning has made
+significant developments in language tasks such as
+conversational language understanding (Tur et al.,
+2018), spoken/text-based dialog systems (Celikyil-
+maz et al., 2018), and natural language generation
+from images (He and Deng, 2018). Large Lan-
+guage models can be viewed as the natural pro-
+gression away from the rigid rule-based systems
+that we’ve had since the 1950’s (Chiticariu et al.,
+2013), continuing the empiricist mentality of sta-
+tistical natural language processing without the po-
+tentially costly and context-specific activity of fea-
+ture engineering (Collobert et al., 2011). However,
+with large corporations touting their ever-growing,
+state-of-the-art models under closed-source code
+and payment walls, it could be seen that these
+large language models are becoming less acces-
+sible. Some organizations have acknowledged the
+potential harms that deep learning models could
+cause by establishing ethical frameworks (Ashurst
+et al., 2022) (Weidinger et al., 2022), but there are
+still growing concerns regarding accessibility and
+the result of false/irreproducible science (Kapoor
+and Narayanan, 2022).
+This criticism for empiricist methods is not new
+in linguistics-based science, in that Chomsky’s
+Poverty of the Stimulus Argument (Stich, 1978)
+has a rich history of discussion and debate amongst
+linguists, scientists, and philosophers (Laurence
+and Margolis, 2001). In this work, we will briefly
+introduce this debate over language learning be-
+tween nativists and empiricists, relate these topics
+to research in natural language processing, and dis-
+ucss how the current state of this research is rein-
+forcing an imbalance between the two perspectives.
+We intend to deliver a neutral ground of analysis,
+as we agree that a hybrid approach for NLP re-
+search can lead to strong results. The current bias
+towards the highly-popular, but inaccessible em-
+piricist methods utilizing LLMs could lead to a
+new wave of nativism in natural language process-
+ing work, following a large backlash against such
+empirical methods.
+2
+Background
+We now provide a holistic background on the lin-
+guistic and scientific developments that encompass
+this issue.
+arXiv:2301.05272v1  [cs.CL]  12 Jan 2023
+
+2.1
+The Three Waves of Modern NLP
+We will give a brief background on the three main
+waves of modern natural language processing re-
+search: the rule-based theories popularized by
+Noam Chomsky (Chomsky, 1965), the statistics-
+based empiricist experiments (Jelinek, 1976), and
+today’s popular methodology of deep learning for
+natural language processing (Collobert et al., 2011).
+The first wave is considered to be under a na-
+tivist perspective (Laurence and Margolis, 2001),
+whereas the latter waves are in support of an em-
+piricist lens (Frank et al., 2019).
+2.1.1
+Rule-based NLP
+The concept of viewing language as a static sys-
+tem of rules to determine interpretation has been
+present as early as the 1830’s (Humboldt, 1836).
+Noam Chomsky popularized this perspective in the
+domain of linguistics as a challenge to an exist-
+ing overbearance of empiricist methods (Chomsky,
+1956b; Laurence and Margolis, 2001).
+This rule-based approach to linguistics domi-
+nated the field for decades, following Chomsky’s
+mutliple works emphasizing and reinforcing this
+doctrine (Chomsky, 1956a, 1957, 1963, 1965;
+Chomsky and Halle, 1968). Being based in proposi-
+tional logic and a fixed content, rule-based methods
+are arguably rather accessible to researchers with
+limited resources. These methods continued to be
+prevalent in the field until the 1970’s, when statisti-
+cal methods were proven to be very useful.
+2.1.2
+Statistical NLP
+The roots of statistical language processing stem
+from Andrey Markov’s efforts in computing bi-
+gram and trigram probabilities (Jurafsky and Mar-
+tin, 2022) of vowel/consonant predictions using a
+novel as a corpus in 1913 (Markov, 2006). This
+n-gram approach was later applied to predicting
+sequences of English words (Shannon, 1948). This
+popularized the notion of using Markov chains for
+use in a variety of applications within and outside
+of linguistics.
+Chomsky specifically challenged this use of
+finite-state Markov processes, the processes that
+formed n-gram based approaches, to be useless
+in serving as a comprehensive cognitive model
+of grammatical knowledge in humans (Chomsky,
+1956b, 1957; Miller and Chomsky, 1963). This
+hindered the progress of probabilistic approaches
+in linguistics.
+Over a decade later, statistical language process-
+ing was revitalized due in part to a series of success-
+ful experiments using n-gram models for speech
+recognition (Baker, 1975a,b; Jelinek, 1976; Bahl
+et al., 1983; Jelinek et al., 1990). These empiricist-
+based experiments showed that Chomsky’s nativist
+theories do not extend to recognizing speech in real
+time as previously proposed (Chomsky and Halle,
+1968).
+This marked a shift towards looking at language
+processing through an empirical lens, where a hy-
+pothesis test primarily guides the experimentation
+process, rather than theoretical insights (Manning
+and Schutze, 1999). After the successful statistical
+speech recognition experiments of the mid 1970’s,
+statistical NLP reigned as the dominant approach
+for decades.
+2.1.3
+ML-based NLP
+Researchers soon began to use shallow neural net-
+works to reinforce statistical methodologies in NLP.
+In the late 2000’s, the advent of deeper neural net-
+works for NLP began to stir when scalable, hierar-
+chical language models (Morin and Bengio, 2005;
+Mnih and Hinton, 2008) and increased computing
+power became available for use by researchers.
+Alongside these developments, researchers be-
+came tiresome of having to hand-engineer features
+for neural networks to learn from, as this can be a
+costly and rather context-specific task (Collobert
+et al., 2011). In was in the 2010’s that deep learning
+became known more globally (LeCun et al., 2015),
+with NLP being a highly prominent application
+for deep neural networks. This sparked the current
+practice of training large language models in efforts
+to create a general model for many language tasks
+(Srivastava et al., 2022). In essence, the empiri-
+cist era of NLP has persisted to today through the
+evolution of deep learning practices. Some appli-
+cations of deep learning outside of language have
+even used empiricist terms such as tabula rasa very
+openly (Silver et al., 2017). The use of deep neural
+networks for language tasks has been confirmed to
+reinforce empircist ideology (Frank et al., 2019).
+3
+Deep Learning Can Be Inaccessible
+Deep learning as a science has been under fire for a
+number of reasons. While there have been encour-
+aging results across many application domains of
+deep learning and positive insights about their role
+in advancing empiricism (Buckner, 2018), deep
+learning has garnered skepticsm from both in and
+
+outside of its community (Marcus, 2018; Buckner,
+2019).
+These critcisms of deep NLP can stem from a
+lack of open sourcing of model code and also data
+(Klein et al., 2017; Fadel et al., 2019; Chen et al.,
+2021; Guo et al., 2022; Xu et al., 2022). These
+issues are not exclusive to language processing,
+as other domains have reasons to leave aspects of
+their experimentation private or inaccessible when
+publishing (Siegle et al., 2015; Suresha et al., 2018;
+Farooq and Hafeez, 2020; Zuin et al., 2020; Guo
+et al., 2022).
+We now focus on issues with closed-source large
+language models due to their popularity and the re-
+cent claims of greater intelligence (even sentience),
+as opposed to other models (y Arcas, 2022).
+4
+Potential Harms
+4.1
+Potential Harms of Open-Sourcing LLMs
+To offer a well-rounded argument in favor of open-
+sourcing LLMs, we will briefly cover some intu-
+itions behind close-sourcing them in terms of po-
+tential harms.
+LLMs could be repurposed for malicious pur-
+poses, particularly in generative tasks. LLMs have
+been seen to learn negative human biases/patterns
+in speech such as hate speech, discrimination, and
+the promotion of misinformation (Schramowski
+et al., 2022). If a powerful, pre-trained LLM is
+made open source, then it could be repurposed as
+an engine to cause harm across the internet at great
+scale (Weidinger et al., 2022). It could also be ar-
+gued that open sourcing LLM code that has been
+deployed to end-users could pose security risks
+(Chang et al., 2020).
+We counter the argument of potential LLM mis-
+use by malicious parties by arguing that such mod-
+els or derivatives of such should not be published
+in any form, open or closed source. We argue that
+LLM experimental papers that indicate such po-
+tential to cause harm at scale should be filtered
+out at the publication review stage, something that
+has been discussed in the deep learning community
+as of late (Ashurst et al., 2022). We also counter
+the security concern argument by saying that this
+could hold true for all open source software that is
+deployable, not just LLMs.
+4.2
+Potential Harms from Continued
+Close-Sourcing of LLMs
+We argue that there are more potential harms in the
+continued prevalence of close sourced LLM code
+than the potential harms of open sourcing them.
+4.2.1
+Nativist Biases
+Given that LLM experiments are becoming so large,
+costly, and complex, it is difficult to argue that an in-
+dependent researcher can stake a claim in this sub-
+field. With top publication venues focusing heavily
+on empiricist experimentation (Russell and Norvig,
+2021), researchers outside the typical corporate
+scope of research could be incentivized to explore
+nativist, rule-based approaches to solve problems
+in the NLP domain. If it is the empiricist group’s
+better interest to foster growth in their methodolo-
+gies and not opposing methods, steps should be
+taken in order to make their approaches accessible.
+Also, for hybrid methods to function, an ML-based
+solution should be made accessible to combine
+with the ruleset from the nativist side. This trend
+could be fostering a new generation of Chomsky-
+following nativist NLP researchers, which would
+not bode well for empiricists if the public begins to
+lose interest in deep learning methods for NLP.
+4.2.2
+Lack of Reproducibility
+We mention reproducibility and will further clarify
+its meaning due to an also recent, yet broader prob-
+lem in deep learning research, the reproducibility
+crisis (Kapoor and Narayanan, 2022). Not only are
+large language models becoming difficult to repro-
+duce, results from other areas of ML are becoming
+difficult to produce (de Freitas Pereira et al., 2022;
+Towers et al., 2022). Initiatives to measure repro-
+ducibility across publication venues have been cre-
+ated, such as the ML Reproducibility Challenge.
+LLM experiments have been specifically reviewed
+to have a questionable about of reproducibility
+(Crane, 2018; Wieling et al., 2018; Cahyawijaya
+et al., 2022; Silva et al., 2022). There is also im-
+plied to be a significant amount of computational
+irreproducibility of LLM experimentation, given
+model complexity and data, however, we leave this
+exploration for future work.
+There is some hope in the form of positive re-
+producibility reports in deep learning (Gibson and
+Cano, 2022). However, this growing amount of
+“bad press” for deep learning, specifically LLMs,
+could cause the public to begin distrusting LLM
+
+research. This, again, could trigger a revisiting of
+Chomsky’s rule-based theories of language.
+4.2.3
+Issues in NLP Education
+Given the previously mentioned issues, this lack
+of accessibility could affect the education of NLP
+methods. If students do not have access to code
+of LLMs, it could be difficult for them to learn to
+implement complex language model code of their
+own and learn to keep up with the state of the art. A
+lack of reproducibility could also be disenfranchis-
+ing to a young, empircist NLP researcher, leading
+them to pursue nativist approaches. These issues
+could reinforce the use of statistical, pre-deep learn-
+ing techniques in the classroom, but it is difficult to
+argue that publication venues are interested in shal-
+low neural network experimentation at this time.
+These issues combine to form an uneven playing
+field for students to study NLP in empiricist and
+hybrid forms. After studying NLP formally, they
+may be inclined to commit to nativist methods or
+even reinforce the popularity of them at scale.
+5
+Potential Solution
+We ask that publication venues merit open source
+LLM experiments significantly higher than they
+do currently. We believe that this would mitigate
+the issues discussed previously in this work. There
+seem to be developments occuring now in the deep
+learning publication space to help implement this in
+a proper form of governance (Ashurst et al., 2022).
+6
+Conclusion
+In this work, we provided a comprehensive history
+of natural language processing methodologies over
+roughly the past century. We then used this nar-
+rative to lead into today’s deep learning practices
+used in language processing, and current issues in
+an excessive closed sourcing of code for LLMs.
+It is our hope that this work inspires researchers
+and reviewers to champion open source language
+model code in order to pave the way for a more
+balanced research space.
+References
+Carolyn Ashurst, Emmie Hine, Paul Sedille, and Alexis
+Carlier. 2022. Ai ethics statements: Analysis and
+lessons learnt from neurips broader impact state-
+ments. In 2022 ACM Conference on Fairness, Ac-
+countability, and Transparency, pages 2047–2056.
+Lalit R Bahl, Frederick Jelinek, and Robert L Mer-
+cer. 1983. A maximum likelihood approach to con-
+tinuous speech recognition. IEEE transactions on
+pattern analysis and machine intelligence, (2):179–
+190.
+James Baker. 1975a. The dragon system–an overview.
+IEEE Transactions on Acoustics, speech, and signal
+Processing, 23(1):24–29.
+James K. Baker. 1975b.
+Stochastic Modeling as a
+Means of Automatic Speech Recognition. Ph.D. the-
+sis, USA.
+Cameron Buckner. 2018. Empiricism without magic:
+Transformational abstraction in deep convolutional
+neural networks. Synthese, 195(12):5339–5372.
+Cameron
+Buckner.
+2019.
+Deep
+learning:
+A
+philosophical introduction.
+Philosophy compass,
+14(10):e12625.
+Samuel Cahyawijaya, Alham Fikri Aji, Holy Lovenia,
+Genta Indra Winata, Bryan Wilie, Rahmad Mahen-
+dra, Fajri Koto, David Moeljadi, Karissa Vincentio,
+Ade Romadhony, et al. 2022. Nusacrowd: A call
+for open and reproducible nlp research in indonesian
+languages. arXiv preprint arXiv:2207.10524.
+Asli Celikyilmaz, Li Deng, and Dilek Hakkani-Tür.
+2018. Deep learning in spoken and text-based dia-
+log systems. In Deep Learning in Natural Language
+Processing, pages 49–78. Springer.
+Heng Chang, Yu Rong, Tingyang Xu, Wenbing Huang,
+Honglei Zhang, Peng Cui, Wenwu Zhu, and Jun-
+zhou Huang. 2020. A restricted black-box adversar-
+ial framework towards attacking graph embedding
+models. In Proceedings of the AAAI Conference on
+Artificial Intelligence, volume 34, pages 3389–3396.
+Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan,
+Henrique Ponde de Oliveira Pinto, Jared Kaplan,
+Harri Edwards, Yuri Burda, Nicholas Joseph, Greg
+Brockman, et al. 2021.
+Evaluating large lan-
+guage models trained on code.
+arXiv preprint
+arXiv:2107.03374.
+Laura Chiticariu, Yunyao Li, and Frederick Reiss. 2013.
+Rule-based information extraction is dead! long live
+rule-based information extraction systems! In Pro-
+ceedings of the 2013 conference on empirical meth-
+ods in natural language processing, pages 827–832.
+Noam Chomsky. 1956a. The logical structure of lin-
+guistic theory. Synthese, 40(2):317–352.
+Noam Chomsky. 1956b. Three models for the descrip-
+tion of language. IRE Transactions on information
+theory, 2(3):113–124.
+Noam Chomsky. 1957. Syntactic structures. In Syntac-
+tic Structures. De Gruyter Mouton.
+Noam Chomsky. 1963. Formal properties of grammars.
+Handbook of Math. Psychology, 2:328–418.
+
+Noam Chomsky. 1965. Aspects of the theory of syntax.
+Multilingual Matters: MIT Press.
+Noam Chomsky and Morris Halle. 1968. The sound
+pattern of english.
+Ronan Collobert, Jason Weston, Léon Bottou, Michael
+Karlen, Koray Kavukcuoglu, and Pavel Kuksa.
+2011.
+Natural language processing (almost) from
+scratch.
+Journal of machine learning research,
+12(ARTICLE):2493–2537.
+Matt Crane. 2018. Questionable answers in question
+answering research: Reproducibility and variability
+of published results. Transactions of the Association
+for Computational Linguistics, 6:241–252.
+Tiago de Freitas Pereira, Dominic Schmidli, Yu Linghu,
+Xinyi Zhang, Sébastien Marcel, and Manuel Gün-
+ther. 2022.
+Eight years of face recognition re-
+search: Reproducibility, achievements and open is-
+sues. arXiv, (2208.04040).
+Ali Fadel, Ibraheem Tuffaha, Mahmoud Al-Ayyoub,
+et al. 2019.
+Arabic text diacritization using deep
+neural networks. In 2019 2nd international confer-
+ence on computer applications & information secu-
+rity (ICCAIS), pages 1–7. IEEE.
+Muhammad Farooq and Abdul Hafeez. 2020. Covid-
+resnet:
+A deep learning framework for screen-
+ing of covid19 from radiographs.
+arXiv preprint
+arXiv:2003.14395.
+Stefan L Frank,
+Padraic Monaghan,
+and Chara
+Tsoukala. 2019. Neural network models of language
+acquisition and processing.
+In Human language:
+From genes and brain to behavior, pages 277–293.
+MIT Press.
+Perry Gibson and José Cano. 2022.
+Productive
+reproducible workflows for dnns:
+A case study
+for industrial defect detection.
+arXiv preprint
+arXiv:2206.09359.
+Yinpeng Guo, Liangyou Li, Xin Jiang, and Qun Liu.
+2022.
+Freetransfer-x: Safe and label-free cross-
+lingual transfer from off-the-shelf models.
+arXiv
+preprint arXiv:2206.06586.
+Xiaodong He and Li Deng. 2018.
+Deep learning in
+natural language generation from images. In Deep
+Learning in Natural Language Processing, pages
+289–307. Springer.
+W von Humboldt. 1836.
+Uber die verschiedenheit
+des menschlichen sprachbaues. Berlin: Konigliche
+Akademie der Wissenschaften.
+Fred Jelinek, B Merialdo, S Roukos, M Strauss, et al.
+1990. Self-organized language modeling for speech
+recognition. In Readings in speech recognition.
+Frederick Jelinek. 1976. Continuous speech recogni-
+tion by statistical methods. Proceedings of the IEEE,
+64(4):532–556.
+Daniel Jurafsky and James H Martin. 2022.
+Speech
+and language processing: An introduction to natural
+language processing, computational linguistics, and
+speech recognition (draft of january 12, 2022).
+Sayash Kapoor and Arvind Narayanan. 2022. Leakage
+and the reproducibility crisis in ml-based science.
+arXiv preprint arXiv:2207.07048.
+Guillaume Klein, Yoon Kim, Yuntian Deng, Jean Senel-
+lart, and Alexander M Rush. 2017. Opennmt: Open-
+source toolkit for neural machine translation. arXiv
+preprint arXiv:1701.02810.
+Stephen Laurence and Eric Margolis. 2001.
+The
+poverty of the stimulus argument. British Journal
+for the Philosophy of Science, 52(2).
+Yann LeCun, Yoshua Bengio, Geoffrey Hinton, et al.
+2015. Deep learning. Nature, 521 (7553), 436-444.
+Christopher Manning and Hinrich Schutze. 1999.
+Foundations of statistical natural language process-
+ing. MIT press.
+Gary Marcus. 2018.
+Deep learning: A critical ap-
+praisal. arXiv preprint arXiv:1801.00631.
+Andre˘ı Andreevich Markov. 2006. An example of sta-
+tistical investigation of the text eugene onegin con-
+cerning the connection of samples in chains. Science
+in Context, 19(4):591–600.
+George A Miller and Noam Chomsky. 1963. Finitary
+models of language users. handbook of mathemati-
+cal psychology, vol. by r. duncan luce, robert r. bush,
+and eugene galanter, 419–91.
+Andriy Mnih and Geoffrey E Hinton. 2008. A scalable
+hierarchical distributed language model. Advances
+in neural information processing systems, 21.
+Frederic Morin and Yoshua Bengio. 2005. Hierarchi-
+cal probabilistic neural network language model. In
+International workshop on artificial intelligence and
+statistics, pages 246–252. PMLR.
+Stuart Russell and Peter Norvig. 2021. Artificial Intel-
+ligence: A Modern Approach. Pearson.
+Patrick Schramowski, Cigdem Turan, Nico Andersen,
+Constantin A Rothkopf, and Kristian Kersting. 2022.
+Large pre-trained language models contain human-
+like biases of what is right and wrong to do. Nature
+Machine Intelligence, 4(3):258–268.
+Claude Elwood Shannon. 1948. A mathematical the-
+ory of communication.
+The Bell system technical
+journal, 27(3):379–423.
+Joshua H Siegle, Gregory J Hale, Jonathan P Newman,
+and Jakob Voigts. 2015. Neural ensemble communi-
+ties: open-source approaches to hardware for large-
+scale electrophysiology. Current opinion in neurobi-
+ology, 32:53–59.
+
+Marília Costa Rosendo Silva, Felipe Alves Siqueira,
+João Pedro Mantovani Tarrega, João Vitor Pataca
+Beinotti, Augusto Sousa Nunes, Miguel de Mattos
+Gardini, Vinícius Adolfo Pereira da Silva, Nádia
+Félix Felipe da Silva, and André Carlos Ponce de
+Leon Ferreira de Carvalho. 2022.
+No pattern, no
+recognition: a survey about reproducibility and dis-
+tortion issues of text clustering and topic modeling.
+arXiv preprint arXiv:2208.01712.
+David Silver, Julian Schrittwieser, Karen Simonyan,
+Ioannis Antonoglou, Aja Huang, Arthur Guez,
+Thomas Hubert, Lucas Baker, Matthew Lai, Adrian
+Bolton, et al. 2017. Mastering the game of go with-
+out human knowledge. nature, 550(7676):354–359.
+Aarohi Srivastava, Abhinav Rastogi, Abhishek Rao,
+Abu Awal Md Shoeb, Abubakar Abid, Adam Fisch,
+Adam R Brown, Adam Santoro, Aditya Gupta,
+Adrià Garriga-Alonso, et al. 2022.
+Beyond the
+imitation game: Quantifying and extrapolating the
+capabilities of language models.
+arXiv preprint
+arXiv:2206.04615.
+Stephen P Stich. 1978. Empiricism, innateness, and
+linguistic universals. Philosophical Studies: An In-
+ternational Journal for Philosophy in the Analytic
+Tradition, 33(3):273–286.
+S Suresha, L Kidzi´nski, E Halilaj, GE Gold, and
+SL Delp. 2018.
+Automated staging of knee os-
+teoarthritis severity using deep neural networks. Os-
+teoarthritis and Cartilage, 26:S441.
+David Towers,
+Matthew Forshaw,
+Amir Atapour-
+Abarghouei, and Andrew Stephen McGough. 2022.
+Long-term reproducibility for neural architecture
+search. arXiv preprint arXiv:2207.04821.
+Gokhan Tur, Asli Celikyilmaz, Xiaodong He, Dilek
+Hakkani-Tür, and Li Deng. 2018.
+Deep learning
+in conversational language understanding. In Deep
+Learning in Natural Language Processing, pages
+23–48. Springer.
+Laura Weidinger, Jonathan Uesato, Maribeth Rauh,
+Conor
+Griffin,
+Po-Sen
+Huang,
+John
+Mellor,
+Amelia Glaese, Myra Cheng, Borja Balle, Atoosa
+Kasirzadeh, et al. 2022. Taxonomy of risks posed
+by language models. In 2022 ACM Conference on
+Fairness, Accountability, and Transparency, pages
+214–229.
+Martijn Wieling, Josine Rawee, and Gertjan van Noord.
+2018. Reproducibility in computational linguistics:
+are we willing to share? Computational Linguistics,
+44(4):641–649.
+Frank F Xu, Uri Alon, Graham Neubig, and Vincent Jo-
+sua Hellendoorn. 2022. A systematic evaluation of
+large language models of code. In Proceedings of
+the 6th ACM SIGPLAN International Symposium on
+Machine Programming, pages 1–10.
+Blaise Agüera y Arcas. 2022. Do large language mod-
+els understand us? Daedalus, 151(2):183–197.
+Gianluca Zuin, Adriano Veloso, João Cândido Porti-
+nari, and Nivio Ziviani. 2020. Automatic tag recom-
+mendation for painting artworks using diachronic de-
+scriptions. In 2020 International Joint Conference
+on Neural Networks (IJCNN), pages 1–8. IEEE.
+

1NE4T4oBgHgl3EQfzQ1E/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf,len=404
+page_content='Inaccessible Neural Language Models Could  Reinvigorate Linguistic Nativism  Patrick Perrine  California Polytechnic State University  1 Grand Ave, San Luis Obispo, CA, 93410  paperrin@calpoly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='edu  Abstract  Large Language Models (LLMs) have been mak-  ing big waves in the machine learning community  within the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The impressive scalabil-  ity of LLMs due to the advent of deep learning can  be seen as a continuation of empiricist lingusitic  methods, as opposed to rule-based linguistic meth-  ods that are grounded in a nativist perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Cur-  rent LLMs are generally inaccessible to resource-  constrained researchers, due to a variety of factors  including closed source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This work argues  that this lack of accessibility could instill a nativist  bias in researchers new to computational linguis-  tics, given that new researchers may only have rule-  based, nativist approaches to study to produce new  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Also, given that there are numerous critics  of deep learning claiming that LLMs and related  methods may soon lose their relevancy, we spec-  ulate that such an event could trigger a new wave  of nativism in the language processing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  To prevent such a dramatic shift and placing favor  in hybrid methods of rules and deep learning, we  call upon researchers to open source their LLM  code wherever possible to allow both empircist and  hybrid approaches to remain accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  1  Introduction  Large Language Models (LLMs) have been a pop-  ular topic of research among the academic com-  munity (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The promise of  a near-general purpose neural model for a variety  of language processing tasks is indeed an attrac-  tive one (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Deep learning has made  significant developments in language tasks such as  conversational language understanding (Tur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=',  2018), spoken/text-based dialog systems (Celikyil-  maz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2018), and natural language generation  from images (He and Deng, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Large Lan-  guage models can be viewed as the natural pro-  gression away from the rigid rule-based systems  that we’ve had since the 1950’s (Chiticariu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=',  2013), continuing the empiricist mentality of sta-  tistical natural language processing without the po-  tentially costly and context-specific activity of fea-  ture engineering (Collobert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' However,  with large corporations touting their ever-growing,  state-of-the-art models under closed-source code  and payment walls, it could be seen that these  large language models are becoming less acces-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Some organizations have acknowledged the  potential harms that deep learning models could  cause by establishing ethical frameworks (Ashurst  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022) (Weidinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022), but there are  still growing concerns regarding accessibility and  the result of false/irreproducible science (Kapoor  and Narayanan, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  This criticism for empiricist methods is not new  in linguistics-based science, in that Chomsky’s  Poverty of the Stimulus Argument (Stich, 1978)  has a rich history of discussion and debate amongst  linguists, scientists, and philosophers (Laurence  and Margolis, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In this work, we will briefly  introduce this debate over language learning be-  tween nativists and empiricists, relate these topics  to research in natural language processing, and dis-  ucss how the current state of this research is rein-  forcing an imbalance between the two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  We intend to deliver a neutral ground of analysis,  as we agree that a hybrid approach for NLP re-  search can lead to strong results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The current bias  towards the highly-popular, but inaccessible em-  piricist methods utilizing LLMs could lead to a  new wave of nativism in natural language process-  ing work, following a large backlash against such  empirical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2  Background  We now provide a holistic background on the lin-  guistic and scientific developments that encompass  this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='05272v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='CL]  12 Jan 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1  The Three Waves of Modern NLP  We will give a brief background on the three main  waves of modern natural language processing re-  search: the rule-based theories popularized by  Noam Chomsky (Chomsky, 1965), the statistics-  based empiricist experiments (Jelinek, 1976), and  today’s popular methodology of deep learning for  natural language processing (Collobert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  The first wave is considered to be under a na-  tivist perspective (Laurence and Margolis, 2001),  whereas the latter waves are in support of an em-  piricist lens (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1  Rule-based NLP  The concept of viewing language as a static sys-  tem of rules to determine interpretation has been  present as early as the 1830’s (Humboldt, 1836).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky popularized this perspective in the  domain of linguistics as a challenge to an exist-  ing overbearance of empiricist methods (Chomsky,  1956b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Laurence and Margolis, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  This rule-based approach to linguistics domi-  nated the field for decades, following Chomsky’s  mutliple works emphasizing and reinforcing this  doctrine (Chomsky, 1956a, 1957, 1963, 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Chomsky and Halle, 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Being based in proposi-  tional logic and a fixed content, rule-based methods  are arguably rather accessible to researchers with  limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' These methods continued to be  prevalent in the field until the 1970’s, when statisti-  cal methods were proven to be very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='2  Statistical NLP  The roots of statistical language processing stem  from Andrey Markov’s efforts in computing bi-  gram and trigram probabilities (Jurafsky and Mar-  tin, 2022) of vowel/consonant predictions using a  novel as a corpus in 1913 (Markov, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This  n-gram approach was later applied to predicting  sequences of English words (Shannon, 1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This  popularized the notion of using Markov chains for  use in a variety of applications within and outside  of linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Chomsky specifically challenged this use of  finite-state Markov processes, the processes that  formed n-gram based approaches, to be useless  in serving as a comprehensive cognitive model  of grammatical knowledge in humans (Chomsky,  1956b, 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Miller and Chomsky, 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This  hindered the progress of probabilistic approaches  in linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Over a decade later, statistical language process-  ing was revitalized due in part to a series of success-  ful experiments using n-gram models for speech  recognition (Baker, 1975a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Jelinek, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Bahl  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Jelinek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' These empiricist-  based experiments showed that Chomsky’s nativist  theories do not extend to recognizing speech in real  time as previously proposed (Chomsky and Halle,  1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  This marked a shift towards looking at language  processing through an empirical lens, where a hy-  pothesis test primarily guides the experimentation  process, rather than theoretical insights (Manning  and Schutze, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' After the successful statistical  speech recognition experiments of the mid 1970’s,  statistical NLP reigned as the dominant approach  for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='3  ML-based NLP  Researchers soon began to use shallow neural net-  works to reinforce statistical methodologies in NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  In the late 2000’s, the advent of deeper neural net-  works for NLP began to stir when scalable, hierar-  chical language models (Morin and Bengio, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Mnih and Hinton, 2008) and increased computing  power became available for use by researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Alongside these developments, researchers be-  came tiresome of having to hand-engineer features  for neural networks to learn from, as this can be a  costly and rather context-specific task (Collobert  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In was in the 2010’s that deep learning  became known more globally (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2015),  with NLP being a highly prominent application  for deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This sparked the current  practice of training large language models in efforts  to create a general model for many language tasks  (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In essence, the empiri-  cist era of NLP has persisted to today through the  evolution of deep learning practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Some appli-  cations of deep learning outside of language have  even used empiricist terms such as tabula rasa very  openly (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The use of deep neural  networks for language tasks has been confirmed to  reinforce empircist ideology (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  3  Deep Learning Can Be Inaccessible  Deep learning as a science has been under fire for a  number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' While there have been encour-  aging results across many application domains of  deep learning and positive insights about their role  in advancing empiricism (Buckner, 2018), deep  learning has garnered skepticsm from both in and  outside of its community (Marcus, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Buckner,  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  These critcisms of deep NLP can stem from a  lack of open sourcing of model code and also data  (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Fadel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' These  issues are not exclusive to language processing,  as other domains have reasons to leave aspects of  their experimentation private or inaccessible when  publishing (Siegle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Suresha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Farooq and Hafeez, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Zuin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Guo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  We now focus on issues with closed-source large  language models due to their popularity and the re-  cent claims of greater intelligence (even sentience),  as opposed to other models (y Arcas, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  4  Potential Harms  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1  Potential Harms of Open-Sourcing LLMs  To offer a well-rounded argument in favor of open-  sourcing LLMs, we will briefly cover some intu-  itions behind close-sourcing them in terms of po-  tential harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  LLMs could be repurposed for malicious pur-  poses, particularly in generative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' LLMs have  been seen to learn negative human biases/patterns  in speech such as hate speech, discrimination, and  the promotion of misinformation (Schramowski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' If a powerful, pre-trained LLM is  made open source, then it could be repurposed as  an engine to cause harm across the internet at great  scale (Weidinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' It could also be ar-  gued that open sourcing LLM code that has been  deployed to end-users could pose security risks  (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  We counter the argument of potential LLM mis-  use by malicious parties by arguing that such mod-  els or derivatives of such should not be published  in any form, open or closed source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' We argue that  LLM experimental papers that indicate such po-  tential to cause harm at scale should be filtered  out at the publication review stage, something that  has been discussed in the deep learning community  as of late (Ashurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' We also counter  the security concern argument by saying that this  could hold true for all open source software that is  deployable, not just LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='2  Potential Harms from Continued  Close-Sourcing of LLMs  We argue that there are more potential harms in the  continued prevalence of close sourced LLM code  than the potential harms of open sourcing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='1  Nativist Biases  Given that LLM experiments are becoming so large,  costly, and complex, it is difficult to argue that an in-  dependent researcher can stake a claim in this sub-  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' With top publication venues focusing heavily  on empiricist experimentation (Russell and Norvig,  2021), researchers outside the typical corporate  scope of research could be incentivized to explore  nativist, rule-based approaches to solve problems  in the NLP domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' If it is the empiricist group’s  better interest to foster growth in their methodolo-  gies and not opposing methods, steps should be  taken in order to make their approaches accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Also, for hybrid methods to function, an ML-based  solution should be made accessible to combine  with the ruleset from the nativist side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This trend  could be fostering a new generation of Chomsky-  following nativist NLP researchers, which would  not bode well for empiricists if the public begins to  lose interest in deep learning methods for NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='2  Lack of Reproducibility  We mention reproducibility and will further clarify  its meaning due to an also recent, yet broader prob-  lem in deep learning research, the reproducibility  crisis (Kapoor and Narayanan, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Not only are  large language models becoming difficult to repro-  duce, results from other areas of ML are becoming  difficult to produce (de Freitas Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Towers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Initiatives to measure repro-  ducibility across publication venues have been cre-  ated, such as the ML Reproducibility Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  LLM experiments have been specifically reviewed  to have a questionable about of reproducibility  (Crane, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Wieling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Cahyawijaya  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' There is also im-  plied to be a significant amount of computational  irreproducibility of LLM experimentation, given  model complexity and data, however, we leave this  exploration for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  There is some hope in the form of positive re-  producibility reports in deep learning (Gibson and  Cano, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' However, this growing amount of  “bad press” for deep learning, specifically LLMs,  could cause the public to begin distrusting LLM  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' This, again, could trigger a revisiting of  Chomsky’s rule-based theories of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='3  Issues in NLP Education  Given the previously mentioned issues, this lack  of accessibility could affect the education of NLP  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' If students do not have access to code  of LLMs, it could be difficult for them to learn to  implement complex language model code of their  own and learn to keep up with the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' A  lack of reproducibility could also be disenfranchis-  ing to a young, empircist NLP researcher, leading  them to pursue nativist approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' These issues  could reinforce the use of statistical, pre-deep learn-  ing techniques in the classroom, but it is difficult to  argue that publication venues are interested in shal-  low neural network experimentation at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  These issues combine to form an uneven playing  field for students to study NLP in empiricist and  hybrid forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' After studying NLP formally, they  may be inclined to commit to nativist methods or  even reinforce the popularity of them at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  5  Potential Solution  We ask that publication venues merit open source  LLM experiments significantly higher than they  do currently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' We believe that this would mitigate  the issues discussed previously in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' There  seem to be developments occuring now in the deep  learning publication space to help implement this in  a proper form of governance (Ashurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  6  Conclusion  In this work, we provided a comprehensive history  of natural language processing methodologies over  roughly the past century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' We then used this nar-  rative to lead into today’s deep learning practices  used in language processing, and current issues in  an excessive closed sourcing of code for LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  It is our hope that this work inspires researchers  and reviewers to champion open source language  model code in order to pave the way for a more  balanced research space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  References  Carolyn Ashurst, Emmie Hine, Paul Sedille, and Alexis  Carlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Ai ethics statements: Analysis and  lessons learnt from neurips broader impact state-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In 2022 ACM Conference on Fairness, Ac-  countability, and Transparency, pages 2047–2056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Lalit R Bahl, Frederick Jelinek, and Robert L Mer-  cer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' A maximum likelihood approach to con-  tinuous speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' IEEE transactions on  pattern analysis and machine intelligence, (2):179–  190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  James Baker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1975a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The dragon system–an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  IEEE Transactions on Acoustics, speech, and signal  Processing, 23(1):24–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  James K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Baker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1975b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Stochastic Modeling as a  Means of Automatic Speech Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' the-  sis, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Cameron Buckner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Empiricism without magic:  Transformational abstraction in deep convolutional  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Synthese, 195(12):5339–5372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Cameron  Buckner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Deep  learning:  A  philosophical introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Philosophy compass,  14(10):e12625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Samuel Cahyawijaya, Alham Fikri Aji, Holy Lovenia,  Genta Indra Winata, Bryan Wilie, Rahmad Mahen-  dra, Fajri Koto, David Moeljadi, Karissa Vincentio,  Ade Romadhony, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Nusacrowd: A call  for open and reproducible nlp research in indonesian  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='10524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Asli Celikyilmaz, Li Deng, and Dilek Hakkani-Tür.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Deep learning in spoken and text-based dia-  log systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Deep Learning in Natural Language  Processing, pages 49–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Heng Chang, Yu Rong, Tingyang Xu, Wenbing Huang,  Honglei Zhang, Peng Cui, Wenwu Zhu, and Jun-  zhou Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' A restricted black-box adversar-  ial framework towards attacking graph embedding  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Proceedings of the AAAI Conference on  Artificial Intelligence, volume 34, pages 3389–3396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan,  Henrique Ponde de Oliveira Pinto, Jared Kaplan,  Harri Edwards, Yuri Burda, Nicholas Joseph, Greg  Brockman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Evaluating large lan-  guage models trained on code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv preprint  arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='03374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Laura Chiticariu, Yunyao Li, and Frederick Reiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Rule-based information extraction is dead!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' long live  rule-based information extraction systems!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Pro-  ceedings of the 2013 conference on empirical meth-  ods in natural language processing, pages 827–832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1956a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The logical structure of lin-  guistic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Synthese, 40(2):317–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1956b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Three models for the descrip-  tion of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' IRE Transactions on information  theory, 2(3):113–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Syntactic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Syntac-  tic Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' De Gruyter Mouton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Formal properties of grammars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Handbook of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Psychology, 2:328–418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Aspects of the theory of syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Multilingual Matters: MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Noam Chomsky and Morris Halle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' The sound  pattern of english.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Ronan Collobert, Jason Weston, Léon Bottou, Michael  Karlen, Koray Kavukcuoglu, and Pavel Kuksa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Natural language processing (almost) from  scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Journal of machine learning research,  12(ARTICLE):2493–2537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Matt Crane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Questionable answers in question  answering research: Reproducibility and variability  of published results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Transactions of the Association  for Computational Linguistics, 6:241–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Tiago de Freitas Pereira, Dominic Schmidli, Yu Linghu,  Xinyi Zhang, Sébastien Marcel, and Manuel Gün-  ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Eight years of face recognition re-  search: Reproducibility, achievements and open is-  sues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' arXiv, (2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='04040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Ali Fadel, Ibraheem Tuffaha, Mahmoud Al-Ayyoub,  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Arabic text diacritization using deep  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In 2019 2nd international confer-  ence on computer applications & information secu-  rity (ICCAIS), pages 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Muhammad Farooq and Abdul Hafeez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Covid-  resnet:  A deep learning framework for screen-  ing of covid19 from radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv preprint  arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='14395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Stefan L Frank,  Padraic Monaghan,  and Chara  Tsoukala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Neural network models of language  acquisition and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  In Human language:  From genes and brain to behavior, pages 277–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Perry Gibson and José Cano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Productive  reproducible workflows for dnns:  A case study  for industrial defect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='09359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Yinpeng Guo, Liangyou Li, Xin Jiang, and Qun Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Freetransfer-x: Safe and label-free cross-  lingual transfer from off-the-shelf models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv  preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='06586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Xiaodong He and Li Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Deep learning in  natural language generation from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Deep  Learning in Natural Language Processing, pages  289–307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  W von Humboldt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Uber die verschiedenheit  des menschlichen sprachbaues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Berlin: Konigliche  Akademie der Wissenschaften.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Fred Jelinek, B Merialdo, S Roukos, M Strauss, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Self-organized language modeling for speech  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Readings in speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Frederick Jelinek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Continuous speech recogni-  tion by statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Proceedings of the IEEE,  64(4):532–556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Daniel Jurafsky and James H Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Speech  and language processing: An introduction to natural  language processing, computational linguistics, and  speech recognition (draft of january 12, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Sayash Kapoor and Arvind Narayanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Leakage  and the reproducibility crisis in ml-based science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='07048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Guillaume Klein, Yoon Kim, Yuntian Deng, Jean Senel-  lart, and Alexander M Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Opennmt: Open-  source toolkit for neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' arXiv  preprint arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='02810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Stephen Laurence and Eric Margolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  The  poverty of the stimulus argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' British Journal  for the Philosophy of Science, 52(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Yann LeCun, Yoshua Bengio, Geoffrey Hinton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Nature, 521 (7553), 436-444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Christopher Manning and Hinrich Schutze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Foundations of statistical natural language process-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Gary Marcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Deep learning: A critical ap-  praisal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' arXiv preprint arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='00631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Andre˘ı Andreevich Markov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' An example of sta-  tistical investigation of the text eugene onegin con-  cerning the connection of samples in chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Science  in Context, 19(4):591–600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  George A Miller and Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Finitary  models of language users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' handbook of mathemati-  cal psychology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' duncan luce, robert r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' bush,  and eugene galanter, 419–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Andriy Mnih and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' A scalable  hierarchical distributed language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Advances  in neural information processing systems, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Frederic Morin and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Hierarchi-  cal probabilistic neural network language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In  International workshop on artificial intelligence and  statistics, pages 246–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Stuart Russell and Peter Norvig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Artificial Intel-  ligence: A Modern Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Pearson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Patrick Schramowski, Cigdem Turan, Nico Andersen,  Constantin A Rothkopf, and Kristian Kersting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Large pre-trained language models contain human-  like biases of what is right and wrong to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Nature  Machine Intelligence, 4(3):258–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Claude Elwood Shannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' A mathematical the-  ory of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  The Bell system technical  journal, 27(3):379–423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Joshua H Siegle, Gregory J Hale, Jonathan P Newman,  and Jakob Voigts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Neural ensemble communi-  ties: open-source approaches to hardware for large-  scale electrophysiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Current opinion in neurobi-  ology, 32:53–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Marília Costa Rosendo Silva, Felipe Alves Siqueira,  João Pedro Mantovani Tarrega, João Vitor Pataca  Beinotti, Augusto Sousa Nunes, Miguel de Mattos  Gardini, Vinícius Adolfo Pereira da Silva, Nádia  Félix Felipe da Silva, and André Carlos Ponce de  Leon Ferreira de Carvalho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  No pattern, no  recognition: a survey about reproducibility and dis-  tortion issues of text clustering and topic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='01712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  David Silver, Julian Schrittwieser, Karen Simonyan,  Ioannis Antonoglou, Aja Huang, Arthur Guez,  Thomas Hubert, Lucas Baker, Matthew Lai, Adrian  Bolton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Mastering the game of go with-  out human knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' nature, 550(7676):354–359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Aarohi Srivastava, Abhinav Rastogi, Abhishek Rao,  Abu Awal Md Shoeb, Abubakar Abid, Adam Fisch,  Adam R Brown, Adam Santoro, Aditya Gupta,  Adrià Garriga-Alonso, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Beyond the  imitation game: Quantifying and extrapolating the  capabilities of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='04615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Stephen P Stich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Empiricism, innateness, and  linguistic universals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Philosophical Studies: An In-  ternational Journal for Philosophy in the Analytic  Tradition, 33(3):273–286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  S Suresha, L Kidzi´nski, E Halilaj, GE Gold, and  SL Delp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Automated staging of knee os-  teoarthritis severity using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Os-  teoarthritis and Cartilage, 26:S441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  David Towers,  Matthew Forshaw,  Amir Atapour-  Abarghouei, and Andrew Stephen McGough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Long-term reproducibility for neural architecture  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='04821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Gokhan Tur, Asli Celikyilmaz, Xiaodong He, Dilek  Hakkani-Tür, and Li Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Deep learning  in conversational language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Deep  Learning in Natural Language Processing, pages  23–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Laura Weidinger, Jonathan Uesato, Maribeth Rauh,  Conor  Griffin,  Po-Sen  Huang,  John  Mellor,  Amelia Glaese, Myra Cheng, Borja Balle, Atoosa  Kasirzadeh, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Taxonomy of risks posed  by language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In 2022 ACM Conference on  Fairness, Accountability, and Transparency, pages  214–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Martijn Wieling, Josine Rawee, and Gertjan van Noord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Reproducibility in computational linguistics:  are we willing to share?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Computational Linguistics,  44(4):641–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Frank F Xu, Uri Alon, Graham Neubig, and Vincent Jo-  sua Hellendoorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' A systematic evaluation of  large language models of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In Proceedings of  the 6th ACM SIGPLAN International Symposium on  Machine Programming, pages 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Blaise Agüera y Arcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Do large language mod-  els understand us?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Daedalus, 151(2):183–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content='  Gianluca Zuin, Adriano Veloso, João Cândido Porti-  nari, and Nivio Ziviani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' Automatic tag recom-  mendation for painting artworks using diachronic de-  scriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' In 2020 International Joint Conference  on Neural Networks (IJCNN), pages 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf'}

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+ 
+1 
+The 3D Structural Phenotype of the Glaucomatous Optic Nerve 
+Head and its Relationship with The Severity of Visual Field Damage   
+ 
+Fabian A. Braeu1,2,3, Thanadet Chuangsuwanich1,3, Tin A. Tun4,5, Shamira A. Perera4,5, Rahat Husain4, 
+Aiste Kadziauskiene6,7, Leopold Schmetterer4,5,8-12, Alexandre H. Thiéry13, George Barbastathis2,14, Tin 
+Aung3,4,5, and Michaël J.A. Girard1,5,12 
+ 
+1. 
+Ophthalmic Engineering & Innovation Laboratory, Singapore Eye Research Institute, Singapore 
+National Eye Centre, Singapore 
+2. 
+Singapore-MIT Alliance for Research and Technology, Singapore 
+3. 
+Yong Loo Lin School of Medicine, National University of Singapore, Singapore 
+4. 
+Singapore Eye Research Institute, Singapore National Eye Centre, Singapore 
+5. 
+Duke-NUS Graduate Medical School, Singapore 
+6. 
+Clinic of Ears, Nose, Throat and Eye Diseases, Institute of Clinical Medicine, Faculty of Medicine, 
+Vilnius University, Vilnius, Lithuania 
+7. 
+Center of Eye Diseases, Vilnius University Hospital Santaros Klinikos, Vilnius, Lithuania 
+8. 
+SERI-NTU Advanced Ocular Engineering (STANCE), Singapore, Singapore 
+9. 
+School of Chemistry, Chemical Engineering and Biotechnology, Nanyang Technological University 
+Singapore 
+10. Department of Clinical Pharmacology, Medical University of Vienna, Austria 
+11. Center for Medical Physics and Biomedical Engineering, Medical University of Vienna, Austria 
+12. Institute of Molecular and Clinical Ophthalmology, Basel, Switzerland 
+13. Department of Statistics and Applied Probability, National University of Singapore, Singapore 
+14. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 
+Massachusetts 02139, USA 
+ 
+Keywords:  
+Geometric deep learning, glaucoma, artificial intelligence, optic nerve 
+head, PointNet 
+ 
+Word count: 
+ 
+4,971 (Manuscript Text) 
+ 
+ 
+339 (Abstract) 
+Tables: 
+ 
+1 
+Figures: 
+ 
+4 
+Conflict of Interest: 
+ 
+MJAG and AHT are the co-founders of the AI start-up company Abyss 
+Processing Pte Ltd   
+ 
+ 
+ 
+Corresponding Author:   
+Michaël J.A. Girard 
+ 
+ 
+Ophthalmic Engineering & Innovation Laboratory (OEIL)  
+ 
+ 
+Singapore Eye Research Institute (SERI)  
+ 
+ 
+The Academia, 20 College Road  
+ 
+ 
+Discovery Tower Level 6,  
+ 
+ 
+Singapore 169856  
+ 
+ 
+[email protected]  
+                                             
+https://www.ophthalmic.engineering 
+ 
+
+ 
+2 
+Abstract  
+Purpose: To describe the 3D structural changes in both connective and neural tissues of the 
+optic nerve head (ONH) that occur concurrently at different stages of glaucoma using 
+traditional and AI-driven approaches. 
+Design: Retrospective cross-sectional study. 
+Methods: We included 213 normal, 204 mild glaucoma (mean deviation [MD] ≥ -6.00 dB), 118 
+moderate glaucoma (MD of -6.01 to -12.00 dB), and 118 advanced glaucoma patients (MD < 
+-12.00 dB). All subjects had their ONHs imaged in 3D with Spectralis optical coherence 
+tomography. To describe the 3D structural phenotype of glaucoma as a function of severity, 
+we used two different approaches: (1) We extracted ‘human-defined’ 3D structural 
+parameters of the ONH (total of 10) including retinal nerve fiber layer (RNFL) thickness, 
+minimum rim width, lamina cribrosa (LC) shape and depth at different stages of glaucoma; 
+(2) we also employed a geometric deep learning method (i.e. PointNet) to identify the most 
+important 3D structural features that differentiate ONHs from different glaucoma severity 
+groups without any human input. 
+Results: We observed that the majority of ONH structural changes occurred in the early 
+glaucoma stage, followed by a plateau effect in the later stages. Using PointNet, we also found 
+that 3D ONH structural changes were present in both neural and connective tissues. 
+Specifically, 57% (normal to mild glaucoma), 39% (mild to moderate glaucoma), and 53% 
+(moderate to advanced glaucoma) of ONH landmarks that showed major structural changes 
+were located in neural tissues with the remaining located in connective tissues. In both 
+approaches, we observed that structural changes were more prominent in the superior and 
+inferior quadrant of the ONH, particularly in the RNFL, the prelamina, and the LC. As the 
+
+ 
+3 
+severity of glaucoma increased, these changes became more diffuse (i.e. widespread), 
+particularly in the LC. 
+Conclusions: In this study, we were able to uncover complex 3D structural changes of the 
+ONH in both neural and connective tissues as a function of glaucoma severity. We hope to 
+provide new insights into the complex pathophysiology of glaucoma that might help clinicians 
+in their daily clinical care. 
+ 
+
+ 
+4 
+Introduction 
+Evaluation of structural changes of the optic nerve head (ONH) – the main site of 
+damage in glaucoma – is a crucial step in diagnosing and monitoring glaucoma [1, 2]. The 
+complex three-dimensional (3D) morphological changes occurring in glaucomatous ONHs can 
+be captured and quantified by optical coherence tomography (OCT) – a fast, high-resolution, 
+quantitative, and non-invasive 3D imaging modality [3]. 
+In current medical practice, several investigations are conducted to assess neural tissue 
+health. These tests involve both a functional (e.g., visual field testing) and a structural 
+assessment of glaucomatous damage. The latter is typically achieved by measuring the 
+thickness of the retinal nerve fiber layer (RNFL) via OCT [4-6]. Researchers have further 
+investigated the association between other neural structural parameters with glaucomatous 
+visual field damage, such as the thickness of the ganglion cell complex (GCC) [7, 8] and Bruch’s 
+membrane opening - minimum rim width (BMO-MRW) [9, 10].  
+However, recent research has indicated that the pathophysiology of glaucoma is 
+multifaceted and cannot purely be characterized as damage to retinal ganglion cells: (1) 
+Brooks et al. reported that the characteristic “glaucomatous cupping” of the ONH cannot 
+solely be explained by neural tissue loss [11]; (2) Quigley et al. found that glaucomatous 
+changes of the lamina cribrosa (LC) precede visual field damage [12]; and (3) Yang et al. 
+suggested that ONH connective tissue deformations are the primary cause of retinal ganglion 
+cell axonal injury [13]. These studies indicate that the pathophysiology of glaucoma should 
+consider the involvement of the biomechanics, mechanobiology, remodeling, and potential 
+mechanical breakdown of ONH connective tissues. Given this new understanding, researchers 
+have begun to investigate the association between ONH connective tissue changes and 
+
+ 
+5 
+glaucoma severity through connective tissue parameters extracted from OCT images of the 
+ONH. Examples of such parameters include the LC depth (LCD) and the LC global shape index 
+(LC-GSI) [14], the thickness of the peripapillary choroid [15], the scleral canal opening [16], 
+and the peripapillary scleral angle representing the amount of bowing of the ONH [17]. 
+However, no study has yet provided a comprehensive analysis of 3D structural changes of 
+both the connective and neural tissues of the ONH that occur concurrently at different stages 
+of glaucoma. 
+Therefore, the aim of this study was to describe the 3D structural phenotype of 
+glaucoma as a function of severity by: (1) Extracting neural and connective tissue ONH 
+parameters from segmented 3D OCT scans and investigating their differences between 
+glaucoma severity groups; (2) Using 3D point clouds representing the complex structure of 
+the ONH as an input for a geometric deep learning technique (i.e. PointNet [18]) that allows 
+us to identify the major 3D structural changes of the ONH with glaucoma severity. Overall, we 
+hope that our work leads to a better understanding of the pathophysiology of glaucoma that 
+might improve the diagnosis and prognosis of glaucoma. 
+ 
+Methods 
+Patient Recruitment 
+ 
+This retrospective study involved a total of 414 subjects with glaucoma and 213 
+controls without glaucoma from two different cohorts: (1) 541 subjects of Chinese ethnicity 
+were recruited at the Singapore National Eye Centre (SNEC) as part of their standard clinical 
+care and (2) 112 subjects of European descent were recruited at the Vilnius University 
+Hospital Santaros Klinikos as part of a prospective observational study. All subjects gave 
+
+ 
+6 
+written informed consent and the study adhered to the tenets of the Declaration of Helsinki 
+and was approved by the institutional review board of the respective institutions (SingHealth 
+Centralized Institutional review board, Singapore and Vilnius Regional Biomedical Research 
+Ethics Committee, Lithuania). 
+Standard Automated Perimetry 
+ 
+All subjects had their visual field (VF) assessed by standard automated perimetry (SAP; 
+Swedish Interactive Threshold Algorithm standard 24-2 or 30-2 program; Humphrey Field 
+Analyzer II-750i, Carl Zeiss Meditec). Subjects with a non-reliable VF examination that was 
+defined using the criteria of a false-positive error rate greater than 15% [19] and a fixation 
+loss greater than 33% [19, 20] were excluded from the study. 
+Definition of Glaucoma and Glaucoma Severity Groups 
+ 
+Glaucomatous eyes were defined as those with vertical cup-disc ratio (VCDR) > 0.7 
+and/or neuroretinal rim narrowing with repeatable glaucomatous VF defects and non-
+occludable angles on gonioscopy whereas non-glaucomatous (normal) eyes were those with 
+an IOP < 21 mmHg and normal VF examinations. Subjects with corneal abnormalities that 
+potentially can reduce the quality of the OCT scans and with ONH disorders other than 
+glaucoma were excluded from the studies. 
+Based upon the mean deviation (MD) of the 24-2 or 30-2 VF, all glaucoma subjects 
+were further split into three glaucoma severity groups [21]: (1) mild glaucoma (MD ≥ -6.00 
+dB); (2) moderate glaucoma (MD of -6.01 to -12.00 dB); and (3) advanced glaucoma (MD < -
+12.00 dB). Even though this classification has its limitations [22], it remains a standard and 
+can be used as a good first indicator for staging functional damage. More information on the 
+
+ 
+7 
+demographics of the four groups (i.e. normal, mild, moderate, and advanced) can be found 
+in Table 1. 
+Optical Coherence Tomography Imaging 
+ 
+Each patient from both cohorts had their ONH imaged with the same spectral domain 
+OCT device (Spectralis, Heidelberg Engineering, Germany). All OCT scans (horizontal raster 
+scans) covered an area of 15x10 centered on the ONH and the number of B-scans varied 
+between 49 and 97 B-scans (distance between B-scans varied from approximately 35 to 70 
+µm) with 384 A-scans per B-scan (approximately 11.5 µm between A-scans) and 496 pixels 
+per A-scan (axial resolution of 3.87 µm/pixel). Images were acquired using signal averaging, 
+eye tracking, and the enhanced depth imaging modality of the Spectralis OCT device. 
+Describing the Structural Phenotype of Glaucoma as a Function of Glaucoma 
+Severity 
+ 
+In the following sections, we introduce two different approaches to study the complex 
+structural changes of the ONH as a function of glaucoma severity. In the first, we performed 
+a comprehensive 3D structural analysis of the ONH using ‘human-defined’ 3D structural 
+parameters of the ONH (total of 10 parameters) describing the morphologies of both neural 
+and connective tissues. In the second, we used a relatively recent geometric deep learning 
+method (i.e. PointNet) to discover important 3D structural features differentiating ONHs from 
+different glaucoma severity groups. An overview of both approaches is shown in Figure 1. 
+Approach 1 for Describing the Structural Phenotype of Glaucoma – ONH 
+Parameters 
+ 
+For this approach, all ONH tissues were segmented in 3D (from the OCT scans), from 
+which all ONH structural parameters were automatically extracted.   
+
+ 
+8 
+AI-based Segmentation of ONH Tissues. We automatically segmented all raw OCT 
+volume scans of the ONH using REFLECTIVITY (Reflectivity, Abyss Processing Pte Ltd, 
+Singapore) – a software that was developed from advances in AI-based ONH segmentation 
+[23] (see Figure 1a, b). More specifically, we automatically labelled the following ONH tissue 
+groups: (1) the retinal nerve fiber layer (RNFL) and the prelamina tissue (PLT); (2) the ganglion 
+cell inner plexiform layer (GCL+IPL); (3) all other retinal layers (ORL); (4) the retinal pigment 
+epithelium (RPE) with Bruch’s membrane (BM) and the BM opening (BMO) points; (5) the 
+choroid; (6) the OCT-visible part of the peripapillary sclera including the scleral flange; and (7) 
+the OCT-visible part of the LC. In almost all OCT volume scans, the posterior boundaries of the 
+sclera and LC were not visible and could therefore not be segmented. 
+Automated extraction of ONH parameters. Using the software REFLECTIVITY, we 
+extracted the following parameters: (1) the average RNFL thickness (RNFLT) in each octant 
+(i.e. temporal [T], superior-temporal [ST], superior [S], superior-nasal [SN], nasal [N], inferior-
+nasal [IN], inferior [I], and inferior-temporal [IT]) calculated at a distance of 1.5 times the BMO 
+radius (BMOR) from the centre of BMO; (2) the average minimum rim width (MRW) in each 
+octant defined as the minimum distance from a BMO point to a point on the inner limiting 
+membrane (ILM); (3) the average ganglion cell complex thickness (GCCT) in each octant 
+evaluated at the same location than the RNFLT; (4) the average choroidal thickness (ChT) in 
+each octant at the same distance as that used for the RNFLT; (5) the prelamina depth (PLD) 
+defined as the distance from the BMO center to a point on the ILM (perpendicular to the BMO 
+plane); (6) the minimum prelamina thickness (MPT); (7) the LC depth (LCD) defined as the 
+distance from the BMO centre to a point on the anterior LC boundary (perpendicular to the 
+BMO plane); (8) the LC global shape index (LC-GSI) that summarizes the shape of the anterior 
+LC boundary into a single number [24]; (9) the peripapillary scleral angle (PPSA) representing 
+
+ 
+9 
+the amount of scleral bowing and defined as the angle between two parallel lines to the 
+anterior scleral boundary in the nasal-temporal plane; and (10) the BMO area defined as the 
+area of the best-fit ellipse to the BMO points. A visualization of the extracted ONH parameters 
+is shown in Figure 1c. 
+ 
+Statistical analysis. All parameters were compared across all 4 groups (normal, mild, 
+moderate, and advanced). All statistical analyses were performed using R (version 4.2.1) and 
+RStudio (version 2022.07.1 for macOS). ONH parameters that were extracted in each octant 
+were reported as mean  standard deviation and single valued ONH parameters were 
+presented as box plots. One-way ANOVA with post-hoc Tukey HSD test was used for the 
+comparisons. P value for significance was set at <0.05. 
+Approach 2 for Describing the Structural Phenotype of Glaucoma – PointNet 
+ 
+PointNet, a deep neural network from the group of geometric deep learning 
+algorithms, can learn from complex 3D shapes, such as that of the ONH, if they are 
+represented as 3D point clouds. In contrast to our first approach, which relied on ‘human-
+defined’ ONH parameters, PointNet allows us to identify important structural landmarks that 
+can differentiate ONHs from the four different glaucoma severity groups, without previous 
+inputs or guidance.  
+ 
+Representation of the ONH structure as 3D point cloud. We described the structure 
+of a given ONH as a 3D point cloud which then was used as input to PointNet. To do so, we 
+first identified the anterior boundaries of all tissue layers in the segmented OCT scan. Each 
+anterior boundary voxel was then represented as a 3D point (see Figure 1d). The final point 
+cloud consisted of about 20,000 points for each ONH (see Figure 1e). Additionally, for each 
+point, we extracted the local tissue thickness (minimum distance between anterior and 
+posterior boundary). In summary, we assigned four values to every point: its position in the 
+
+ 
+10 
+3D space ([x, y, z]-coordinate) and its local tissue thickness (not applicable for the sclera and 
+LC). To homogenize the data across all ONHs, the centre of BMO was set as origin of the 
+coordinate system [x=0, y=0, z=0] and the normal of BMO plane (best-fit plane to the BMO 
+points) was aligned with the axial direction of the scan. The more interested reader is referred 
+to our previous publication on geometric deep learning for glaucoma diagnosis [25]. 
+ 
+Glaucoma severity classification. PointNet was specifically designed to process and 
+learn from 3D point clouds such as the one shown in Figure 1. We used the same architecture 
+as in the original publication [18], except that we implemented a max pooling layer of 
+dimension 256. To identify important 3D structural features of the ONH at different stages of 
+glaucoma, we trained three PointNet classification networks to differentiate between: (1) 
+normal and mild glaucoma subjects (normal-mild); (2) mild and moderate glaucoma subjects 
+(mild-moderate); and (3) moderate and advanced glaucoma subjects (moderate-advanced).  
+To assess the performance of the three binary classification networks, we split each 
+respective dataset (i.e. normal-mild, mild-moderate, and moderate-advanced) in training 
+(70%), validation (15%), and test (15%) sets. To improve performance and reduce overfitting, 
+we used data augmentation techniques such as random cropping, random rotations, random 
+rigid translations, random sampling (i.e. randomly picking a subset of points from the input 
+point cloud), oversampling to reduce data imbalance, and additive Gaussian noise where 
+applicable. A five-fold cross validation study was performed (using the train and validation 
+set) to tune hyperparameters and we reported the area under the receiver operating 
+characteristic curves (AUCs) of the model with the best performing hyperparameters as mean 
+± standard deviation. All models were trained on a Nvidia RTX A5000 GPU card until optimum 
+performance was reached in the validation set.  
+
+ 
+11 
+Identification of important 3D structural features of the ONH. The specific 
+architecture of PointNet inherently allowed us to identify regions of the ONH important for 
+the differentiation of different glaucoma severity groups by extracting all points that 
+contributed to the final classification score – the so-called critical points. For each 
+classification group (i.e. normal-mild, mild-moderate, and moderate-advanced), we extracted 
+critical points from all ONHs of the respective test set (networks trained on respective training 
+set using tuned hyperparameters). Comparing the locations of these points between the 
+three groups allowed us to draw conclusion on the characteristic 3D structural changes of the 
+ONH at different stages of glaucoma.  
+Visualization of critical points. To better visualize the location of the resulting critical 
+points, we first constructed an average ONH geometry (represented by the average anterior 
+boundaries of each segmented tissue) for each of the three classification groups, i.e. normal-
+mild, mild-moderate, and moderate-advanced. For each group, we then projected the critical 
+points (closest point projection) onto their corresponding anterior tissue boundary of the 
+respective average ONH geometry and visualized them as 3D point cloud density maps. A 
+density measure for each point was obtained by counting the neighbouring points within a 
+75 μm radius sphere. Since all critical points were projected on an average ONH geometry, 
+such a density map should highlight landmarks of the ONH that exhibit distinct 3D structural 
+changes between the different stages of glaucoma (represented as a cluster of red points in 
+the point cloud density maps). 
+ 
+ 
+
+ 
+12 
+Results 
+Approach 1 – Statistical Analysis of ONH Parameters 
+We observed that the majority of ONH structural changes occurred in the early 
+glaucoma stage (normal to mild). These changes were also the most substantial in terms of 
+their size or magnitude. Specifically, we noted a decrease in average RNFLT (average over all 
+sectors) from 112  26 µm to 83  29 µm (Figure 2a), a decrease in average MRW from 256  
+60 µm to 169  55 µm (Figure 2b), a decrease in average GCCT from 154  26 µm to 124  30 
+µm (Figure 2c), no change in average ChT (Figure 2d), an increase in PLD from 136  195 µm 
+to 288  199 µm (Figure 2e), a decrease in MPT from 146  116 µm to 63  70 µm (Figure 2f), 
+an increase in LCD from 410  109 µm to 468  132 µm (Figure 2g), a decrease in LC-GSI from 
+-0.37  0.42 to -0.61  0.33 (Figure 2h), an increase in PPSA from 5.4  4.6 degree to 9.5  6.2 
+degree (Figure 2i), and an increase in BMOA from 2.15  0.5 mm2 to 2.28  0.5 mm2 (Figure 
+2j). 
+Following substantial structural changes of the ONH in the early stage of glaucoma, 
+most ONH parameters showed a plateau effect, with little change from mild to moderate 
+glaucoma. Only RNFLT (average), GCCT (average), and MRW (average) showed a significant 
+decrease from 83  29 to 71  30 µm, 124  30 to 111  32 µm, and 169  55 to 159  56 µm, 
+respectively.  
+In the later stages of glaucoma (moderate to advanced), we observed significant 
+structural changes of the ONH, but they were much less pronounced in terms of their 
+magnitude compared to those seen in the early stages. In detail, the average RNFLT decreased 
+from 71  30 µm to 50  25 µm (Figure 2a), the average MRW decreased from 159  56 µm 
+to 126  46 µm (Figure 2b), the average GCCT decreased from 111  32 µm to 88  27 µm 
+
+ 
+13 
+(Figure 2c), the LCD increased from 459  121 to 502  147 µm (Figure 2g), and the BMOA 
+decreased from 2.30  0.58 mm2 to 2.12  0.42 mm2 (Figure 2j). The ChT (Figure 2d), the PLD 
+(Figure 2e), the MPT (Figure 2f), the LC-GSI (Figure 2h), and the PPSA (Figure 2i) showed no 
+significant change. 
+If we were to examine regional variations, we noted that structural changes of the 
+RNFLT, MRW, and GCCT were more pronounced (higher in magnitude) in both the superior 
+and inferior octants of the ONH. This was true throughout all stages of glaucoma. In these 
+sectors, we observed that the decrease in MRW slowed as glaucoma severity increased. 
+Specifically, in the early stage of glaucoma (normal to mild), MRW decreased in the superior 
+octant from 295  64 µm to 192  58 µm while in the later stage (moderate to advanced), the 
+decrease was smaller from 179  58 µm to 133  49 µm (Figure 2b). In contrast, RNFLT and 
+GCCT decreased linearly as glaucoma severity increased. In the early stage of glaucoma 
+(normal to mild), RNFLT and GCCT in the superior octant decreased from 163  31 to 122  
+34 µm and 200  31 to 160  34 µm, respectively, while in the later stage (moderate to 
+advanced), the decrease was from 102  35 to 61  31 µm and 141  35 to 99  32 µm (Figure 
+2a, 2c). With the exception of the inferior octant of the ONH, we did not observe any 
+significant changes in the ChT with glaucoma severity (Figure 2d). 
+Approach 2 – Performance Assessment 
+ 
+Using PointNet, we were able to differentiate ONHs from different glaucoma severity 
+groups. The normal-mild glaucoma classification showed the best performance (AUC: 0.94  
+0.02), followed by the moderate-advanced (AUC: 0.80  0.04) and mild-moderate glaucoma 
+classification (AUC: 0.68  0.08). 
+
+ 
+14 
+Approach 2 – Changes of Important 3D Structural Features of the ONH with 
+Glaucoma Severity 
+For each classification task (i.e. normal-mild, mild-moderate, and moderate-
+advanced), we pooled all critical points from all ONHs (test set), mapped them onto the 
+corresponding average ONH geometry, and displayed them as a 3D point cloud density map 
+for all ONH tissues (Figure 3), or separately for each ONH tissue (Figure 4).  
+In general, we observed that critical points were present in both, neural (normal-mild: 
+57%, mild-moderate: 39%, moderate-advanced: 53%) and connective tissues (normal-mild: 
+43%, mild-moderate: 61%, moderate-advanced: 47%). More specifically, most of the critical 
+points were located in the RNFL+PLT (normal-mild: 53%, mild-moderate: 37%, moderate-
+advanced: 47%), the sclera (normal-mild: 17%, mild-moderate: 15%, moderate-advanced: 
+11%), and the LC (normal-mild: 23%, mild-moderate: 43%, moderate-advanced: 31%). In 
+contrast, we observed almost no critical points in the other tissue layers, such as the GCC+IPL, 
+ORL, RPE, Choroid. 
+On a tissue level, we found that the critical points from the RNFL of all three 
+classification tasks formed an hourglass pattern with points mainly located in the superior 
+and inferior quadrant. In addition, in the normal-mild glaucoma classification, critical points 
+from the RNFL were mostly located around the neuro-retinal rim whereas in the moderate-
+advanced glaucoma classification, these points moved more outwards to the peripheral 
+region of the ONH. Interestingly, we also found that in the normal-mild and mild-moderate 
+classification most of the critical points from the LC were located near the LC insertion zone 
+in the superior (normal-mild) and superior + inferior quadrant (mild-moderate) whereas in 
+
+ 
+15 
+the moderate-advanced classification, critical points were more spread out over the entire 
+LC.  
+ 
+Discussion 
+ 
+In this study, we were able to describe the 3D structural phenotype of glaucoma as a 
+function of severity using two separate approaches. In the first, we extracted ‘human-defined’ 
+3D structural parameters of the ONH and compared them across four different groups: 
+normal, mild, moderate, and advanced. In the second, we represented the complex structure 
+of the ONH as a 3D point cloud and used PointNet to uncover the structural landmarks that 
+were the most affected by glaucoma severity without any human input. Overall, we found 
+that the structural features of both neural and connective tissues contributed to the 
+structural phenotype of glaucoma, and that each of our proposed method could provide its 
+own unique knowledge.   
+ 
+In this study, we found that after substantial structural changes of the ONH in the early 
+stage of glaucoma (normal to mild), almost all ONH parameters reached a plateau, with less 
+change in the later stages (mild to moderate and moderate to advanced). This is in good 
+agreement with previous studies that investigated the structure-function relationship and 
+reported a considerable structural loss before any functional VF defects were detectable [26-
+28]. Some of these studies suggested a “tipping point” in the early stage of glaucoma (at about 
+– 3 dB MD) from which onwards even small structural changes were associated with a 
+relatively large decrease in MD value [26, 28]. One should also keep in mind that MD values 
+are usually reported on a logarithmic scale (non-linear scale). For instance, a shift in MD value 
+from 0 to -6 dB would imply a much larger loss in visual sensitivity compared to a shift from -
+
+ 
+16 
+6 to -12 dB on a linear scale [29]. Therefore, the observed plateau effect might be a result of 
+reporting MD values on a logarithmic scale. However, further research is needed to verify 
+such a hypothesis. 
+ 
+Furthermore, we found that critical points were present in both neural (normal-mild: 
+57%, mild-moderate: 39%, moderate-advanced: 53%) and connective tissues (normal-mild: 
+43%, mild-moderate: 61%, moderate-advanced: 47%) at all stages of glaucoma indicating that 
+the structural changes caused by glaucoma affected both types of tissue in the ONH. Our 
+findings are in line with previous research that suggested that the pathophysiology of 
+glaucoma is complex and cannot purely be characterized as a damage to the neural tissue in 
+the ONH (i.e. retinal ganglion cells) [11-13]. Despite these recent findings, current glaucoma 
+tests focus on assessing neural tissue health, ignoring any glaucomatous structural changes 
+of connective tissue in the ONH. In the future, the development of more comprehensive tests 
+that consider structural changes in both, neural and connective tissues, could potentially 
+improve the diagnosis and prognosis of glaucoma. 
+Additionally, we found that most of the critical points (normal-mild: 93%, mild-
+moderate: 95%, moderate-advanced: 89%) were concentrated in the RNFL+PLT, sclera, and 
+LC. PointNet only focuses on the major structural changes of the optic nerve head, and since 
+we limited the number of critical points to 256, only the ONH landmarks with significant 3D 
+structural changes will be highlighted in the point cloud density maps. Therefore, the fact that 
+there are almost no critical points present in the GCC+IPL, ORL, RPE, and choroid does not 
+necessarily imply that these tissues do not exhibit any structural changes in glaucoma. 
+However, our findings suggest that any structural changes in these tissues are likely to be 
+smaller in magnitude compared to the structural changes observed in the RNFL, sclera, and 
+LC. 
+
+ 
+17 
+In both approaches, we found that structural changes of neural tissues were more 
+prominent in the inferior and superior quadrants of the ONH over all stages of glaucoma. This 
+is in accordance with many previous studies (including our recent study in glaucoma diagnosis 
+[25]) that reported significant structural changes of glaucomatous ONHs in these quadrants 
+[30, 31]. In addition, Wang et al. reported a progressive nasalization of the central retinal 
+vessel trunk (CRVT) with glaucoma severity [32]. One might argue that the location of some 
+of the critical points from the RNFL coincides with the location of the CRVT and its branches 
+indicating changes in the CRVT location with disease progression. However, further research 
+is needed to confirm such speculations. 
+Furthermore, we found that the decline in MRW slowed, whereas RNFLT decreased 
+linearly as glaucoma severity increased. This suggests that neural tissue changes in the early 
+stage of glaucoma (normal to mild) are more pronounced around the optic disc (i.e. MRW), 
+in contrast to the later stages of glaucoma (mild to moderate and moderate to advanced), 
+where such changes move to the periphery of the ONH (i.e. RNFLT). Interestingly, we found a 
+similar trend in the distribution of critical points from the RNFL. In the early glaucoma group 
+(normal-mild), critical points were mostly located around the neuro-retinal rim. These critical 
+points (with their local tissue thickness) might act as a surrogate measurement for MRW. In 
+the more severe glaucoma groups (i.e. mild-moderate and moderate-advanced), critical 
+points from the RNFL moved to more peripheral regions of the ONH and thus closer to where 
+the RNFLT was measured. Up to date, there is no common consent on whether RNFLT or 
+MRW is better correlated with VF damage (i.e. glaucoma severity). Some studies favored 
+RNFLT [10, 33] whereas others reported better performance of MRW [30, 34]. In addition, 
+Gmeiner et al. reported that depending on the stage of glaucoma and the major site of 
+glaucomatous damage (peripheral or central), RNFLT might be superior to MRW and vice 
+
+ 
+18 
+versa suggesting that morphological changes of the glaucomatous ONH are diverse and may 
+depend on various factors [33]. Therefore, when assessing ONH structural changes, it might 
+be important to analyze the entire region of the ONH (peripheral and central) with its complex 
+3D morphology as it was done with PointNet. 
+We found that a considerable number of critical points were extracted from the sclera 
+over all stages of glaucoma, suggesting significant and progressive structural changes of the 
+sclera with glaucoma severity. In addition, and in line with a previous study [17], we found 
+that the PPSA, representative for the bending of the sclera in the nasal-temporal plane, is 
+significantly larger in mild glaucoma compared to normal eyes, however, no significant 
+differences were found between the later stages of the disease. Considering the presence of 
+critical points from the sclera in all stages of glaucoma, one might speculate that a single 
+parameter like the PPSA is not enough to capture the complex 3D structural changes of the 
+sclera with glaucoma severity and further research is needed to quantify such changes.  
+Furthermore, we found that most of the LC critical points were located in the region of 
+the LC insertion zone over all stages of glaucoma. However, the major site of these critical 
+points changed from the superior quadrant (normal-mild) to the superior + inferior quadrant 
+(mild-moderate) to a more diffuse distribution over all quadrants (moderate-advanced).  
+Previous studies reported morphological changes of the LC with glaucoma severity reflected 
+by a change in LC depth [35], LC curvature [36], and LC-GSI [14]. In addition, local LC defects 
+or alterations like posterior movement of the LC insertion zones [37] and LC disinsertions [38] 
+were observed in glaucomatous eyes. However, none of the studies reported structural 
+changes of the LC insertion zone with glaucoma severity. Our findings suggest that assessing 
+morphological changes of the glaucomatous LC, especially in the region of the LC insertion 
+zone, could be useful in monitoring disease progression (in conjunction with other ONH 
+
+ 
+19 
+parameters like the RNFLT). However, further longitudinal studies are necessary to unravel 
+the complex 3D structural changes of the LC with glaucoma severity. 
+In this study, several limitations warrant further discussion. First, although the overall 
+sample size was fairly large, however, subjects were unevenly distributed over the glaucoma 
+severity groups (normal: 213, mild: 204, moderate: 118, advanced: 118). In addition, the 
+Caucasian subgroup had no healthy controls that might introduce a bias in both, the 
+comparison of ONH parameters and the learning process of PointNet. Therefore, our findings 
+might not be easily transferable to other populations. In the future, we want to investigate 
+possible differences in structural changes of the ONH with glaucoma severity between 
+different ethnic groups.  
+Second, we used MD values of the 24-2 or 30-2 VF to determine glaucoma severity, 
+however, standard automated perimetry is subjective and sometimes underestimate disease 
+severity [22]. Recent studies suggest chromatic pupillometry [39] or electroretinogram [40] 
+as an objective way to assess functional loss in glaucomatous eyes. However, these devices 
+have their own limitations and a future study has to show whether our findings would change 
+when using a different staging system.  
+Third, the accuracy of the extracted ONH parameters and the extracted point clouds to 
+represent local structural features of the ONH depends on the performance of the 
+segmentation algorithm. Even though the segmentation software that we used in this study 
+(Reflectivity, Abyss Processing Pte Ltd, Singapore) was tested and validated on a large cohort 
+of glaucomatous and non-glaucomatous ONHs at different stages of glaucoma, one should 
+keep in mind that the choice of the segmentation algorithm might have an impact on the 
+results.  
+
+ 
+20 
+Fourth, although we found that many ONH parameters showed significant differences 
+between glaucoma severity groups, the cross-sectional nature of our data limits causal 
+inferences. As a result, our findings might differ from longitudinal studies that follow 
+individual patients over a certain period of time. In the future, we aim to validate our findings 
+by applying our herein developed approaches to a longitudinal dataset. 
+ Fifth, the differentiation of ONHs from the mild and moderate glaucoma severity group 
+was the most challenging task and resulted in a rather small AUC of 0.68  0.08 (PointNet). 
+The moderate performance of PointNet might be due to the plateau effect that we observed 
+after substantial structural changes in the early stage of glaucoma. In the future, we could 
+consider the MD value as a continuous variable and predict its “true” value, instead of a binary 
+classification, as this might give us a boost in performance.  
+In summary, we successfully described the 3D structural phenotype of glaucoma as a 
+function of glaucoma severity by: (1) a “traditional” approach based on extracted ONH 
+parameters and (2) a more recently introduced approach based on critical points extracted 
+by PointNet. We showed that ONH structural changes are not limited to neural tissues but 
+occurred in both, neural and connective tissues simultaneously. In addition, we identified a 
+major site of 3D morphological change of the ONH that might potentially be worth monitoring 
+in the future - the region around the LC insertion zone. With this study, we hope to provide 
+new insights into the complex pathophysiology of glaucoma that might help clinicians in their 
+daily clinical care. 
+ 
+Acknowledgment 
+
+ 
+21 
+We acknowledge funding from (1) the donors of the National Glaucoma Research, a 
+program of the BrightFocus Foundation, for support of this research (G2021010S [MJAG]); (2) 
+SingHealth Duke-NUS Academic Medicine Research Grant (SRDUKAMR21A6 [MJAG]); (3) the 
+“Retinal Analytics through Machine learning aiding Physics (RAMP)" project that is supported 
+by the National Research Foundation, Prime Minister’s Office, Singapore under its Intra-
+Create Thematic Grant “Intersection Of Engineering And Health” - NRF2019-THE002-0006 
+awarded to the Singapore MIT Alliance for Research and Technology (SMART) Centre 
+[MJAG/AT/GB]. (4) the “Tackling & Reducing Glaucoma Blindness with Emerging Technologies 
+(TARGET)” project that is supported by the National Medical Research Council (NMRC), 
+Singapore (MOH-OFLCG21jun-0003 [MJAG]). 
+ 
+ 
+
+ 
+22 
+References 
+ 
+ 
+1. 
+Bussel, I.I., G. Wollstein, and J.S. Schuman, OCT for glaucoma diagnosis, screening 
+and detection of glaucoma progression. British Journal of Ophthalmology, 2014. 
+98(Suppl 2): p. ii15-ii19. 
+2. 
+Robin, T.A., et al., Performance of community-based glaucoma screening using 
+Frequency Doubling Technology and Heidelberg Retinal Tomography. Ophthalmic 
+Epidemiol, 2005. 12(3): p. 167-78. 
+3. 
+Lavinsky, F., et al., The Future of Imaging in Detecting Glaucoma Progression. 
+Ophthalmology, 2017. 124(12s): p. S76-s82. 
+4. 
+Gonzalez-Hernandez, M., et al., Structure–function relationship depends on 
+glaucoma severity. British Journal of Ophthalmology, 2009. 93(9): p. 1195. 
+5. 
+Medeiros, F.A., et al., The Structure and Function Relationship in Glaucoma: 
+Implications for Detection of Progression and Measurement of Rates of Change. 
+Investigative Ophthalmology & Visual Science, 2012. 53(11): p. 6939-6946. 
+6. 
+Hood, D.C., et al., Detecting glaucoma with only OCT: Implications for the clinic, 
+research, screening, and AI development. Progress in Retinal and Eye Research, 2022. 
+90: p. 101052. 
+7. 
+Kim, N.R., et al., Structure–Function Relationship and Diagnostic Value of Macular 
+Ganglion Cell Complex Measurement Using Fourier-Domain OCT in Glaucoma. 
+Investigative Ophthalmology & Visual Science, 2010. 51(9): p. 4646-4651. 
+8. 
+Kim, Y.J., et al., Comparative study of macular ganglion cell complex thickness 
+measured by spectral-domain optical coherence tomography in healthy eyes, eyes 
+with preperimetric glaucoma, and eyes with early glaucoma. Japanese Journal of 
+Ophthalmology, 2014. 58(3): p. 244-251. 
+9. 
+Danthurebandara, V.M., et al., Enhanced Structure–Function Relationship in 
+Glaucoma With an Anatomically and Geometrically Accurate Neuroretinal Rim 
+Measurement. Investigative Ophthalmology & Visual Science, 2015. 56(1): p. 98-105. 
+10. 
+Amini, N., et al., Structure-Function Relationships in Perimetric Glaucoma: 
+Comparison of Minimum-Rim Width and Retinal Nerve Fiber Layer Parameters. 
+Investigative Ophthalmology & Visual Science, 2017. 58(11): p. 4623-4631. 
+11. 
+Brooks, D.E., et al., Functional and structural analysis of the visual system in the 
+rhesus monkey model of optic nerve head ischemia. Invest Ophthalmol Vis Sci, 2004. 
+45(6): p. 1830-40. 
+12. 
+Quigley, H.A., et al., Morphologic changes in the lamina cribrosa correlated with 
+neural loss in open-angle glaucoma. Am J Ophthalmol, 1983. 95(5): p. 673-91. 
+13. 
+Yang, H., et al., The connective tissue phenotype of glaucomatous cupping in the 
+monkey eye - Clinical and research implications. Progress in Retinal and Eye 
+Research, 2017. 59: p. 1-52. 
+14. 
+Tan, N.Y.Q., et al., Changes in the Anterior Lamina Cribrosa Morphology with 
+Glaucoma Severity. Scientific Reports, 2019. 9(1): p. 6612. 
+15. 
+Vianna, J.R., et al., Serial Changes in Lamina Cribrosa Depth and Neuroretinal 
+Parameters in Glaucoma: Impact of Choroidal Thickness. Ophthalmology, 2017. 
+124(9): p. 1392-1402. 
+16. 
+Bellezza, A.J., et al., Deformation of the Lamina Cribrosa and Anterior Scleral Canal 
+Wall in Early Experimental Glaucoma. Investigative Ophthalmology & Visual Science, 
+2003. 44(2): p. 623-637. 
+
+ 
+23 
+17. 
+Wang, X., et al., Peripapillary sclera exhibits a v-shaped configuration that is more 
+pronounced in glaucoma eyes. Br J Ophthalmol, 2022. 106(4): p. 491-496. 
+18. 
+Charles, R.Q., et al. PointNet: Deep Learning on Point Sets for 3D Classification and 
+Segmentation. in 2017 IEEE Conference on Computer Vision and Pattern Recognition 
+(CVPR). 2017. 
+19. 
+Atalay, E., et al., Pattern of Visual Field Loss in Primary Angle-Closure Glaucoma 
+Across Different Severity Levels. Ophthalmology, 2016. 123(9): p. 1957-64. 
+20. 
+Keltner, J.L., et al., Classification of visual field abnormalities in the ocular 
+hypertension treatment study. Arch Ophthalmol, 2003. 121(5): p. 643-50. 
+21. 
+Mills, R.P., et al., Categorizing the Stage of Glaucoma From Pre-Diagnosis to End-
+Stage Disease. American Journal of Ophthalmology, 2006. 141(1): p. 24-30. 
+22. 
+De Moraes, C.G., et al., Association of Macular Visual Field Measurements With 
+Glaucoma Staging Systems. JAMA Ophthalmology, 2019. 137(2): p. 139-145. 
+23. 
+Devalla, S.K., et al., Towards label-free 3D segmentation of optical coherence 
+tomography images of the optic nerve head using deep learning. Biomed Opt 
+Express, 2020. 11(11): p. 6356-6378. 
+24. 
+Thakku, S.G., et al., A Global Shape Index to Characterize Anterior Lamina Cribrosa 
+Morphology and Its Determinants in Healthy Indian Eyes. Investigative 
+Ophthalmology & Visual Science, 2015. 56(6): p. 3604-3614. 
+25. 
+Braeu, F.A., et al. Geometric Deep Learning to Identify the Critical 3D Structural 
+Features of the Optic Nerve Head for Glaucoma Diagnosis. 2022. arXiv:2204.06931. 
+26. 
+Park, K.-H., et al., Bruch's membrane opening-minimum rim width and visual field 
+loss in glaucoma: a broken stick analysis. International journal of ophthalmology, 
+2018. 11(5): p. 828-834. 
+27. 
+Jonas, J.B. and A.E. Gründler, Correlation between mean visual field loss and 
+morphometric optic disk variables in the open-angle glaucomas. Am J Ophthalmol, 
+1997. 124(4): p. 488-97. 
+28. 
+Wollstein, G., et al., Retinal nerve fibre layer and visual function loss in glaucoma: the 
+tipping point. Br J Ophthalmol, 2012. 96(1): p. 47-52. 
+29. 
+Liebmann, K., C.G. De Moraes, and J.M. Liebmann, Measuring Rates of Visual Field 
+Progression in Linear Versus Nonlinear Scales: Implications for Understanding the 
+Relationship Between Baseline Damage and Target Rates of Glaucoma Progression. J 
+Glaucoma, 2017. 26(8): p. 721-725. 
+30. 
+Chauhan, B.C., et al., Enhanced Detection of Open-angle Glaucoma with an 
+Anatomically Accurate Optical Coherence Tomography–Derived Neuroretinal Rim 
+Parameter. Ophthalmology, 2013. 120(3): p. 535-543. 
+31. 
+Mwanza, J.-C., et al., Ability of Cirrus HD-OCT Optic Nerve Head Parameters to 
+Discriminate Normal from Glaucomatous Eyes. Ophthalmology, 2011. 118(2): p. 241-
+248.e1. 
+32. 
+Wang, M., et al., Relationship Between Central Retinal Vessel Trunk Location and 
+Visual Field Loss in Glaucoma. American journal of ophthalmology, 2017. 176: p. 53-
+60. 
+33. 
+Gmeiner, J.M.D., et al., Comparison of Bruch's Membrane Opening Minimum Rim 
+Width and Peripapillary Retinal Nerve Fiber Layer Thickness in Early Glaucoma 
+Assessment. Investigative Ophthalmology & Visual Science, 2016. 57(9): p. OCT575-
+OCT584. 
+
+ 
+24 
+34. 
+Muth, D.R. and C.W. Hirneiß, Structure–Function Relationship Between Bruch's 
+Membrane Opening–Based Optic Nerve Head Parameters and Visual Field Defects in 
+Glaucoma. Investigative Ophthalmology & Visual Science, 2015. 56(5): p. 3320-3328. 
+35. 
+Park, S.C., et al., Lamina cribrosa depth in different stages of glaucoma. Invest 
+Ophthalmol Vis Sci, 2015. 56(3): p. 2059-64. 
+36. 
+Lee, S.H., et al., Diagnostic Power of Lamina Cribrosa Depth and Curvature in 
+Glaucoma. Investigative Ophthalmology & Visual Science, 2017. 58(2): p. 755-762. 
+37. 
+Yang, H., et al., Posterior (outward) migration of the lamina cribrosa and early 
+cupping in monkey experimental glaucoma. Investigative ophthalmology & visual 
+science, 2011. 52(10): p. 7109-7121. 
+38. 
+Takayama, K., et al., Three-Dimensional Imaging of Lamina Cribrosa Defects in 
+Glaucoma Using Swept-Source Optical Coherence Tomography. Investigative 
+Ophthalmology & Visual Science, 2013. 54(7): p. 4798-4807. 
+39. 
+Najjar, R.P., et al., Handheld chromatic pupillometry can accurately and rapidly 
+reveal functional loss in glaucoma. British Journal of Ophthalmology, 2021: p. 
+bjophthalmol-2021-319938. 
+40. 
+Sarossy, M., et al., Prediction of glaucoma severity using parameters from the 
+electroretinogram. Scientific Reports, 2021. 11(1): p. 23886. 
+ 
+ 
+ 
+
+ 
+25 
+Figures  
+ 
+ 
+ 
+Figure 1. Overview of two approaches to describe the 3D structural phenotype of glaucoma 
+as a function of severity. Approach 1 was based on the comparison of well-established ONH 
+parameters between different glaucoma severity groups (a-c). Approach 2 leverages on 
+
+ 
+26 
+geometric deep learning to identify important 3D landmarks of the ONH to differentiate ONHs 
+at different stages of glaucoma. By looking at the changes of these critical 3D structural 
+features with glaucoma severity, we were able to draw conclusions about the complex 3D 
+structural changes of the ONH taking place at different stages of glaucoma (a, b, d, and e). 
+ 
+ 
+
+ 
+27 
+ 
+
+ 
+28 
+Figure 2. Summary of statistical analysis of automatically extracted ONH parameters. RNFLT, 
+MRW, GCCT, and ChT are shown as sector plots (T: temporal, ST: superior-temporal, S: 
+superior, SN: superior-nasal, N: nasal, NI: nasal-inferior, and I: inferior sector) with values for 
+each group given as average  standard deviation. Non-sectorial parameters are presented 
+as boxplots. A significant difference between two groups (p<0.05) was indicated with a * 
+(determined by post-hoc Tukey HSD tests). 
+ 
+ 
+ 
+ 
+ 
+Figure 3. Critical points resulting from the three classification tasks: normal-mild, mild-
+moderate, and moderate advanced. From left to right column: 3D, en face (top), and sagittal 
+(side) view. Surfaces represent the average anterior tissue boundaries for each respective 
+dataset: RNFL+PLT (red), GCL+IPL (green), ORL (blue), RPE (yellow), choroid (purple), sclera 
+(cyan), and LC (orange). Red colored critical points correspond to ONH regions with high 
+importance for the differentiation of the respective glaucoma severity groups. 
+ 
+
+ 
+29 
+ 
+1 
+ 
+2 
+Figure 4. En face (top) view layer by layer comparison (columns) of critical points at different stages of glaucoma severity (rows). Critical points 
+3 
+are presented as point cloud density maps with colours indicating the number of neighbouring points within a sphere with a radius of 75 µm. 
+ 
+4 
+
+ 
+30 
+Tables 
+5 
+ 
+6 
+Table 1. Summary of glaucoma severity groups. 
+7 
+ 
+8 
+ 
+NORMAL 
+(N=213) 
+MILD 
+(N=204) 
+MODERATE 
+(N=118) 
+ADVANCED 
+(N=118) 
+P* 
+AGE, YEARS 
+63.36 (6.99) 
+66.9 (6.42) 
+68.05 (7.11) 
+68.52 (7.69) 
+<0.001 
+SEX, FEMALE 
+126 (59.15) 
+91 (44.61) 
+49 (41.52) 
+43 (36.44) 
+<0.001 
+RACE 
+ 
+ 
+ 
+ 
+ 
+     CHINESE 
+213 
+178 
+97 
+53 
+<0.001 
+     CAUCASIAN 
+0 
+26 
+21 
+65 
+MD, DB 
+-1.41 (2.11) 
+-3.35 (1.95) 
+-8.16 (2.35) 
+-18.64 (5.31) 
+<0.001 
+Data are in mean (standard deviation) or n (%) as appropriate. 
+9 
+MD = mean deviation of the 24-2 or 30-2 visual field test. 
+10 
+*Comparison between the four groups using Fisher’s exact test (for sex and race) and ANOVA 
+11 
+(for age and MD). 
+12 
+

49FKT4oBgHgl3EQf9y5B/content/tmp_files/2301.11955v1.pdf.txt

@@ -0,0 +1,1414 @@
+Statistical whitening of neural populations with gain-modulating interneurons
+Lyndon R. Duong * 1 David Lipshutz * 2 David J. Heeger 1 Dmitri B. Chklovskii 2 3 Eero P. Simoncelli 1 2
+Abstract
+Statistical whitening transformations play a fun-
+damental role in many computational systems,
+and may also play an important role in biologi-
+cal sensory systems. Individual neurons appear
+to rapidly and reversibly alter their input-output
+gains, approximately normalizing the variance of
+their responses. Populations of neurons appear
+to regulate their joint responses, reducing correla-
+tions between neural activities. It is natural to see
+whitening as the objective that guides these be-
+haviors, but the mechanism for such joint changes
+is unknown, and direct adjustment of synaptic in-
+teractions would seem to be both too slow, and in-
+sufficiently reversible. Motivated by the extensive
+neuroscience literature on rapid gain modulation,
+we propose a recurrent network architecture in
+which joint whitening is achieved through modu-
+lation of gains within the circuit. Specifically, we
+derive an online statistical whitening algorithm
+that regulates the joint second-order statistics of a
+multi-dimensional input by adjusting the marginal
+variances of an overcomplete set of interneuron
+projections. The gains of these interneurons are
+adjusted individually, using only local signals,
+and feed back onto the primary neurons. The net-
+work converges to a state in which the responses
+of the primary neurons are whitened. We demon-
+strate through simulations that the behavior of the
+network is robust to poor conditioning or noise
+when the gains are sign-constrained, and can be
+generalized to achieve a form of local whitening
+in convolutional populations, such as those found
+throughout the visual or auditory system.
+*Equal contribution 1Center for Neural Science, New York Uni-
+versity, New York, NY 2Center for Computational Neuroscience,
+Flatiron Institute, New York, NY 3Neuroscience Institute, New
+York University School of Medicine, New York, NY. Correspon-
+dence to: Lyndon R. Duong <[email protected]>, David
+Lipshutz <dlipshutz@flatironinstitute.org>.
+Under review.
+1. Introduction
+Statistical whitening transformations, in which multi-
+dimensional inputs are decorrelated and normalized to have
+unit variance, are common in statistical signal processing
+and machine learning systems. For example, they provide a
+common step in statistical factorization methods (Hyv¨arinen
+& Oja, 2000) and are often used as a preprocessing step
+for training deep networks (Krizhevsky, 2009). Empiri-
+cal evidence shows that statistical whitening improves un-
+supervised feature learning (Coates et al., 2011). More
+recently, self-supervised learning methods have used sta-
+tistical whitening or related decorrelation transformations
+to prevent representational collapse (Ermolov et al., 2021;
+Zbontar et al., 2021; Bardes et al., 2021; Hua et al., 2021).
+Whitening in neural networks is often performed in the of-
+fline setting. However, online methods are useful, especially
+when the inputs are from dynamic environments.
+In early sensory systems, which receive inputs from dy-
+namic environments, changes in sensory input statistics
+induce rapid changes in the input-output gains of single
+neurons, allowing cells to normalize their output variance
+(Fairhall et al., 2001; Nagel & Doupe, 2006). This is hypoth-
+esized to enable maximal information transmission (Barlow,
+1961; Laughlin, 1981; Fairhall et al., 2001). At the popu-
+lation level, whitening and related adaptive decorrelation
+transformations have been reported in sensory areas such
+as the early visual cortex of cats (Benucci et al., 2013) and
+the olfactory bulb in zebrafish (Friedrich, 2013; Wanner &
+Friedrich, 2020) and mice (Giridhar et al., 2011; Gschwend
+et al., 2015). However, the mechanisms underlying such
+whitening behaviors are unknown, and would seem to re-
+quire coordination among all pairs of neurons, as opposed to
+the single-neuron case which relies only on gain rescaling.
+Here, motivated by the large neuroscience literature on rapid
+gain modulation, we propose a novel recurrent network ar-
+chitecture for statistical whitening that exclusively relies on
+gain modulation. In particular, we introduce a novel objec-
+tive for statistical whitening that is expressed solely in terms
+of the marginal variances of an overcomplete representation
+of the input signal. We derive a recurrent circuit to optimize
+the objective, and show that it corresponds to a network
+comprising primary neurons and an auxiliary population of
+interneurons with scalar gain modulation. Importantly, the
+arXiv:2301.11955v1  [q-bio.NC]  27 Jan 2023
+
+Statistical whitening of neural populations with gain-modulating interneurons
+Figure 1. Schematic of a recurrent statistical whitening network with 2 primary neurons and 3 interneurons. Left: 2D Scatter plot of the
+(non-Gaussian) network inputs x = (x1, x2) whose covariance is the ellipse. Center: Primary neurons, whose outputs are y = (y1, y2),
+receive external feedforward inputs, x, and recurrent feedback inputs from an auxiliary population of interneurons, − �3
+i=1 giziwi.
+Linear projection vectors {w1, w2, w3} ∈ R2 encode non-negative feedforward synaptic weights connecting the primary neurons to
+interneuron i = 1, 2, 3 (symmetric weights are used for feedback connections). The weights are shown in the left and right panels with
+corresponding colors. Inset: The ith interneuron (e.g. here i = 2) receives input zi = w⊤
+i y, which is multiplied by its gain gi to produce
+output gizi. Its gain, gi, is adjusted s.t. ∆gi ∝ z2
+i − 1. The dark arrow indicates that the gain update operates on a slower time scale.
+Right: Scatter plots of the whitened network outputs y. Outputs have unit variance along all wi’s, which is equivalent to having identity
+covariance matrix, i.e., Cyy = IN (black circle).
+network operates online, and its responses converge to the
+classical ZCA whitening solution without supervision or
+backpropagation. To demonstrate potential applications of
+this framework, we show that gain modulation serves as an
+implicit gating mechanism, which facilitates fast context-
+dependent whitening. Further, we show how non-negative
+gain modulation provides a novel approach for dealing with
+ill-conditioned or noisy data. Finally, we relax the overcom-
+pleteness constraint in our objective and provide a method
+for local decorrelation of convolutional populations.
+2. A novel objective for ZCA whitening
+Consider a neural network with N primary neurons. For
+each t = 1, 2, . . . , let xt and yt be N-dimensional vectors
+whose components respectively denote the inputs and out-
+puts of the primary neurons at time t, Figure 1. Without loss
+of generality we assume the inputs xt are centered.
+2.1. Conventional objective
+Statistical whitening aims to linearly transform inputs xt so
+that the covariance of the outputs yt is identity, i.e.,
+Cyy = ⟨yty⊤
+t ⟩t = IN,
+(1)
+where ⟨·⟩t denotes the expectation operator over t, and IN
+denotes the N × N identity matrix (see Appendix A for a
+list of notation used in this work).
+It is well known that whitening is not unique: any orthog-
+onal rotation of a random vector with identity covariance
+matrix also has identity covariance matrix. There are sev-
+eral common choices to resolve this rotational ambiguity,
+each with their own advantages (Kessy et al., 2018). Here,
+we focus on the popular whitening transformation called
+Zero-phase Component Analysis (ZCA) whitening or Ma-
+halanobis whitening, which is the whitening transformation
+that minimizes the mean-squared error between the inputs
+and the whitened outputs (alternatively, the one whose trans-
+formation matrix is symmetric). Mathematically, the ZCA-
+whitened outputs are the optimal solution to the minimiza-
+tion problem
+min
+{yt}⟨∥xt − yt∥2
+2⟩t
+s.t.
+⟨yty⊤
+t ⟩t = IN,
+(2)
+where ∥ · ∥2 denotes the Euclidean norm on RN. Assuming
+the covariance of the inputs Cxx := ⟨xtx⊤
+t ⟩t is positive
+definite, the unique solution to the optimization problem
+in Equation 2 is yt = C−1/2
+xx
+xt for t = 1, 2, . . . , where
+C−1/2
+xx
+is the inverse matrix square root of Cxx.
+Equation 2 provides a starting point for deriving online ZCA
+whitening algorithms that can be implemented with recur-
+rent neural networks that learn by updating their synaptic
+weights (Pehlevan & Chklovskii, 2015).
+2.2. A novel objective using marginal statistics
+We formulate a novel objective for learning the ZCA whiten-
+ing transform via gain modulation. Our innovation exploits
+the fact that a random vector has identity covariance matrix
+(i.e., Equation 1 holds) if and only if it has unit marginal
+
+g12
+W2,1
+9222
+1
+1
+W:
+92之。
+ W2,2
+Close-up of
+ gain-modulated interneuron
+Input:(1, 2Statistical whitening of neural populations with gain-modulating interneurons
+variance along all possible 1D projections (a form of to-
+mography; see Related Work). We can derive a tighter
+statement, that holds for a finite but overcomplete set of at
+least K ≥ KN := N(N + 1)/2 distinct axes (‘overcom-
+plete’ simply means that the number of axes exceeds the
+dimensionality of the input, i.e., K > N). Intuitively, this
+equivalence holds because an N × N symmetric matrix has
+KN degrees of freedom, so the marginal variances along
+K ≥ KN distinct axes are sufficient to constrain the N ×N
+(symmetric) covariance matrix. We formalize this equiva-
+lence in the following proposition, whose proof is provided
+in Appendix B.
+Proposition 2.1. Fix K ≥ KN. Suppose w1, . . . , wK ∈
+RN are unit vectors1 such that
+span({w1w⊤
+1 , . . . , wKw⊤
+K}) = SN,
+(3)
+where SN denotes the KN-dimensional vector space of N ×
+N symmetric matrices. Then Equation 1 holds if and only if
+the projection of yt onto each unit vector w1, . . . , wK has
+unit variance, i.e.,
+⟨(w⊤
+i yt)2⟩t = 1
+for
+i = 1, . . . , K.
+(4)
+Assuming Equation 3 holds, we can interpret the set of
+vectors {w1, . . . , wK} as a frame (i.e., an overcomplete
+basis; Casazza et al., 2013) in RN such that the covariance
+of the outputs Cyy can be computed from the variances of
+the K-dimensional projection onto the set of frame vectors.
+Thus, we can replace the whitening constraint in Equation 2
+with the equivalent marginal variance constraint to obtain
+the following objective:
+min
+{yt}⟨∥xt − yt∥2
+2⟩t
+s.t.
+Equation 4 holds.
+(5)
+3. A recurrent neural network with gain
+adaptation for ZCA whitening
+In this section, we derive an online algorithm for solving
+the optimization problem in Equation 5 and map the algo-
+rithm onto a recurrent neural network with gain modulation.
+We first introduce Lagrange multipliers to enforce the con-
+straints, which transforms the minimization problem into a
+minimax problem. We then solve the minimax problem by
+taking stochastic gradient steps.
+Assume we have an overcomplete frame {w1, . . . , wK} in
+RN satisfying Equation 3. We concatenate the frame vectors
+into an N ×K matrix W := [w1, . . . , wK]. In our network,
+primary neurons project onto the layer of K interneurons
+with the synaptic weights representing matrix W. Then,
+the post-synaptic currents in interneurons at time t encode
+1The unit-length assumption is without loss of generality and
+is imposed here for notational convenience.
+the K-dimensional vector zt := W⊤yt (Figure 1). We
+emphasize that the synaptic weight matrix W will remain
+fixed in our whitening algorithm.
+3.1. Enforcing the marginal variance constraints with
+scalar gains
+We introduce Lagrange multipliers g1, . . . , gK ∈ R to en-
+force the K constraints in Equation 4. We concatenate
+the Lagrange multipliers into the K-dimensional vector
+g := [g1, . . . , gK]⊤ ∈ RK, and formulate the problem as a
+saddle point optimization,
+max
+g
+min
+{yt}⟨ℓ(xt, yt, g)⟩t,
+(6)
+where ℓ(x, y, g) := ∥x − y∥2
+2 +
+K
+�
+i=1
+gi
+�
+(w⊤
+i y)2 − 1
+�
+.
+Here, we have interchanged the order of maximization over
+g and minimization over yt, which is justified because
+ℓ(xt, yt, g) is convex in yt and linear in g, see Appendix C.
+In our neural network implementation, gi will correspond
+to the multiplicative gain associated with the ith interneuron,
+so that its output at time t is gizi,t (Figure 1, Inset). From
+Equation 6, we see that the gain of the ith interneuron, gi,
+enforces the marginal variance of yt along the axis spanned
+by wi to be unity. Importantly, the gains are not hyper-
+parameters, but rather they are optimization variables which
+promote statistical whitening of {yt}, preventing the neural
+outputs from trivially matching the inputs {xt}.
+3.2. Deriving recurrent neural network update rules
+To solve Equation 6 in the online setting, we assume there
+is a time-scale separation between ‘fast’ neural dynamics
+and ‘slow’ gain updates, so that at each time step the neural
+dynamics equilibrate before the gains are adjusted. This al-
+lows us to perform the inner minimization over {yt} before
+the outer maximization over the gains. In biological neural
+networks, this is justifiable because a given neuron’s activa-
+tions (i.e. action potential firing) operate on a much more
+rapid time-scale than its intrinsic input-output gain, which
+is driven by slower processes such as changes in calcium
+ion concentration gradients (Ferguson & Cardin, 2020).
+3.2.1. FAST NEURAL ACTIVITY DYNAMICS
+For each time step t = 1, 2, . . . , we minimize the objective
+ℓ(xt, yt, g) over yt by recursively running gradient-descent
+steps to equilibrium:
+yt ← yt − γ
+2 ∇yℓ(xt, yt(τ), g)
+= yt + γ {xt − W(g ◦ zt) − yt} ,
+(7)
+where γ > 0 is a small constant, the circle ‘◦’ denotes the
+Hadamard (element-wise) product, g ◦ zt is a vector of K
+
+Statistical whitening of neural populations with gain-modulating interneurons
+gain-modulated interneuron outputs, and we assume the
+primary cell outputs are initialized at zero.
+We see from the right-hand-side of Equation 7 that the ‘fast’
+dynamics of the primary neurons are driven by three terms
+(inside the curly braces): i) constant feedforward external
+input xt; ii) recurrent gain-modulated feedback from in-
+terneurons −W(g ◦ zt); and iii) a leak term −yt. Because
+the neural activity dynamics are linear, we can analytically
+solve for their equilibrium (i.e. steady-state), ¯yt, by setting
+the update in Equation 7 to zero:
+¯yt =
+�
+IN + W diag (g) W⊤�−1 xt
+=
+�
+IN +
+K
+�
+i=1
+giwiw⊤
+i
+�−1
+xt,
+(8)
+where diag (g) denotes the K × K diagonal matrix whose
+(i, i)th entry is gi, for i = 1, . . . , K. The equilibrium feed-
+forward interneuron inputs are then given by
+¯zt = W⊤¯yt.
+(9)
+The gain-modulated outputs of the K interneurons, g ◦ zt,
+are then projected back onto the primary cells via symmetric
+weights, −W (Figure 1).
+3.2.2. SLOW GAIN DYNAMICS
+After the fast neural activities reach steady-state, the in-
+terneuron gains are updated by taking a stochastic gradient-
+ascent step with respect to g:
+g ← g + η
+2∇gℓ(xt, ¯yt, g)
+= g + η
+�¯z◦2
+t − 1
+�
+,
+(10)
+where η
+>
+0 is the learning rate, the superscript
+‘◦2’ denotes the element-wise squaring operation (i.e.,
+¯z◦2
+t
+= [¯z2
+t,1, . . . , ¯z2
+t,K]⊤) and 1 = [1, . . . , 1]⊤ is the K-
+dimensional vector of ones2. Remarkably, the update to the
+ith interneuron’s gain gi (Equation 10) depends only on the
+online estimate of the variance of its equilibrium input ¯z2
+t,i,
+and its distance away from the target variance, 1. Networks
+such as these which adapt using only local signals to each
+interneuron are suitable candidates for hardware implemen-
+tations using low-power neuromorphic chips (Pehlevan &
+Chklovskii, 2019). Thus, although statistical whitening in-
+herently requires a joint transformation in response to joint
+statistics, our recurrent network solution operates solely
+using single-neuron gain changes in response to marginal
+statistics.
+2Appendix D generalizes the gain update to allowing for
+temporal-weighted averaging of the variance over past samples.
+3.2.3. ONLINE UNSUPERVISED ALGORITHM
+By combining Equations 7 – 10, we arrive at our online
+recurrent neural network algorithm for statistical whitening
+via gain modulation (Algorithm 1). We also provide batched
+and offline versions of the algorithm in Appendix E.
+Algorithm 1 Online ZCA whitening via gain modulation
+1: Input: Centered inputs x1, x2, · · · ∈ RN
+2: Initialize: W ∈ RN×K; g ∈ RK; η, γ > 0
+3: for t = 1, 2, . . . do
+4:
+yt ← 0
+5:
+{Run yt and zt dynamics to equilibrium}
+6:
+while not converged do
+7:
+zt ← W⊤yt
+8:
+yt ← yt + γ {xt − W(g ◦ zt) − yt}
+9:
+end while
+10:
+g ← g + η
+�
+z◦2
+t − 1
+�
+{Update gains}
+11: end for
+There are a few points worth noting about this network:
+• The weights W remain fixed in Algorithm 1. Rather,
+the gains g adapt to statistically whiten the outputs.
+This allows the whitening to be easily adjusted and
+reversed, by simply returning the gains to their default
+states.
+• While the objective is effectively in the form of an auto-
+encoding loss function involving an ℓ2 reconstruction
+term (Eq. 6), the recurrent network never explicitly
+reconstructs its inputs.
+• Since all recurrent dynamics are linear, it is possible to
+bypass the inner loop representing the fast dynamics
+of the primary cells (lines 6 – 9 of Algorithm 1), by
+directly computing the equilibrium responses of ¯yt,
+and ¯z directly (Eqs. 8, 9).
+4. Numerical experiments and applications
+We provide different applications of our recurrent ZCA
+whitening network via gain modulation. In particular, we
+emphasize that gain adaptation is distinct from, while also
+complementary to, a synaptic weight learning. We therefore
+side-step the goal of learning the frame W, and assume it
+is known. This allows us to decouple and analyze the gen-
+eral properties of our proposed gain modulation framework,
+independently from the choice of frame.
+4.1. Gain modulation: a new solution to ZCA
+whitening
+We first demonstrate that our algorithm succeeds in yielding
+statistically whitened outputs. We simulated a network with
+interneuron weights, W, as illustrated in Figure 1 (N=2,
+
+Statistical whitening of neural populations with gain-modulating interneurons
+Figure 2. Network from Figure 1 (with corresponding colors;
+N=2, K=KN=3, η=2E-3) whitening to two randomly gener-
+ated statistical contexts online (10K steps each). Top: Marginal
+variances (log scale) measured by interneurons approach 1 over
+time. Middle: Dynamics of interneuron gains, which are applied
+to zi before feeding back onto the primary cells. Dashed lines are
+optimal gains (Appendix F). Bottom: Whitening error over time.
+K=KN=3). Figure 2 shows network adaptation to inputs
+from two contexts with randomly generated underlying in-
+put covariances Cxx (10K gain update steps each). As up-
+date steps progress, all marginal variances converge to unity,
+as expected from the objective (top panel). To achieve ZCA
+whitening at equilibrium, then IN +�K
+i=1 giwiw⊤
+i = C1/2
+xx
+(Equation 8). When the number of interneurons satisfies
+K=KN, the optimal gains to achieve ZCA whitening can
+be solved analytically (see Appendix F for details). These
+are displayed as dashed lines in the (middle panel). We
+found that the network successfully adapted to the two ran-
+dom statistical contexts, and converged to the optimal set
+of gains to achieve whitened yt (Figure 2). Accordingly,
+the whitening error, as measured by the Frobenius norm be-
+tween Cyy and IN, approached zero (bottom panel). Thus,
+with each interneuron monitoring their respective marginal
+input variances z2
+i , and re-scaling their input-output gains to
+modulate feedback onto the primary neurons, the network
+succeeded in adapting to each context and yielded whitened
+outputs.
+4.2. Rate of convergence depends on frame W
+Thus far, we have assumed the frame, W, was fixed and
+known (e.g., optimized through pre-training or long time-
+scale development). This distinguishes our method from
+existing ZCA whitening methods, which typically operate
+by estimating the eigenvectors of the data. By contrast, our
+network obviates learning the principal axes of the data
+altogether, and instead uses a statistical sampling approach
+along a fixed set of measurement axes.
+If the number of interneurons K=KN, their gains will de-
+scend the gradient of the objective (Equation 10), and by
+Proposition Theorem 2.1, the outputs will become whitened.
+We were interested in how effectively the network whitened
+randomly sampled inputs with fixed input covariance de-
+pending on its initialization. Figure 3 summarizes an empir-
+ical convergence test of 100 networks where N = 2 with
+three different kinds of frame W ∈ RN×KN : i) with i.i.d.
+Gaussian entries (‘Random’); ii) through an optimization
+procedure that finds a frame whose columns have mini-
+mum mutual coherence and cover the ambient space (‘Opti-
+mized’); and iii) a frame whose first N columns were the
+eigenvectors of the data and the remaining KN −N columns
+were random Gaussian entries (‘Spectral’). For clarity, we
+have removed the effects of sampling stochasticity by run-
+ning the offline version of our network, which assumes
+having direct access to the input covariance (Appendix E);
+the online version was qualitatively similar.
+Figure 3. Convergence depends on qualitative structure of W.
+Networks each had N=2, K=KN=3, η=5E-3. Shaded error
+regions are standard errors over the 100 repeats.
+The Spectral frame defines a bound on achievable perfor-
+mance, converging much faster than the Random and Op-
+timized frames. This is because the interneuron axes were
+aligned with the input’s principal axes, and a simple gain
+scaling along those directions is the optimal whitening solu-
+tion. Interestingly, we found that networks with optimized
+weights systematically converged faster than randomly-
+initialized frames. These results indicate that the choice
+of frame does in fact play an important role in the effective-
+ness of our algorithm. Namely, increased coverage of the
+space by the frame vectors facilitates whitening with our
+gain re-scaling mechanism. The random sampling approach
+has little hope of scaling to high dimensional inputs, and the
+green line in Figure 3 shows that one would benefit from
+aligning the frame vectors to the principal axes of the inputs.
+4.3. Implicit gating via gain modulation
+Motivated by the findings in Figure 3, we wished to demon-
+strate a way in which our adaptive gain modulation net-
+work could complement or augment a network in which
+context-dependent weights have already been learned. We
+performed an experiment involving a network with ‘pre-
+trained’ W (N=6, K=KN=21) whitening inputs from
+
+100
+2
+ICy-Inl
+2F
+10-2
+10-6
+0
+10000
+20000
+Stepw
+Random
+Optimized
+10-1
+Spectral
+10-2
+10-3
+0
+200
+400
+600
+800
+1000
+StepStatistical whitening of neural populations with gain-modulating interneurons
+Figure 4. Gains can act as an implicit gating mechanism. Top:
+Whitening error over time with a network (N=6; KN=21; η=1E-
+3) adapting to 2 alternating statistical contexts A and B, with
+different input covariances for 10K steps each. W was initialized
+as a Spectral frame, with the first 2N columns set to be the eigen-
+vectors of covariances of contexts A and B, respectively. Bottom:
+Gains can be seen to act as switches for context, gating the spectral
+components to optimally whiten each context.
+two alternating statistical contexts, A and B, for 10K steps
+each. The frame was constructed such that the first and
+second N columns were the eigenvectors of context A and
+B’s covariance, respectively, and the remaining K − 2N
+columns’ elements were random i.i.d. Gaussian. Figure 4
+(top panel) shows that the network adaptively whitens the
+inputs from each successive context. Surprisingly, upon
+closer inspection to the K interneurons’ gains over time
+(bottom panel) showed that they approximately served to
+‘select’ the frame vectors corresponding to the eigenvectors
+of each respective condition (as indicated by the blue/red in-
+tensity on the figure). Our gain modulation framework thus
+serves as an effective means of gating context-dependent
+information without an explicit context signal.
+4.4. Normalizing ill-conditioned data
+When inputs are low-rank, Cxx is ill-conditioned (Fig-
+ure 5A), and whitening can amplify directions of small
+variance that are due to noise. In this section, we show how
+our gain-modulating network can be simply modified to han-
+dle these types of inputs. To prevent amplification of inputs
+below a certain threshold, we can replace the unit marginal
+variance equality constraints with upper bound constraints:
+⟨(w⊤
+i yt)2⟩t ≤ 1
+for
+i = 1, . . . , K.
+(11)
+Our modified network objective then becomes
+min
+{yt}⟨∥xt − yt∥2
+2⟩t
+s.t.
+Equation 11 holds.
+(12)
+Figure 5. Two networks (N=2, K=3, η=0.02) whitening ill-
+conditioned inputs. A: Outputs without whitening. 2D scatterplot
+of a non-Gaussian density whose underlying signal lies close to
+a latent 1D axis. The signal magnitude along that axis is denoted
+by the colors. The covariance matrix is depicted as a black ellipse.
+Gray dashed lines are axes spanned by W (here chosen to be an
+equi-angular frame). B: ZCA whitening boosts small-amplitude
+noise lying along the uninformative direction. C: Modulating gains
+according to Eq. 14 rescales the data without amplifying noise. D:
+Gains updated with Eq. 10 (solid) vs. Eq. 14 (dashed).
+Intuitively, if the projected variance along a given direction
+is already less than or equal to unity, then it will not affect
+the overall loss. To enforce the upper bound constraints,
+we introduce gains as Lagrange multipliers as before, but
+restrict the domain of g to be the non-negative orthant RK
++ ,
+resulting in non-negative optimal gains:
+max
+g∈RK
++
+min
+{yt}⟨ℓ(xt, yt, g)⟩t,
+(13)
+where ℓ(x, y, g) is defined as in Equation 6. At each time
+step t, we optimize Equation 13 by first taking gradient-
+descent steps with respect to yt, resulting in the same neu-
+ral dynamics (Equation 7) and equilibrium solution (Equa-
+tion 8) as before. After the neural activities equilibrate, we
+take a projected gradient-ascent step with respect to g:
+g ← ⌊g + η(¯z◦2
+t − 1)⌋
+(14)
+where ⌊·⌋ denotes the element-wise half-wave rectification
+operation that projects its inputs onto the positive orthant
+RK
++ , i.e., ⌊v⌋ := [max(v1, 0), . . . , max(vK, 0)]⊤.
+We simulated a network with gains set to either updates
+using unconstrained gains (Equation 10), or rectified gains
+(Equation 14), and observed that these two models con-
+verged to two different solutions (Figure 5B, C). When
+
+10-2
+Context A
+Context B
+Context A
+Context B
+10-3
+10-4
+10-5
+0
+Interneuron index
+5
+10
+15
+20
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Step (x10)
+-0.20
+-0.15
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+0.20
+GainA
+B
+2
+2 
+1 
+1 
+0
+0
+-1 -
+-1 
+-2 
+-2 
+1
+T
+-
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+y1
+y1
+C
+D
+0.75
+2
+0.50
+0.25
+1 
+0.00
+0
+9
+-0.25
+-0.50
+-1 
+-0.75
+gi
+-2
+-1.00
+Lgi]
+-2
+-1
+0
+1
+2
+0
+100
+200
+300
+y1
+StepStatistical whitening of neural populations with gain-modulating interneurons
+gi was not constrained to be non-negative, the network
+achieved global whitening, as before. By contrast, the gains
+constrained to be non-negative converged to different val-
+ues altogether, with one of them converging to zero rather
+than becoming negative. The whitening error for this net-
+work unsurprisingly converged to a non-zero value with the
+non-negative gain constraint. Thus, with a non-negative
+constraint, the network failed to fully whiten y, but in doing
+so, it did not amplify the noise. In Appendix G we show
+additional cases that provide further geometric intuition on
+differences between ZCA whitening and non-negative gain
+constrained ZCA whitening with our network.
+4.5. Gain modulation enables local spatial
+decorrelation
+The requirement of KN interneurons to ensure a statisti-
+cally white output becomes prohibitively costly for high-
+dimensional inputs due to the number of interneurons scal-
+ing as O(N 2). This led us to ask: how many interneurons
+are needed in practice? For natural sensory inputs such
+as images, it is well known that inter-pixel correlation is
+highly structured, decaying as a function of distance. Using
+a Gaussian random walk, we simulated gaze fixation and
+micro-saccadic eye movements, drawing 12×12 patch sam-
+ples from a natural image (Figure 6A; Hateren & Schaaf,
+1998). We did this for different randomly selected regions
+of the image (colors). The content of each region is quite
+different, but the inter-pixel correlation within each context
+fell rapidly with distance (Figure 6B).
+We relaxed the O(N 2) marginal variance constraint to in-
+stead target whitening of spatially local neighborhoods of
+primary neurons with image patch inputs. That is, the frame
+W spanned K < KN axes in RN, but was constructed such
+that overlapping neighborhoods of 4 × 4 primary neurons
+were decorrelated, each by a population of interneurons that
+was ‘overcomplete’ with respect to that neighborhood (see
+Appendix H for frame construction details). Importantly,
+taking into account convolutional structure dramatically re-
+duces the interneuron complexity from O(N 2) → O(N)
+(Appendix H). This frame is still overcomplete (K > N),
+but because K<KN, we no longer guarantee at equilibrium
+that Cyy = IN.
+After running this local whitening network on the inputs
+drawn from the red context, we found that (Figure 6C): i)
+inter-pixel correlations drop within the region specified by
+the local neighborhood; and ii) surprisingly, correlations at
+longer-range are dramatically reduced. Accordingly, the co-
+variance eigenspectrum of the locally whitened outputs was
+significantly flatter compared to the inputs (Figure 6D left
+vs. right columns). We also provide a 1D example in Ap-
+pendix H. We remark that this empirical result is not at all
+obvious – that whitening individual overlapping neighbor-
+hoods of neurons should produce a more globally whitened
+output covariance. Indeed, studying whether and when a
+globally whitened solution is possible from whitening of
+spatial overlapping neighborhoods is an interesting problem
+that is worth pursuing.
+5. Related work
+5.1. Biologically plausible whitening networks
+Biological circuits operate in the online setting and, due
+to physical constraints, learn exclusively using local sig-
+nals. Therefore, to plausibly model neural computation, a
+neural network model must operate in the online setting
+(i.e., streaming data) and use local learning rules. There
+are a few existing normative models of statistical whiten-
+ing and related transformations; however, these models use
+synaptic plasticity mechanisms (i.e., changing W) to adapt
+to changing input statistics (Pehlevan & Chklovskii, 2015;
+Pehlevan et al., 2017; Chapochnikov et al., 2021; Lipshutz
+et al., 2022). Adaptation of neural population responses to
+changes in sensory inputs statistics occurs rapidly, on the
+order of seconds (Benucci et al., 2013; Wanner & Friedrich,
+2020), so it could potentially be accounted for by short-
+term synaptic plasticity, which operates on the timescale of
+tens of milliseconds to minutes (Zucker et al., 2002), but
+not by long-term synaptic plasticity, which operates on the
+timescale of minutes or longer (Martin et al., 2000). Here,
+we explore the alternative hypothesis that modulation of
+neural gains, which operates on the order of tens of mil-
+liseconds to minutes (Fairhall et al., 2001), facilitates rapid
+adaptation of neural populations to changing input statistics.
+5.2. Tomography and “sliced” density measurements
+Our leveraging of 1D projections to compute the ZCA
+whitening transform is reminiscent of approaches taken in
+the field of tomography. Geometrically, our method repre-
+sents an ellipsoid (i.e., the N dimensional covariance ma-
+trix) using noisy 1D projections of the ellipsoid onto axes
+spanned by frame vectors (i.e., estimates of the marginal
+variances). This is a special case of reconstruction problems
+that have been studied in geometric tomography (Karl et al.,
+1994; Gardner, 1995). An important distinction between
+tomographic reconstruction and our solution to ZCA whiten-
+ing is that we are not using the 1D projections to reconstruct
+the multi-dimensional inputs; instead, we are utilizing the
+univariate measurements to transform the ellipsoid into a
+new shape (a hyper-sphere, in the case of whitening).
+In optimal transport, “sliced” methods offer a way to mea-
+sure otherwise intractable p-Wasserstein distances in high
+dimensions (Bonneel et al., 2015), thereby enabling its use
+in optimization loss functions. Sliced methods compute
+Wasserstein distance by repeatedly taking series of 1D pro-
+jections of two densities, then computing the expectation
+
+Statistical whitening of neural populations with gain-modulating interneurons
+Figure 6. Local spatial whitening. A) Large grayscale image from which 12×12 image patch samples are drawn. Colors represent
+random-walk sampling from regions of the image corresponding to contexts with different underlying statistics. Six samples from
+each context are shown below. B) Without whitening, mean pairwise output pixel correlations decay rapidly with spatial distance in
+each context, suggesting that local whitening may be effective. C) Pairwise output pixel correlation of patches from the red context
+before (gray) and after global (black dots) vs. convolutional whitening with overlapping 4×4 neighborhoods (red). Shaded regions
+represent standard deviations. D) Top: Expected correlation matrices of all flattened patches of the red context before whitening, and after
+global/local ZCA whitening. Correlation and not covariance matrices are displayed here to facilitate comparison; all panels use the same
+color scale. Bottom: Corresponding covariance eigenspectra.
+over all 1D Wasserstein distances, for which there exists
+an analytic solution. Notably, the 2-Wasserstein distance
+between a 1D zero-mean Gaussian with variance σ2 and a
+standard normal (i.e. white) density is
+W2
+�
+N
+�
+0, σ2�
+; N (0, 1)
+�
+= ∥σ − 1∥ .
+Comparing this with the rule by which we update each in-
+terneuron gain, gi ← gi + η((w⊤
+i ¯yt)2 − 1) (Equation 10),
+reveals striking similarity between our recurrent neural net-
+work and methods optimizing using sliced Wasserstein dis-
+tances. However, distinguishing characteristics of our ap-
+proach include: 1) minimizing distance between univariate
+variances rather than standard deviations; 2) the directions
+along which we compute slices (columns of W) are fixed,
+whereas sliced methods typically compute a new set of
+random projections at each optimization step; 3) most im-
+portantly, our network operates online, and minimizes sliced
+variance distances without backpropagation.
+6. Discussion
+We have derived a novel family of recurrent models for
+whitening, which use gain modulation to transform joint
+second-order statistics of their inputs based on marginal
+variance measurements. We showed that, given sufficiently
+many marginal measurements along unique axes, the net-
+work will produce ZCA whitened outputs. In particular,
+our objective (Equation 5) provides an elegant way to think
+about the classical problem of statistical whitening, and
+draws connections to old concepts in tomography and trans-
+port theory. The framework developed here is flexible, with
+several generalizations or extensions that we omitted due
+to space limitations. For example, by replacing the unity
+marginal variance constraint by a set of target variances
+differing from 1, the network can be used to transform (i.e.
+transport) its input density to one matching the correspond-
+ing (non-white) covariance.
+Modulating feature gains has proven effective in adapting
+pre-trained neural networks to novel inputs with out-of-
+training distribution statistics (Ball´e et al., 2020; Duong
+et al., 2022; Mohan et al., 2021). In fact, adaptive gain
+modulation is an old concept in neuroscience which we
+believe would be of importance to the broader machine
+learning community. In real neural networks, there exist
+several computational processes operating concurrently at
+different time-scales. Examples include synaptic weights
+encoding long-term information, while faster processes like
+gain modulation facilitate rapid adaptation to different con-
+texts. Indeed, the demonstrations in this study were largely
+agnostic to the exact structure of the weights W, and in-
+stead focused on the computational role of adaptive gain
+modulation itself. We showed how gains can adaptively
+decorrelate a network’s outputs without modifying its pre-
+trained weights in an online setting. Specifically, we showed
+that gain modulation: 1) enables fast switching between pre-
+learned context-dependent weight regimes; 2) can be used
+in conjunction with properly-aligned interneuron projec-
+tion weights to handle ill-conditioned inputs; and 3) reduce
+
+A
+1.0
+lation
+Unwhitened
+Global whitening
+Local whitening
+0.5
+Correl:
+0.0
+7
+7
+0
+6
+12
+Distance (px)
+102
+1.0 b
+ Unwhitened
+ Locally whitened
+101↓
+1011
+1011
+igenvalue
+ Globally whitened
+Correlation
+10°1
+ oOT
+10°1
+0.5
+10-1 -
+10-11
+LEF
+10-2 1
+10-2 1
+T
+0.0
+.......
+72
+144
+1
+72
+144
+1
+72
+144
+1
+Eigenvector index
+0
+6
+12
+Distance (px)Statistical whitening of neural populations with gain-modulating interneurons
+long-range dependencies by modifying local signals.
+Feature whitening and decorrelation has become an im-
+portant objective constraint in self-supervised contrastive
+learning methods to help prevent representational collapse
+(Bardes et al., 2021; Zbontar et al., 2021; Ermolov et al.,
+2021). We believe that the networks developed in this study,
+motivated by extensive neuroscience research on rapid gain
+modulation, provide an effective whitening solution for
+these methods – particularly in regimes which prioritize
+streaming data, and networks designed for low-power con-
+sumption hardware.
+References
+Ball´e, J., Chou, P. A., Minnen, D., Singh, S., Johnston, N.,
+Agustsson, E., Hwang, S. J., and Toderici, G. Nonlinear
+Transform Coding. arXiv:2007.03034 [cs, eess, math],
+2020.
+Bardes, A., Ponce, J., and LeCun, Y. VICReg: Variance-
+invariance-covariance regularization for self-supervised
+learning. arXiv preprint arXiv:2105.04906, 2021.
+Barlow, H. B. Possible Principles Underlying the Transfor-
+mations of Sensory Messages. In Sensory Communica-
+tion, pp. 216–234. The MIT Press, 1961.
+Benucci, A., Saleem, A. B., and Carandini, M. Adapta-
+tion maintains population homeostasis in primary visual
+cortex. Nature neuroscience, 16(6):724–729, 2013.
+Bonneel, N., Rabin, J., Peyr´e, G., and Pfister, H. Sliced and
+Radon Wasserstein Barycenters of Measures. Journal of
+Mathematical Imaging and Vision, 51(1):22–45, January
+2015. ISSN 0924-9907, 1573-7683.
+Boyd, S. and Vandenberghe, L. Convex optimization. Cam-
+bridge university press, 2004.
+Casazza, P. G., Kutyniok, G., and Philipp, F. Introduction
+to Finite Frame Theory. In Casazza, P. G. and Kutyniok,
+G. (eds.), Finite Frames, pp. 1–53. Birkh¨auser Boston,
+Boston, 2013. ISBN 978-0-8176-8372-6 978-0-8176-
+8373-3.
+Chapochnikov, N. M., Pehlevan, C., and Chklovskii, D. B.
+Normative and mechanistic model of an adaptive circuit
+for efficient encoding and feature extraction. bioRxiv,
+2021.
+Coates, A., Ng, A., and Lee, H. An analysis of single-
+layer networks in unsupervised feature learning. In Pro-
+ceedings of the fourteenth international conference on
+artificial intelligence and statistics, pp. 215–223. JMLR
+Workshop and Conference Proceedings, 2011.
+Duong, L. R., Li, B., Chen, C., and Han, J.
+Multi-
+rate adaptive transform coding for video compression.
+arXiv:2210.14308 [eess.IV], October 2022.
+Ermolov, A., Siarohin, A., Sangineto, E., and Sebe, N.
+Whitening for self-supervised representation learning. In
+International Conference on Machine Learning, pp. 3015–
+3024. PMLR, 2021.
+Fairhall, A. L., Lewen, G. D., and Bialek, W. Efficiency
+and ambiguity in an adaptive neural code. Nature, 412:
+787–792, 2001.
+Ferguson, K. A. and Cardin, J. A. Mechanisms underlying
+gain modulation in the cortex. Nature Reviews Neuro-
+science, 21(2):80–92, 2020. ISSN 1471-0048.
+Friedrich, R. W. Neuronal computations in the olfactory
+system of zebrafish. Annual review of neuroscience, 36:
+383–402, 2013.
+Gardner, R. J. Geometric tomography, volume 58. Cam-
+bridge University Press Cambridge, 1995.
+Giridhar, S., Doiron, B., and Urban, N. N.
+Timescale-
+dependent shaping of correlation by olfactory bulb lateral
+inhibition. Proceedings of the National Academy of Sci-
+ences, 108(14):5843–5848, 2011.
+Gschwend, O., Abraham, N. M., Lagier, S., Begnaud, F.,
+Rodriguez, I., and Carleton, A. Neuronal pattern separa-
+tion in the olfactory bulb improves odor discrimination
+learning. Nature Neuroscience, 18(10):1474–1482, 2015.
+Hateren, J. H. v. and Schaaf, A. v. d. Independent component
+filters of natural images compared with simple cells in
+primary visual cortex. Proceedings: Biological Sciences,
+265(1394):359–366, Mar 1998.
+Hua, T., Wang, W., Xue, Z., Ren, S., Wang, Y., and Zhao, H.
+On feature decorrelation in self-supervised learning. In
+Proceedings of the IEEE/CVF International Conference
+on Computer Vision, pp. 9598–9608, 2021.
+Hyv¨arinen, A. and Oja, E. Independent component analysis:
+algorithms and applications. Neural networks, 13(4-5):
+411–430, 2000.
+Karl, W. C., Verghese, G. C., and Willsky, A. S. Recon-
+structing Ellipsoids from Projections. CVGIP: Graphi-
+cal Models and Image Processing, 56(2):124–139, 1994.
+ISSN 1049-9652.
+Kessy, A., Lewin, A., and Strimmer, K. Optimal whitening
+and decorrelation. The American Statistician, 72(4):309–
+314, 2018.
+Krizhevsky, A. Learning multiple layers of features from
+tiny images. Master’s thesis, University of Toronto, 2009.
+
+Statistical whitening of neural populations with gain-modulating interneurons
+Laughlin, S.
+A Simple Coding Procedure Enhances a
+Neuron’s Information Capacity. Zeitschrift fur Natur-
+forschung. C, Journal of biosciences, pp. 910–2, 1981.
+Lipshutz, D., Pehlevan, C., and Chklovskii, D. B.
+In-
+terneurons accelerate learning dynamics in recurrent neu-
+ral networks for statistical adaptation. arxiv preprint
+arxiv:2209.10634, 2022.
+Martin, S., Grimwood, P. D., and Morris, R. G. Synaptic
+plasticity and memory: an evaluation of the hypothesis.
+Annual Review of Neuroscience, 23(1):649–711, 2000.
+Mohan, S., Vincent, J. L., Manzorro, R., Crozier, P. A.,
+Simoncelli, E. P., and Fernandez-Granda, C. Adaptive
+Denoising via GainTuning. arXiv:2107.12815 [cs.CV],
+July 2021.
+Nagel, K. I. and Doupe, A. J. Temporal Processing and
+Adaptation in the Songbird Auditory Forebrain. Neuron,
+51(6):845–859, September 2006.
+Pehlevan, C. and Chklovskii, D. B. A normative theory
+of adaptive dimensionality reduction in neural networks.
+Advances in Neural Information Processing Systems, 28,
+2015.
+Pehlevan, C. and Chklovskii, D. B. Neuroscience-Inspired
+Online Unsupervised Learning Algorithms: Artificial
+Neural Networks. IEEE Signal Processing Magazine,
+36(6):88–96, November 2019. ISSN 1053-5888, 1558-
+0792.
+Pehlevan, C., Sengupta, A. M., and Chklovskii, D. B. Why
+do similarity matching objectives lead to hebbian/anti-
+hebbian networks? Neural Computation, 30(1):84–124,
+2017.
+Wanner, A. A. and Friedrich, R. W. Whitening of odor
+representations by the wiring diagram of the olfactory
+bulb. Nature neuroscience, 23(3):433–442, 2020.
+Zbontar, J., Jing, L., Misra, I., LeCun, Y., and Deny, S. Bar-
+low twins: Self-supervised learning via redundancy reduc-
+tion. In International Conference on Machine Learning,
+pp. 12310–12320. PMLR, 2021.
+Zucker, R. S., Regehr, W. G., et al. Short-term synaptic
+plasticity. Annual Review of Physiology, 64(1):355–405,
+2002.
+
+Statistical whitening of neural populations with gain-modulating interneurons
+A. Notation
+For N ≥ 2, let KN := N(N + 1)/2. Let RN denote N-dimensional Euclidean space equipped with the Euclidean norm,
+denoted ∥·∥2. Let RN
++ denote the non-negative orthant in RN. Given K ≥ 2, let RN×K denote the set of N ×K real-valued
+matrices and SN denote the set of N × N symmetric matrices.
+Matrices are denoted using bold uppercase letters (e.g., M) and vectors are denoted using bold lowercase letters (e.g., v).
+Given a matrix M, Mij denotes the entry of M located at the ith row and jth column. Let 1 = [1, . . . , 1]⊤ denote the
+N-dimensional vector of ones. Let IN denote the N × N identity matrix.
+Given vectors v, w ∈ RN, define their Hadamard product by v ◦ w := (v1w1, . . . , vNwN) ∈ RN. Define v◦2 :=
+(v2
+1, . . . , v2
+N) ∈ RN. Define diag(v) to be the N × N diagonal matrix whose (i, i)th entry is equal to vi, for i = 1, . . . , N.
+Let ⟨·⟩t denote expectation over t = 1, 2, . . . .
+The diag (·) operator, similar to numpy.diag() or MATLAB’s diag(), can either: 1) map a vector in RK to the
+diagonal of a K × K zeros matrix; or 2) map the diagonal entries of a K × K matrix to a vector in RK. The specific
+operation being used should be clear by context.
+B. Proof of Proposition 2.1
+Proof of Proposition 2.1. Suppose Equation 1 holds. Then, for i = 1, . . . , K,
+⟨(w⊤
+i yt)2⟩t = ⟨w⊤
+i yty⊤
+t wi⟩t = w⊤
+i wi = 1.
+Therefore, Equation 4 holds.
+Now suppose Equation 4 holds. Let v ∈ RN be an arbitrary unit vector. Then vv⊤ ∈ SN and by Equation 3, there exist
+g1, . . . , gK ∈ R such that
+vv⊤ = g1w1w⊤
+1 + · · · + gKwKw⊤
+K.
+(15)
+We have
+v⊤⟨yty⊤
+t ⟩tv = Tr(vv⊤⟨yty⊤
+t ⟩t) =
+K
+�
+i=1
+gi Tr(wiw⊤
+i ⟨yty⊤
+t ⟩t) =
+K
+�
+i=1
+gi Tr(wiw⊤
+i ) = Tr(vv⊤) = 1.
+(16)
+The first equality is a property of the trace operator. The second and fourth equalities follows from Equation 15 and the
+linearity of the trace operator. The third equality follows from Equation 3. The final equality holds because v is a unit vector.
+Since Equation 16 holds for every unit vector v ∈ RN, Equation 1 holds.
+C. Saddle point property
+We recall the following minmax property for a function that satisfies the saddle point property (Boyd & Vandenberghe, 2004,
+section 5.4).
+Theorem C.1. Let V ⊆ Rn, W ⊆ Rm and f : V × W → R. Suppose f satisfies the saddle point property; that is, there
+exists (a∗, b∗) ∈ V × W such that
+f(a∗, b) ≤ f(a∗, b∗) ≤ f(a, b∗),
+for all (a, b) ∈ V × W.
+Then
+min
+a∈V max
+b∈W f(a, b) = max
+b∈W min
+a∈V f(a, b) = f(a∗, b∗).
+D. Weighted average update rule for gi
+The update for g in Equation 10 can be generalized to allow for a weighted average over past samples. In particular, the
+general update is given by
+g ← g + η
+�
+1
+Z
+t
+�
+s=1
+γt−sz◦2
+s − 1
+�
+,
+
+Statistical whitening of neural populations with gain-modulating interneurons
+where γ ∈ [0, 1] determines the decay rate and Z := 1 + γ + · · · + γt−1 is a normalizing factor.
+E. Batched and offline algorithms for whitening with RNNs via gain modulation
+In addition to the fully-online algorithm provided in the main text (Algorithm 1), we also provide two variants below. In
+many applications, streaming inputs arrive in batches rather than one at a time (e.g. video streaming frames). Similarly
+for conventional offline stochastic gradient descent training, data is sampled in batches. Algorithm 2 would be one way to
+accomplish this in our framework, where the main difference between the fully online version is taking the mean across
+samples in the batch to yield average gain update ∆g term. Furthermore, in the fully offline setting when the covariance of
+the inputs, Cxx is known, Algorithm 3 presents a way to whiten the covariance directly.
+Algorithm 2 Batched ZCA whitening
+1: Input: Data matrix X ∈ RN×T (assumed centered)
+2: Initialize: W ∈ RN×K; g ∈ RK; η; batch size B
+3: while not converged do
+4:
+XB ← sample batch(X, B){N × B}
+5:
+Yb ← [IN + W diag (g) W⊤]−1XB
+6:
+Zb ← W⊤Yb
+7:
+∆g ← Z◦2
+B − 1 {Subtract 1 from all entries}
+8:
+g ← g + η mean(∆g, axis=1)
+9: end while
+Algorithm 3 Offline ZCA whitening
+1: Input: Input covariance Cxx
+2: Initialize: W ∈ RN×K; g ∈ RK; η
+3: while not converged do
+4:
+M ← [IN + W diag (g) W⊤]−1
+5:
+Cyy ← MCxxM⊤
+6:
+∆g ← diag
+�
+W⊤CyyW
+�
+− 1
+7:
+g ← g + η∆g
+8: end while
+F. Frame factorizations of symmetric matrices
+F.1. Analytic solution for the optimal gains
+Recall that the optimal solution of the ZCA objective in Equation 5 is given by yt = C−1/2
+xx
+xt for t = 1, 2, . . . . In our
+neural circuit with interneurons and gain control, the outputs of the primary neurons at equilibrium is (given in Equation 8,
+but repeated here for clarity)
+¯yt =
+�
+IN + W diag (g) W⊤�−1 xt.
+Therefore, the circuit performs ZCA whitening when the gains g satisfy the relation
+IN + W diag (g) W⊤ = C1/2
+xx .
+(17)
+When K is exactly N(N + 1)/2, we can explicitly solve for the optimal gains ¯g (derived in the next subsection):
+¯g =
+��
+W⊤W
+�◦2�−1 �
+w⊤
+1 C1/2
+xx w1 − 1, . . . , w⊤
+NC1/2
+xx wN − 1
+�⊤
+.
+(18)
+F.2. Deriving optimal gains
+We find it useful to first demonstrate that any matrix C ∈ SN, where SN is the space of symmetric N × N matrices, can be
+factorized as
+W diag (g) W⊤ = C
+(19)
+where W ∈ RN×K is some fixed, arbitrary, frame with K ≥ N(N+1)
+2
+(i.e. a representation that is O(N 2) overcomplete),
+and g ∈ RK is a variable vector encoding information about C. We multiply both sides of Equation 19 from the left and
+right by W⊤ and W, respectively, then take the diagonal3 of the resultant matrices,
+diag
+�
+W⊤W diag (g) W⊤W
+�
+= diag
+�
+W⊤CW
+�
+.
+(20)
+3Similar to commonly-used matrix libraries, the diag (·) operator here is overloaded and can map a vector to a matrix or vice versa.
+See Appendix A for details.
+
+Statistical whitening of neural populations with gain-modulating interneurons
+Finally, employing a simple matrix identity involving the diag (·) operator yields
+(W⊤W)◦2g = diag
+�
+W⊤CW
+�
+,
+(21)
+=⇒ g =
+�
+(W⊤W)◦2�−1 diag
+�
+WT CW
+�
+,
+(22)
+where (·)◦2 denotes element-wise squaring. Thus, any N × N symmetric matrix, can be encoded as a vector, g, with respect
+to an arbitrary fixed frame, W, by solving a standard linear system of K equations of the form Ag = b. Importantly, when
+K = N(N + 1)/2, and the columns of W are not collinear, then the matrix on the LHS, (W⊤W)◦2 ∈ SK
+++, is invertible,
+and the vector g is unique (Appendix B).
+Without loss of generality, assume that the columns of W are unit-norm (otherwise, we can always normalize them by
+absorbing their lengths into the elements of g). Furthermore, assume without loss of generality that C ∈ SN
+++, the set of all
+symmetric positive definite matrices (e.g. covariance, precision, PSD square roots, etc.). When C is a covariance matrix,
+then diag
+�
+W⊤CW
+�
+can be interpreted as a vector of projected variances of C along each axis spanned by W. Therefore,
+Equation 21 states that the vector g is linearly related to the vector of projected variances via the element-wise squared
+frame Gramian, (W⊤W)◦2.
+G. Adaptation with inequality constraint
+In general, the modified objective with rectified gains (Equation 14) does not statistically whiten the inputs x1, x2, . . . ,
+but rather adapts the non-negative gains g1, . . . , gK to ensure that the variances of the outputs y1, y2, . . . in the directions
+spanned by the frame vectors {w1, . . . , wK} are bounded above by unity (Figure 7). This one-sided normalization
+carries interesting implications for how and when the circuit statistically whitens its outputs, which can be compared with
+experimental observations. For instance, the circuit performs ZCA whitening if and only if there are non-negative gains such
+that Equation 17 holds (see, e.g., the top right example in Figure 7), which corresponds to cases such that the matrix C1/2
+xx is
+an element of the following cone (with its vertex translated by IN):
+�
+IN +
+K
+�
+i=1
+giwiw⊤
+i : g ∈ RK
++
+�
+.
+On the other hand, if the variance of an input projection is less than unity — i.e., w⊤
+i Cxxwi ≤ 1 for some i — then the
+corresponding gain gi remains zero. When this is true for all i = 1, . . . , K, the gains all remain zero and the circuit output
+is equal to its input (see, e.g., the bottom middle example of Figure 7).
+Figure 7. Geometric intuition of whitening with/without inequality constraint. Whitening efficacy using non-negative gains depends on W
+and Cxx. For N = 2 and K = 3, examples of covariance matrices Cyy (red ellipses) corresponding to optimal solutions y of objective
+12, for varying input covariance matrices Cxx (black ellipses) and frames W (spanning axes denoted by gray lines). Unit circles, which
+correspond to the identity matrix target covariance, are shown with dashed lines. Each row corresponds to a different frame W and each
+column corresponds to a different input covariance Cxx.
+
+Cxx,1
+Cxx,2
+Cxx,3
+IM
+W2Statistical whitening of neural populations with gain-modulating interneurons
+H. Whitening spatially local neighborhoods
+H.1. Spatially local whitening in 1D
+For an N-dimensional input, we consider a network that whitens spatially local neighborhoods of size M < N. To this end,
+we can construct N filters of the form
+wi = ei,
+i = 1, . . . , N
+and M(N − M+1
+2
+) filters of the form
+w = ei + ej
+√
+2
+,
+i, j = 1, . . . , N,
+1 ≤ |i − j| ≤ M.
+The total number of filters is (M + 1)(N − M
+2 ), so for fixed M the number of filters scales linearly in N rather than
+quadratically.
+We simulated a network comprising N = 10 primary neurons, and a convolutional weight matrix connecting each interneuron
+to spatial neighborhoods of three primary neurons. Given input data with covariance Cxx illustrated in Figure 8A (left
+panel), this modified network succeeded to statistically whiten local neighborhoods of size of primary 3 neurons (right
+panel). Notably, the eigenspectrum (Figure 8B) after local whitening is much closer to being equalized. Furthermore, while
+the global whitening solution produced a flat spectrum as expected, the local whitening network did not amplify the axis
+with very low-magnitude eigenvalues (Figure 8B right panel).
+Figure 8. Statistically adapting local neighborhoods of neurons. A) ˆCxx denotes correlation matrix, which are shown here for display
+purposes only, to facilitate comparisons. Network with 10-dimensional input correlation (left) 10-dimensional output correlation matrix
+after global whitening (middle); and output correlation matrix after statistically whitening local neighborhoods of size 3. The output
+correlation matrix of the locally adapted circuit has block-identity structure along the diagonal. B) Corresponding eigenspectra of
+covariance matrices of unwhitened (left), global whitened (middle), and locally whitened (right) network outputs. The black dashed line
+denotes unity.
+H.2. Filter bank construction in 2D
+Here, we describe one way of constructing a set of convolutional weights for overlapping spatial neighborhoods (e.g. image
+patches) of neurons. Given an n × m input and overlapping neighborhoods of size h × w to be statistically whitened, the
+samples are therefore matrices X ∈ Rn×m. In this case, filters w ∈ R1×n×m can be indexed by pairs of pixels that are in
+the same patch:
+((i, j), (k, ℓ)),
+1 ≤ i ≤ n,
+1 ≤ j ≤ m,
+0 ≤ |i − k| ≤ h,
+0 ≤ |j − ℓ| ≤ w
+
+A
+1.0
+Cyy globally white
+1.0
+yy locally white
+1.0
+0.5
+ 0.5
+- 0.5
+- 0.0
+- 0.0
+- 0.0
+-0.5
+-0.5
+-0.5
+-1.0
+B
+-1.0
+-1.0
+30
+2.5
+2.5
+25
+2.0
+2.0
+20
+1.5
+1.5
+15
+1.0
+1.0
+10
+0.5
+0.5
+5
+0
+0.0
+0.0
+1
+5
+10
+1
+5
+10
+1
+5
+10
+EigenvectorStatistical whitening of neural populations with gain-modulating interneurons
+We can then construct the filters as,
+w(i,j),(k,ℓ)(X) =
+�
+xi,j
+if (i, j) = (k, ℓ),
+xi,j+xk,ℓ
+√
+2
+if (i, j) ̸= (k, ℓ).
+In this case there are
+nm + wh
+�
+(n − w)(m − h) + (n − w)(h + 1)
+2
++ (m − h)(w + 1)
+2
++ (h + 1)(w + 1)
+2
+�
+such filters, so the number of filters required scales linearly with nm rather than quadratically.
+

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+Extreme mass ratio inspirals in galaxies with dark matter halos
+Ning Dai,1, ∗ Yungui Gong,1, † Yang Zhao,1, ‡ and Tong Jiang1, §
+1School of Physics, Huazhong University of Science and Technology,
+1037 LuoYu Rd, Wuhan, Hubei 430074, China
+Using the analytic, static and spherically symmetric metric for a Schwarzschild
+black hole immersed in dark matter (DM) halos with Hernquist type density distri-
+bution, we derive analytic formulae for the orbital period and orbital precession, the
+evolutions of the semi-latus rectum and the eccentricity for eccentric EMRIs with the
+environment of DM halos. We show how orbital precessions are decreased and even
+reverse the direction if the density of DM halo is large enough. The presence of local
+DM halos slows down the decrease of the semi-latus rectum and the eccentricity.
+Comparing the number of orbital cycles with and without DM halos over one-year
+evolution before the merger, we find that DM halos with the compactness as small
+as 10−4 can be detected. By calculating the mismatch between GW waveforms with
+and without DM halos, we show that we can use GWs from EMRIs in the environ-
+ments of galaxies to test the existence of DM halos and detect the compactness as
+small as 10−5.
+I.
+INTRODUCTION
+The first detection of gravitational waves (GWs) from the merger of black hole (BH)
+binary by the LIGO Scientific Collaboration and the Virgo Collaboration in 2015 [1, 2]
+opened a new window for probing gravitational physics and fundamental physics. Since then,
+tens of confirmed GW events have been detected by the ground-based GW observatories [3–
+6]. The ground-based GW observatories are only sensitive to GWs in the frequency range
+of 10 − 103 Hz.
+The space-based GW observatories such as LISA [7], TianQin [8] and
+Taiji [9, 10] will usher a new era in GW astronomy due to their unprecedented accuracy
+and their sensitive range of mHz [11–14]. One particular interesting target of space-based
+∗ [email protected]
+† Corresponding author. [email protected]
+‡ [email protected]
+§ [email protected]
+arXiv:2301.05088v1  [gr-qc]  12 Jan 2023
+
+2
+GW detectors is a stellar-mass compact object (SCO) inspiralling onto a massive black hole
+(MBH), the extreme mass ratio inspirals (EMRIs) [15]. There are 105 − 106 GW cycles in
+the detector band when the SCO inspirals deep inside the strong field region of the MBH,
+and rich information about the spacetime geometry around the MBH is encoded in GW
+waveforms. Therefore, the observations of GWs emitted from EMRIs present us a good
+opportunity for the study of astrophysics, gravity in the strong and nonlinear regions and
+the nature of BHs [15–20].
+Although the property of DM is still a mystery in physics, there are a lot of indirect
+evidence for the existence of dark matter (DM) in the Universe [21–30]. DM may cluster
+at the center of galaxies and around BHs [31–34], and affect the dynamics of binaries and
+hence GWs emitted from them. Since EMRIs are believed to reside in stellar clusters and
+the center of galaxies, so DM may affect the dynamics of EMRIs and the observations of
+GWs from EMRIs, especially those in DM environments may be used to understand the
+astrophysical environment surrounding EMRIs and probably confirm the existence of DM
+and uncover the nature of DM [35–49].
+In the studies of DM effects discussed above, Newtonian approaches to the problems were
+applied and the gravitational effects of DM on the dynamical evolution of EMRIs were mod-
+eled at Newtonian level. In Ref. [50], the authors generalized Einstein clusters [51, 52] to
+include horizons, solved Einstein’s equations sourced by DM halo of Hernquist type density
+distribution [34] with a MBH at its center and obtained analytical formulae for the metric
+of galaxies harboring MBHs. Exact solutions for the geometry of a MBH immersed in DM
+halos with different density distributions were then derived [53, 54]. With the fully relativis-
+tic formalism, it was found that the leading order correction to the ringdown stage induced
+by the external matter and fluxes by orbiting particles is a gravitational redshift, and the
+difference between the number of GW cycles accumulated by EMRIs with and without DM
+halos over one year before the innermost stable circular orbit can reach about 500 [50]. In
+galaxies harboring MBHs, tidal forces and geodesic deviation depend on the masses of the
+DM halos and the typical length scales of the galaxies [55]. Due to the gravitational pull
+of DM halos, the apsidal precession of the geodesic orbits for EMRIs is strongly affected
+and even prograde-to-retrograde drift can occur [56]. In prograde-to-retrograde orbital al-
+terations, GWs show transient frequency phenomena around a critial non-precessing turning
+point [56]. A fully relativistic formalism to study GWs from EMRIs in static, spherically
+
+3
+symmetric spacetimes describing a MBH immersed in generic astrophysical environments
+was established in Ref. [57] and it was shown how the astrophysical environment changes
+GW generation and propagation.
+The above discussions are based on circular motions or eccentric cases without GW
+reaction. In this paper, we study eccentric orbital motions and GWs of EMRIs in galaxies
+with DM environments. The paper is organized as follows. A review of the spacetime of
+galaxies harboring MBHs is given first, then we discuss the geodesic motions of EMRIs in
+the spacetime in Section II. In Section III, we use the ”Numerical Klugde” method [58–
+60] to calculate GWs from eccentric EMRIs in galaxies with DM environments. To assess
+the capability of detecting DM halos with LISA, we calculate the mismatch between GWs
+from EMRIs with and without DM halos along with their signal-to noise (SNR) ratios in
+Section III. We draw conclusions in Section IV. In this paper we use the units G = c = 1.
+II.
+THE MOTIONS OF BINARIES IN THE ENVIRONMENTS OF GALAXIES
+Following [50], we use the Hernquist-type density distribution [34] to describe the profiles
+observed in the bulges and elliptical galaxies
+ρH =
+Mr0
+2πr(r + r0)3,
+(1)
+where M is the total mass of the DM halo, and r0 is the typical lengthscale of a galaxy. The
+energy-momentum tensor of a galaxy harboring a MBH with the mass MBH is assumed to
+be an anisotropic fluid
+T µ
+ν = diag(−ρDM, 0, Pt, Pt),
+(2)
+where the density profile for a MBH residing at the center of the distribution (1) is
+4πρDM = m′
+r2 = 2M(r0 + 2MBH)(1 − 2MBH/r)
+r(r + r0)3
+,
+(3)
+the mass function m(r) is
+m(r) = MBH +
+Mr2
+(r0 + r)2
+�
+1 − 2MBH
+r
+�2
+,
+(4)
+and the tangential pressure Pt is
+2Pt = m(r)ρDM
+r − 2m(r).
+(5)
+
+4
+Obviously, in the absence of the MBH, the density profile (3) reduces to Eq. (1). At large
+distance, r ≫ MBH, the density profile ρDM becomes the Hernquist-type distribution (1) for
+large galaxies with r0 ≫ MBH, ρDM ∼ (M/r0)2/(Mr), so the DM density ρDM is smaller
+if the compactness M/r0 is smaller with fixed M or if M is larger with fixed compactness
+M/r0. Using the following ansatz for the static, spherically symmetric spacetime [50],
+ds2 = −f(r)dt2 +
+dr2
+1 − 2m(r)/r + r2(dθ2 + sin2 θ dφ2),
+(6)
+and solving Einstein equations, we get [50]
+f(r) =
+�
+1 − 2MBH
+r
+�
+eΥ,
+Υ = −π
+�
+M
+ξ + 2
+�
+M
+ξ arctan
+�r + r0 − M
+√Mξ
+�
+,
+ξ = 2r0 − M + 4MBH.
+(7)
+The geometry (6) describes a BH spacetime with an horizon at r = 2MBH and a curvature
+singularity at r = 0, the matter density vanishes at the horizon and the ADM mass of the
+spacetime is M + MBH. In the absence of DM halo, M = 0, the spacetime (6) reduces to
+Schwarzschild BH with mass MBH. In galaxies, the compactness M/r0 can be as large as
+10−4 [32]. In general astrophysical environments the compactness M/r0 is usually small.
+Expanding the function f(r) in Eq. (7) about M/r0 = 0 to the second order we get
+f(r) ≃
+�
+1 − 2MBH
+r
+� �
+1 − 2M
+r0
++ 4M 2
+3r2
+0
++ 2Mr
+r2
+0
++ O[r−3
+0 ]
+�
+=
+�
+1 − 2MBH
+r
+�
+(1 + α + rβ),
+(8)
+where α = −2M/r0 + 4M 2/3r2
+0 and β = 2M/r2
+0.
+Now we consider a MBH in the center of a DM halo and a SCO moving on geodesics
+around the MBH in the equatorial plane (θ = π/2). The geodesic equation is
+duµ
+dτ = 1
+2uαuβ∂µgαβ,
+(9)
+where uα = drα/dτ, τ is the proper time and rα = (t, r, θ, φ). Because the spacetime is
+static and spherically symmetric, from the geodesic equation (9) we obtain two conserved
+quantities u0 = −E/µ and uφ = L/µ,
+u0 = −E/µ = −
+√
+1 + 2ε,
+(10)
+uφ = L/µ = h,
+(11)
+
+5
+where E and L represent the orbital energy and angular momentum of the system, respec-
+tively, and the reduced mass µ is approximately equal to the mass of the SCO. The radial
+equation of motion is
+1 +
+�dr
+dτ
+�2 �
+1 − 2m(r)
+r
+�−1
++ h2
+r2 = 1 + 2ε
+f
+.
+(12)
+For convenience, we introduce the orbital elements, the semi-latus rectum p and the
+eccentricity e, to parameterize the orbital motion,
+r =
+p
+1 + e cos χ,
+(13)
+where χ is a parameter. Rewriting the variables h and ε in terms of p and e, we obtain
+h2 =
+p Rs (1 + α) + p3β (1 − e2)−1
+2(1 + α)
+�
+1 − 1
+2
+Rs
+p (3 + e2)
+�
++ p β
+�
+1 − 2 Rs
+p
+�,
+(14)
+ε = −
+Rs
+2p (1 − e2)
+�
+1 − 2Rs
+p
+�
++ α j + α2 g + β k
+2
+�
+1 − 1
+2
+Rs
+P (3 + e2)
+�
+(1 + α) + p β
+�
+1 − 2 Rs
+p
+�,
+(15)
+where Rs = 2MBH,
+j = −
+�
+1 − 2Rs
+p
+�
++ Rs
+2p
+�
+1 − 4Rs
+p
+�
+(1 − e2),
+g = −
+�
+1 − 2Rs
+p
+�
+− R2
+s
+p2 (1 − e2),
+k = −p(3 + e2)
+2(1 − e2)
+�
+1 − 2Rs
+p
+�
+− 2R2
+s
+p .
+In terms of χ, Eqs. (10) and (11) become
+dφ
+dχ =
+�1
+2
+Rs
+p (1 + α) + 1
+2pβ(1 − e2)−1
+� 1
+2 �1
+2
+Rs
+p
+�
+1 − Rs
+p (3 + e cos χ)
+�
++ α A + 2α2 A + β B
+�− 1
+2
+J1,
+(16)
+dt
+dχ =
+p
+(1 + e cos χ)2
+��
+1 − (1 + e)Rs
+p
+� �
+1 − (1 − e)Rs
+p
+�
++ C
+� 1
+2
+×
+�
+1 − Rs
+p (1 + e cos χ)
+�−1 �1
+2
+Rs
+p
+�
+1 − Rs
+p (3 + e cos χ) + αA + 2��2A + βB
+��− 1
+2
+J2,
+(17)
+
+6
+where
+A = Rs
+p
+�
+1 − Rs
+p (3 + e cos χ)
+�
+,
+B =
+p
+2(1 − e2)(1 + e cos χ)
+�
+2
+�
+1 − Rs
+p
+�
++
+�
+1 − 4Rs
+p
+−
+�Rs
+p
+�2
+(1 − e2)(1 + e cos χ) − Rs
+p e2(1 + cos2 χ)
+��
+,
+C = α
+�
+1 − 1
+2(3 + e2)Rs
+p
+�
++ 1
+2pβ
+�
+1 − 2Rs
+r
+�
+− (αj + α2g + βk),
+J1 =
+�
+1 + α +
+βp
+1 + e cos χ
+� 1
+2 �
+1 − 2Mp/(1 + e cos χ)
+a + p/(1 + e cos χ)2
+�
+1 − Rs
+p (1 + e cos χ)
+� �− 1
+2
+,
+J2 =
+�
+1 + α +
+βp
+1 + e cos χ
+�− 1
+2 �
+1 − 2Mp/(1 + e cos χ)
+a + p/(1 + e cos χ)2
+�
+1 − Rs
+p (1 + e cos χ)
+� �− 1
+2
+.
+Eqs. (16) and (17) can be integrated to obtain φ(χ) and t(χ). Taking different compact-
+ness and mass for the DM halo, using Cartesian coordinate (x, y) = (r cos φ, r sin φ) in the
+equatorial plane, we show the orbits of EMRIs in galaxies with and without DM in Fig.
+1. Due to the gravitational drag of DM halos, the orbits with DM halos are different from
+those without DM. From Fig. 1, we see that for the same value of M, the effect of DM
+halos on the orbital precession is larger if the compactness of the DM halo M/r0 is bigger.
+DM halos decrease the orbital precessions, and can even reverse the direction of precession
+if the density of DM halo ρDM is large enough. The result of retrograde precessions of the
+orbital motion in the spacetime (6) is consistent with that found in [56], and the anomalous
+precessions of binaries in DM environments were also found in [48, 61, 62].
+To probe DM halos and study their impact on the orbits of EMRIs, we calculate the time
+P and the orbital precession ∆φ over one cycle when the orbital parameter χ increases by
+2π,
+T =
+� 2π
+0
+dt
+dχdχ,
+(18)
+∆φ =
+� 2π
+0
+dφ
+dχdχ − 2π.
+(19)
+Expanding Eqs. (16) and (17) about Rs/p = 0 to the second order and substituting the
+
+7
+-100
+-50
+0
+50
+100
+-100
+-50
+0
+50
+100
+x/Rs
+y/Rs
+r0=102M, M=102MBH
+-100
+-50
+0
+50
+-100
+-50
+0
+50
+x/Rs
+y/Rs
+r0=103M, M=102MBH
+-100
+-50
+0
+50
+-100
+-50
+0
+50
+x/Rs
+y/Rs
+r0=102M, M=103MBH
+-100
+-50
+0
+50
+100
+-100
+-50
+0
+50
+100
+x/Rs
+y/Rs
+r0=103M, M=103MBH
+FIG. 1.
+The orbits of EMRIs in galaxies with and without DM halos. The mass of MBHs is set
+as MBH = 106M⊙, the eccentricity e = 0.6, and the semi-latus rectum p = 20Rs. We take the
+compactness M/r0 as 10−2 and 10−3, and the total mass M as 102MBH and 103MBH. The red
+dashed lines show the trajectories with DM and the blue solid lines show the orbits without DM.
+The arrows represent the directions of orbital precessions.
+
+8
+results into Eqs. (18) and (19), we get
+T = 2π
+�
+2p3
+Rs
+1
+(1 − e2)3/2
+�
+1 + 3
+2(1 − e2)Rs
+p + 3
+2(1 − e2)
+�
+1 + 5
+4(1 − e2)
+1
+2
+� �Rs
+p
+�2
++ M
+r0
++ 5M 2
+6r2
+0
++
+Mp
+r2
+0(1 − e2)
+�
+e2 − 11
+2
+�
+− 3Mp2/Rs
+r2
+0(1 − e2)
+�
+,
+(20)
+∆φ = 3πRs
+p + 3π
+8 (18 + e2)
+�Rs
+p
+�2
+−
+2π
+1 − e2
+Mp
+r2
+0
+�
+3 +
+1 + e2 + 2 Rs
+p
+(1 − e2)1/2
+�
+.
+(21)
+The terms with M in the above Eqs. (20) and (21) come from DM halos. In the absence of
+DM, M = 0, the above results (20) and (21) recover those for EMRIs with the central MBH
+being a Schwarzschild BH. The dominant contribution to the period T in Eq. (20) is the first
+term, so T becomes larger as the semi-latus rectum p increases. However, there are positive
+and negative contributions from the local DM halos, the local DM halos may slow down the
+increase of T as p increases because the negative contribution in the last term in Eq. (20)
+and the presence of DM halos helps the increase of T with p if the last negative contribution
+is negligible. From Eq. (21), it is easy to understand that the presence of DM halo decreases
+the orbital procession and even retrogrades the orbital procession if the local density of DM
+halos ρDM ∼ M/r2
+0 is large enough so that the third term dominates over the first two terms.
+As the orbit becomes larger, i.e., the semi-latus rectum p increases, the orbital precession
+decreases and the prograde precession decreases faster in the presence of DM halos because
+the third term due to DM halos in Eq. (21) becomes bigger. With DM halos, the prograde-
+to-retrograde precession transition happens at some critial value of p and then the prograde
+precessions change to retrograde precessions as p increases further; afterwards, the retrograde
+precessions increase as p increases. Choosing different values for the compactness M/r0 and
+the total mass of DM halos M and using Eqs. (20) and (21), we plot the results of the period
+T and the orbital precession ∆φ versus the semi-latus rectum p in Fig. 2. As expected,
+the orbital period T increases with p; the prograde precessions decrease with p and DM
+halos help the decrease. For the case of r0 = 102M and M = 102MBH, the periapsis shifts
+change from prograde precessions to retrograde precessions at p = 60Rs and the retrograde
+precession increases with p when p ≳ 60Rs.
+From the above discussions, we see that the orbital motions of EMRIs are influenced by
+DM halos, and we expect that the effects of local DM halos will leave imprints on GWs so
+that we can probe local DM halos through the observations of GWs emitted from EMRIs.
+
+9
+20
+40
+60
+80
+100
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+p/RS
+<Δϕ>
+90
+91
+0.07
+0.09
+0.11
+r0=103M, M=103MBH
+r0=102M, M=103MBH
+r0=103M, M=102MBH
+r0=102M, M=102MBH
+M=0
+30
+40
+50
+60
+70
+80
+90
+100
+10
+20
+30
+40
+50
+p/Rs
+P/hour
+90
+91
+41
+41.5
+FIG. 2.
+The results of orbital period and precession for EMRIs in galaxies with and without DM.
+The mass of central MBHs is set as MBH = 106M⊙ and the eccentricity e = 0.6. We take the
+compactness M/r0 as 10−2 and 10−3, and the total mass M as 102MBH, 103MBH and M = 0. The
+inserts show the evolution in a short time period.
+III.
+GWS OF EMRIS IN THE ENVIRONMENTS OF GALAXIES
+Using the above results for the orbital motions of EMRIs, we get the leading order energy
+and angular momentum fluxes
+�dE
+dt
+�
+GW
+≃ 32
+5
+�
+µ
+MBH
+�2 �MBH
+p
+�5
+(1 − e2)3/2
+�
+1 + 73
+24e2 + 37
+96e4
+� �
+1 − 6M
+r0
+�
+,
+(22)
+�dL
+dt
+�
+GW
+≃ 32
+5
+�
+µ
+MBH
+�2
+MBH
+�MBH
+p
+�7/2
+(1 − e2)3/2
+�
+1 + 7
+8e2
+� �
+1 − 5M
+r0
+�
+.
+(23)
+The last factors 1 − 6M/r0 and 1 − 5M/r0 are the corrections from DM halos around the
+MBH. Note that the effects of environmental DM halos on the losses of energy and angular
+momentum only depend on the compactness M/r0 and the energy and angular momentum
+fluxes become smaller if the compactness is larger. In the absence of local DM halos, M = 0,
+Eqs. (22) and (23) recover the standard results for eccentric binaries [63, 64]. Applying the
+energy and angular momentum balance equations
+�dE
+dt
+�
+GW
+= −
+�dE
+dt
+�
+orbit
+,
+(24)
+�dL
+dt
+�
+GW
+= −
+�dL
+dt
+�
+orbit
+,
+(25)
+
+10
+we get the leading order evolution of the orbital parameters p(t) and e(t) due to the emission
+of GWs,
+dp
+dt = −64
+5
+µ
+MBH
+�MBH
+p
+�3 �
+1 − e2� 3
+2
+�
+1 + 7
+8e2
+� �
+1 − 5M
+r0
+�
+,
+(26)
+de
+dt = −304
+15
+e
+p
+µ
+MBH
+�MBH
+p
+�3 �
+1 − e2� 3
+2
+�
+1 + 121
+304e2
+� �
+1 − 5M
+r0
+�
+.
+(27)
+Since the right sides of Eqs. (26) and (27) are negative, both the semi-latus rectum p and
+the eccentricity decrease with time due to the radiation of GWs. The presence of local DM
+halos slows down the decrease of p and e, the bigger the compactness M/r0 is, the slower
+the semi-latus rectum p(t) and the eccentricity decrease. In Fig. 3, we show the evolution
+of the orbital parameters p(t) and e(t) due to the emission of GWs. Comparing with the
+astrophysical environments without DM, it takes more time for EMRIs with DM halos to
+evolve from p = 20Rs to p = 3Rs. The larger the compactness M/r0 is, the more time it
+takes. The presence of DM halos also slows down the decrease rate of the eccentricity and
+the final eccentricity is a bit larger with larger compactness.
+r0=102M, e0=0.6
+r0=103M, e0=0.6
+r0=102M, e0=0.2
+r0=103M, e0=0.2
+M=0, e0=0.6
+M=0, e0=0.2
+0
+200
+400
+600
+800
+1000
+0
+5
+10
+15
+20
+t/yr
+p/Rs
+0
+200
+400
+600
+800
+1000
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+t/yr
+e
+FIG. 3.
+The evolution of the orbital parameters p and e from the initial p = 20Rs to p = (3+e)Rs.
+The mass of central MBHs is chosen as MBH = 106M⊙, the mass of the SCO is µ = 10M⊙ and the
+initial eccentricity is chosen as e0 = 0.2, 0.6. We consider two different values for the compactness
+of the DM halo, M/r0 = 10−2 and 10−3. The solid lines correspond to the cases without DM.
+As discussed above, the effects of DM halos will be manifested in GW waveforms. The
+quadrupole formula of GWs is
+hjk = 2
+dL
+¨Ijk,
+(28)
+
+11
+where dL is the luminosity distance between the detector and the source and Ijk is the
+quadrupole moment of EMRIs. The tenser modes h+ and h× in the transverse-traceless
+gauge are given by
+h+ = 1
+2
+�
+ej
+Xek
+X − ej
+Y ek
+Y
+�
+hjk,
+(29)
+h× = 1
+2
+�
+ej
+Xek
+Y − ej
+Y ek
+X
+�
+hjk,
+(30)
+where eX and eY are the orthonormal vectors in the plane that is perpendicular to the
+direction from the detector to the GW source. Plugging the results for the orbital evolution
+obtained above into Eq. (28), we numerically calculate the time-domain GW waveforms.
+The time-domain plus-mode GW waveforms for EMRIs with and without DM halos are
+shown in Fig. 4. From Fig 4, we see that initially the difference between GW waveforms
+with and without DM halos is negligible. One year later, the two waveforms for EMRIs with
+and without DM halos are quite different.
+In order to quantify the impact of DM halo environments on the dephasing of GW
+waveforms, we calculate the number of orbital cycles accumulated from time ti to tf [65–67]
+N(t) =
+� tf
+ti
+˙φ(t)dt.
+(31)
+Over one-year evolution before the merger, the numbers of orbital cycles for EMRIs with
+and without DM halos are NDM and N0 respectively. In Fig 5, we show the difference ∆N =
+NDM − N0 between the number of orbital cycles with and without DM halos accumulated
+over one year before the merger. Following [68], we choose ∆N ∼ 1 rad as the threshold for
+a detectable dephasing. The results show that we can detect the compactness as small as
+≲ 10−4. The results also show that eccentric orbits can help detect DM halos with smaller
+compactness.
+To distinguish the waveforms more accurately, we calculate the mismatch between GW
+signals emitted from EMRIs with and without DM halos. Given two signals h1(t) and h2(t),
+the inner product (h1|h2) is defined as
+(h1|h2) = 2
+� +∞
+0
+˜h1(f)˜h∗
+2(f) + ˜h2(f)˜h∗
+1(f)
+Sh(f)
+df,
+(32)
+where ˜h(f) is the Fourier transformation of the time-domain signal h(t), ˜h∗ denotes the
+complex conjugate of ˜h, and the SNR for the signal h is
+�
+(h|h). For LISA, the one-side
+
+12
+0
+20
+40
+60
+80
+100
+-5
+0
+5
+t/hour
+h+
+At the beginning
+0
+20
+40
+60
+80
+100
+-5
+0
+5
+t/hour
+h+
+365 days later
+r0=102M, M=102MBH
+M=0
+1023×
+1023×
+0
+20
+40
+60
+80
+100
+-5
+0
+5
+t/hour
+h+
+0
+20
+40
+60
+80
+100
+-5
+0
+5
+t/hour
+h+
+r0=103M, M=102MBH
+M=0
+1023×
+1023×
+FIG. 4.
+The time-domain plus mode GW waveforms for EMRIs with and without DM halos.
+The mass of central MBHs is MBH = 106M⊙, the mass of the SCO is µ = 10M⊙, the total mass
+of DM halos M is = 102MBH, the inclination angle ι = π/6, the luminosity distance dL = 1Gpc,
+the initial longitude of pericenter ω0 = 0 and the initial eccentricity e0 = 0.6 at p0 = 20Rs. M = 0
+corresponds to the case without DM halos. The left panels show the initial waveforms. The right
+panels show the waveforms after one year. The top panels are for M/r0 = 10−2 and the bottom
+panels are for M/r0 = 10−3.
+noise power spectral density is [69]
+Sh(f) = Sx
+L2 + 2Sa [1 + cos2(2π fL/c)]
+(2 πf)4L2
+�
+1 +
+�4 × 10−4Hz
+f
+��
+,
+(33)
+where √Sa = 3 × 10−15 m s−2/Hz1/2 is the acceleration noise, √Sx = 1.5 × 10−11 m/Hz1/2
+is the displacement noise and L = 2.5 × 106 km is the arm length of LISA [7]. The overlap
+between two GW signals is quantified as [60]
+O(˜h1, ˜h2) =
+(˜h1|˜h2)
+�
+(˜h1|˜h1)(˜h2|˜h2)
+,
+(34)
+
+ ro=102M, M=102MBH
+- M=0ro=103M, M=102MBH
+M-013
+e0=0
+e0=0.2
+e0=0.4
+e0=0.6
+-5
+-4.5
+-4
+-3.5
+-3
+-2.5
+-2
+300
+250
+200
+150
+100
+50
+0
+Log10[M/r0]
+|Δ|
+FIG. 5.
+The difference between the orbital cycles with and without DM halos ∆N(t) over one-year
+evolution before the merger for different compactness of halos M/r0. The initial eccentricity e0
+is chosen at p0 = 20Rs. The mass of central MBHs is MBH = 106M⊙ and the mass of the SCO
+is µ = 10M⊙. The masses of DM halos are M = 102MBH. The black dashed line corresponds to
+∆N = 1 rad.
+and the mismatch between two signals is defined as
+Mismatch = 1 − Omax(˜h1, ˜h2),
+(35)
+where the maximum is evaluated with respect to time and phase shifts. The mismatch is
+zero if two signals are identical. Two signals are considered experimentally distinguishable if
+their mismatch is larger than d/(2 SNR2), where d = 13 is the number of intrinsic parameters
+of the GW source [70–72]. Considering EMRIs with masses (106+10)M⊙ at dL = 1 Gpc and
+integration time of one year before the coalescence, we calculate the mismatch between GW
+waveforms with and without DM halos and the results with LISA are shown in Fig 6. The
+SNR is about 32 for the GW signals from EMRIs considered above. The initial eccentricity
+e0 is chosen at p0 = 20Rs. As shown in Fig 6, if the compactness of DM halo M/r0 is
+larger, then the mismatch between GW waveforms with and without DM halos is bigger,
+so more compact DM halos can be detected easier with LISA. Again eccentric orbits can
+detect smaller compactness. Therefore, we can use GWs from EMRIs in the environments
+of galaxies to test the existence of DM halos and detect the compactness of the halos M/r0
+as small as 10−5.
+
+14
+e0=0.2
+e0=0.6
+-6
+-5
+-4
+-3
+-2
+-1
+0.001
+0.010
+0.100
+1
+Log10[M/r0]
+Mismatch
+FIG. 6.
+The results of the mismatch between GW waveforms with and without DM halos for
+different compactness M/r0 and initial eccentricity e0. The black dashed line corresponds to the
+threshold d/(2 SNR2) ≈ 0.0072.
+IV.
+CONCLUSIONS AND DISCUSSIONS
+Using the analytic, static and spherically symmetric metric for a Schwarzschild black hole
+immersed in DM halos with Hernquist type density distribution, we derive analytic formulae
+for the orbital period and orbital precession for eccentric EMRIs with the environment of
+DM halos. The results show that the presence of DM halo decreases the orbital procession
+and even retrogrades the orbital procession if the local density of DM halos ρDM ∼ M/r2
+0 is
+large enough. As the orbit becomes larger, the orbital precession decreases and the prograde
+precession decreases faster in the presence of DM halos. With DM halos, the prograde-to-
+retrograde precession transition happens at some critial value of p and then the prograde
+precessions change to retrograde precessions as p increases further; afterwards, the retrograde
+precessions increase as p increases.
+Taking the energy and angular momentum fluxes of GWs into consideration, we derive
+analytic formulae for the evolutions of the semi-latus rectum and the eccentricity.
+The
+presence of local DM halos slows down the decrease of the semi-latus rectum and the eccen-
+tricity. Comparing the numbers of orbital cycles with and without DM halos over one-year
+evolution before the merger, we find that DM halos with the compactness as small as 10−4
+can be detected. By calculating the mismatch between GW waveforms with and without
+DM halos, we show that we can use GWs from EMRIs in the environments of galaxies to
+
+15
+test the existence of DM halos and detect the compactness as small as 10−5. We also find
+that eccentric orbits can help detect DM halos with smaller compactness.
+Binaries in the environments of galaxies are also affected by the dynamical frictions of the
+surrounding medium [73–77], and the accretion of the medium [46, 78, 79]. It is necessary
+to consider the effects of dynamical frictions and accretion when the medium is dense. To
+distinguish the effects of DM halos from other mediums (e.g. accretion disks), or modified
+gravity on GWs, further study is needed [43, 68, 80–82].
+ACKNOWLEDGMENTS
+The computing work in this paper is supported by the Public Service Platform of High
+Performance Computing by Network and Computing Center of HUST. This research is
+supported in part by the National Key Research and Development Program of China under
+Grant No. 2020YFC2201504.
+[1] B. P. Abbott et al. (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary
+Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc].
+[2] B. P. Abbott et al. (LIGO Scientific, Virgo), GW150914: The Advanced LIGO Detectors in
+the Era of First Discoveries, Phys. Rev. Lett. 116, 131103 (2016), arXiv:1602.03838 [gr-qc].
+[3] B. P. Abbott et al. (LIGO Scientific, Virgo), GWTC-1: A Gravitational-Wave Transient
+Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and
+Second Observing Runs, Phys. Rev. X 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE].
+[4] R. Abbott et al. (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed
+by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X 11,
+021053 (2021), arXiv:2010.14527 [gr-qc].
+[5] R. Abbott et al. (LIGO Scientific, VIRGO), GWTC-2.1: Deep Extended Catalog of Com-
+pact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third
+Observing Run, arXiv:2108.01045 [gr-qc].
+[6] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), GWTC-3: Compact Binary Coales-
+cences Observed by LIGO and Virgo During the Second Part of the Third Observing Run,
+
+16
+arXiv:2111.03606 [gr-qc].
+[7] P. Amaro-Seoane et al. (LISA), Laser Interferometer Space Antenna, arXiv:1702.00786 [astro-
+ph.IM].
+[8] J. Luo et al. (TianQin), TianQin: a space-borne gravitational wave detector, Class. Quant.
+Grav. 33, 035010 (2016), arXiv:1512.02076 [astro-ph.IM].
+[9] W.-R. Hu and Y.-L. Wu, The Taiji Program in Space for gravitational wave physics and the
+nature of gravity, Natl. Sci. Rev. 4, 685 (2017).
+[10] Y. Gong, J. Luo, and B. Wang, Concepts and status of Chinese space gravitational wave
+detection projects, Nature Astron. 5, 881 (2021), arXiv:2109.07442 [astro-ph.IM].
+[11] V. Baibhav et al., Probing the nature of black holes: Deep in the mHz gravitational-wave sky,
+Exper. Astron. 51, 1385 (2021), arXiv:1908.11390 [astro-ph.HE].
+[12] P. Amaro-Seoane et al., Astrophysics with the Laser Interferometer Space Antenna,
+arXiv:2203.06016 [gr-qc].
+[13] K. G. Arun et al. (LISA), New horizons for fundamental physics with LISA, Living Rev. Rel.
+25, 4 (2022), arXiv:2205.01597 [gr-qc].
+[14] N. Karnesis et al., The Laser Interferometer Space Antenna mission in Greece White Paper,
+arXiv:2209.04358 [gr-qc].
+[15] S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sopuerta, C. P. L. Berry, E. Berti, P. Amaro-
+Seoane, A. Petiteau, and A. Klein, Science with the space-based interferometer LISA. V:
+Extreme mass-ratio inspirals, Phys. Rev. D 95, 103012 (2017), arXiv:1703.09722 [gr-qc].
+[16] P. Amaro-Seoane, J. R. Gair, M. Freitag, M. Coleman Miller, I. Mandel, C. J. Cutler, and
+S. Babak, Astrophysics, detection and science applications of intermediate- and extreme mass-
+ratio inspirals, Class. Quant. Grav. 24, R113 (2007), arXiv:astro-ph/0703495.
+[17] C. P. L. Berry, S. A. Hughes, C. F. Sopuerta, A. J. K. Chua, A. Heffernan, K. Holley-
+Bockelmann, D. P. Mihaylov, M. C. Miller, and A. Sesana, The unique potential of extreme
+mass-ratio inspirals for gravitational-wave astronomy, arXiv:1903.03686 [astro-ph.HE].
+[18] P. A. Seoane et al., The effect of mission duration on LISA science objectives, Gen. Rel. Grav.
+54, 3 (2022), arXiv:2107.09665 [astro-ph.IM].
+[19] D. Laghi, N. Tamanini, W. Del Pozzo, A. Sesana, J. Gair, S. Babak, and D. Izquierdo-Villalba,
+Gravitational-wave cosmology with extreme mass-ratio inspirals, Mon. Not. Roy. Astron. Soc.
+508, 4512 (2021), arXiv:2102.01708 [astro-ph.CO].
+
+17
+[20] S. McGee, A. Sesana, and A. Vecchio, Linking gravitational waves and X-ray phenomena with
+joint LISA and Athena observations, Nature Astron. 4, 26 (2020), arXiv:1811.00050 [astro-
+ph.HE].
+[21] S. van den Bergh, The Early history of dark matter, Publ. Astron. Soc. Pac. 111, 657 (1999),
+arXiv:astro-ph/9904251.
+[22] V. C. Rubin and W. K. Ford, Jr., Rotation of the Andromeda Nebula from a Spectroscopic
+Survey of Emission Regions, Astrophys. J. 159, 379 (1970).
+[23] V. C. Rubin, N. Thonnard, and W. K. Ford, Jr., Rotational properties of 21 SC galaxies with
+a large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122
+kpc/, Astrophys. J. 238, 471 (1980).
+[24] K. G. Begeman, A. H. Broeils, and R. H. Sanders, Extended rotation curves of spiral galaxies:
+Dark haloes and modified dynamics, Mon. Not. Roy. Astron. Soc. 249, 523 (1991).
+[25] M. Persic, P. Salucci, and F. Stel, The Universal rotation curve of spiral galaxies: 1. The Dark
+matter connection, Mon. Not. Roy. Astron. Soc. 281, 27 (1996), arXiv:astro-ph/9506004.
+[26] E. Corbelli and P. Salucci, The Extended Rotation Curve and the Dark Matter Halo of M33,
+Mon. Not. Roy. Astron. Soc. 311, 441 (2000), arXiv:astro-ph/9909252.
+[27] L. A. Moustakas et al., Strong gravitational lensing probes of the particle nature of dark
+matter, arXiv:0902.3219 [astro-ph.CO].
+[28] R. Massey, T. Kitching, and J. Richard, The dark matter of gravitational lensing, Rept. Prog.
+Phys. 73, 086901 (2010), arXiv:1001.1739 [astro-ph.CO].
+[29] J. Ellis and K. A. Olive, Supersymmetric Dark Matter Candidates, arXiv:1001.3651 [astro-
+ph.CO].
+[30] A. Challinor, CMB anisotropy science: a review, IAU Symp. 288, 42 (2013), arXiv:1210.6008
+[astro-ph.CO].
+[31] L. Sadeghian, F. Ferrer, and C. M. Will, Dark matter distributions around massive black
+holes: A general relativistic analysis, Phys. Rev. D 88, 063522 (2013), arXiv:1305.2619 [astro-
+ph.GA].
+[32] J. F. Navarro, C. S. Frenk, and S. D. M. White, A Universal density profile from hierarchical
+clustering, Astrophys. J. 490, 493 (1997), arXiv:astro-ph/9611107.
+[33] P. Gondolo and J. Silk, Dark matter annihilation at the galactic center, Phys. Rev. Lett. 83,
+1719 (1999), arXiv:astro-ph/9906391.
+
+18
+[34] L. Hernquist, An Analytical Model for Spherical Galaxies and Bulges, Astrophys. J. 356, 359
+(1990).
+[35] N. Yunes, B. Kocsis, A. Loeb, and Z. Haiman, Imprint of Accretion Disk-Induced Migration
+on Gravitational Waves from Extreme Mass Ratio Inspirals, Phys. Rev. Lett. 107, 171103
+(2011), arXiv:1103.4609 [astro-ph.CO].
+[36] B. Kocsis, N. Yunes, and A. Loeb, Observable Signatures of EMRI Black Hole Binaries Embed-
+ded in Thin Accretion Disks, Phys. Rev. D 84, 024032 (2011), arXiv:1104.2322 [astro-ph.GA].
+[37] K. Eda, Y. Itoh, S. Kuroyanagi, and J. Silk, New Probe of Dark-Matter Properties: Gravita-
+tional Waves from an Intermediate-Mass Black Hole Embedded in a Dark-Matter Minispike,
+Phys. Rev. Lett. 110, 221101 (2013), arXiv:1301.5971 [gr-qc].
+[38] C. F. B. Macedo, P. Pani, V. Cardoso, and L. C. B. Crispino, Into the lair: gravitational-wave
+signatures of dark matter, Astrophys. J. 774, 48 (2013), arXiv:1302.2646 [gr-qc].
+[39] K. Eda, Y. Itoh, S. Kuroyanagi, and J. Silk, Gravitational waves as a probe of dark matter
+minispikes, Phys. Rev. D 91, 044045 (2015), arXiv:1408.3534 [gr-qc].
+[40] E. Barausse, V. Cardoso, and P. Pani, Can environmental effects spoil precision gravitational-
+wave astrophysics?, Phys. Rev. D 89, 104059 (2014), arXiv:1404.7149 [gr-qc].
+[41] L. Barack et al., Black holes, gravitational waves and fundamental physics: a roadmap, Class.
+Quant. Grav. 36, 143001 (2019), arXiv:1806.05195 [gr-qc].
+[42] O. A. Hannuksela, K. W. K. Wong, R. Brito, E. Berti, and T. G. F. Li, Probing the existence of
+ultralight bosons with a single gravitational-wave measurement, Nature Astron. 3, 447 (2019),
+arXiv:1804.09659 [astro-ph.HE].
+[43] V. Cardoso and A. Maselli, Constraints on the astrophysical environment of binaries with
+gravitational-wave observations, Astron. Astrophys. 644, A147 (2020), arXiv:1909.05870
+[astro-ph.HE].
+[44] X.-J. Yue and Z. Cao, Dark matter minispike: A significant enhancement of eccentricity for
+intermediate-mass-ratio inspirals, Phys. Rev. D 100, 043013 (2019), arXiv:1908.10241 [astro-
+ph.HE].
+[45] L. Annulli, V. Cardoso, and R. Vicente, Stirred and shaken: Dynamical behavior of boson stars
+and dark matter cores, Phys. Lett. B 811, 135944 (2020), arXiv:2007.03700 [astro-ph.HE].
+[46] A. Derdzinski, D. D’Orazio, P. Duffell, Z. Haiman, and A. MacFadyen, Evolution of gas
+disc–embedded intermediate mass ratio inspirals in the LISA band, Mon. Not. Roy. Astron.
+
+19
+Soc. 501, 3540 (2021), arXiv:2005.11333 [astro-ph.HE].
+[47] L. Zwick, P. R. Capelo, and L. Mayer, Priorities in gravitational waveform modelling for future
+space-borne detectors: vacuum accuracy or environment?, arXiv:2209.04060 [gr-qc].
+[48] N. Dai, Y. Gong, T. Jiang, and D. Liang, Intermediate mass-ratio inspirals with dark matter
+minispikes, Phys. Rev. D 106, 064003 (2022), arXiv:2111.13514 [gr-qc].
+[49] A. Coogan, G. Bertone, D. Gaggero, B. J. Kavanagh, and D. A. Nichols, Measuring the dark
+matter environments of black hole binaries with gravitational waves, Phys. Rev. D 105, 043009
+(2022), arXiv:2108.04154 [gr-qc].
+[50] V. Cardoso, K. Destounis, F. Duque, R. P. Macedo, and A. Maselli, Black holes in galaxies:
+Environmental impact on gravitational-wave generation and propagation, Phys. Rev. D 105,
+L061501 (2022), arXiv:2109.00005 [gr-qc].
+[51] A. Einstein, On a stationary system with spherical symmetry consisting of many gravitating
+masses, Annals Math. 40, 922 (1939).
+[52] A. Geralico, F. Pompi, and R. Ruffini, On Einstein clusters, Int. J. Mod. Phys. Conf. Ser. 12,
+146 (2012).
+[53] R. A. Konoplya and A. Zhidenko, Solutions of the Einstein Equations for a Black Hole Sur-
+rounded by a Galactic Halo, Astrophys. J. 933, 166 (2022), arXiv:2202.02205 [gr-qc].
+[54] K. Jusufi, Black holes surrounded by Einstein clusters as models of dark matter fluid,
+arXiv:2202.00010 [gr-qc].
+[55] J. Liu, S. Chen, and J. Jing, Tidal effects of a dark matter halo around a galactic black hole*,
+Chin. Phys. C 46, 105104 (2022), arXiv:2203.14039 [gr-qc].
+[56] K. Destounis, A. Kulathingal, K. D. Kokkotas, and G. O. Papadopoulos, Gravitational-
+wave imprints of compact and galactic-scale environments in extreme-mass-ratio binaries,
+arXiv:2210.09357 [gr-qc].
+[57] V. Cardoso, K. Destounis, F. Duque, R. Panosso Macedo, and A. Maselli, Gravitational
+Waves from Extreme-Mass-Ratio Systems in Astrophysical Environments, Phys. Rev. Lett.
+129, 241103 (2022), arXiv:2210.01133 [gr-qc].
+[58] J. R. Gair and K. Glampedakis, Improved approximate inspirals of test-bodies into Kerr black
+holes, Phys. Rev. D 73, 064037 (2006), arXiv:gr-qc/0510129.
+[59] J. R. Gair, D. J. Kennefick, and S. L. Larson, Semi-relativistic approximation to gravita-
+tional radiation from encounters with black holes, Phys. Rev. D 72, 084009 (2005), [Erratum:
+
+20
+Phys.Rev.D 74, 109901 (2006)], arXiv:gr-qc/0508049.
+[60] S. Babak, H. Fang, J. R. Gair, K. Glampedakis, and S. A. Hughes, ’Kludge’ gravitational wave-
+forms for a test-body orbiting a Kerr black hole, Phys. Rev. D 75, 024005 (2007), [Erratum:
+Phys.Rev.D 77, 04990 (2008)], arXiv:gr-qc/0607007.
+[61] T. Igata and Y. Takamori, Periapsis shifts in dark matter distribution with a dense core, Phys.
+Rev. D 105, 124029 (2022), arXiv:2202.03114 [gr-qc].
+[62] T. Igata, T. Harada, H. Saida, and Y. Takamori, Periapsis shifts in dark matter distribution
+around a black hole, arXiv:2202.00202 [gr-qc].
+[63] P. Peters and J. Mathews, Gravitational radiation from point masses in a Keplerian orbit,
+Phys. Rev. 131, 435 (1963).
+[64] P. Peters, Gravitational Radiation and the Motion of Two Point Masses, Phys. Rev. 136,
+B1224 (1964).
+[65] E. Berti, A. Buonanno, and C. M. Will, Estimating spinning binary parameters and testing al-
+ternative theories of gravity with LISA, Phys. Rev. D 71, 084025 (2005), arXiv:gr-qc/0411129.
+[66] B. J. Kavanagh, D. A. Nichols, G. Bertone, and D. Gaggero, Detecting dark matter around
+black holes with gravitational waves: Effects of dark-matter dynamics on the gravitational
+waveform, Phys. Rev. D 102, 083006 (2020), arXiv:2002.12811 [gr-qc].
+[67] S. Barsanti, N. Franchini, L. Gualtieri, A. Maselli, and T. P. Sotiriou, Extreme mass-ratio
+inspirals as probes of scalar fields: Eccentric equatorial orbits around Kerr black holes, Phys.
+Rev. D 106, 044029 (2022), arXiv:2203.05003 [gr-qc].
+[68] A. Maselli, N. Franchini, L. Gualtieri, and T. P. Sotiriou, Detecting scalar fields with Extreme
+Mass Ratio Inspirals, Phys. Rev. Lett. 125, 141101 (2020), arXiv:2004.11895 [gr-qc].
+[69] T. Robson, N. J. Cornish, and C. Liu, The construction and use of LISA sensitivity curves,
+Class. Quant. Grav. 36, 105011 (2019), arXiv:1803.01944 [astro-ph.HE].
+[70] E. E. Flanagan and S. A. Hughes, Measuring gravitational waves from binary black hole
+coalescences: 2. The Waves’ information and its extraction, with and without templates,
+Phys. Rev. D 57, 4566 (1998), arXiv:gr-qc/9710129.
+[71] L. Lindblom, B. J. Owen, and D. A. Brown, Model Waveform Accuracy Standards for Grav-
+itational Wave Data Analysis, Phys. Rev. D 78, 124020 (2008), arXiv:0809.3844 [gr-qc].
+[72] A. Buonanno, Y.-b. Chen, and M. Vallisneri, Detection template families for gravitational
+waves from the final stages of binary–black-hole inspirals: Nonspinning case, Phys. Rev. D
+
+21
+67, 024016 (2003), [Erratum: Phys.Rev.D 74, 029903 (2006)], arXiv:gr-qc/0205122.
+[73] S. Chandrasekhar, Dynamical Friction. I. General Considerations: the Coefficient of Dynam-
+ical Friction, Astrophys. J. 97, 255 (1943).
+[74] E. C. Ostriker, Dynamical friction in a gaseous medium, Astrophys. J. 513, 252 (1999),
+arXiv:astro-ph/9810324.
+[75] H. Kim and W.-T. Kim, Dynamical Friction of a Circular-Orbit Perturber in a Gaseous
+Medium, Astrophys. J. 665, 432 (2007), arXiv:0705.0084 [astro-ph].
+[76] D. Traykova, K. Clough, T. Helfer, E. Berti, P. G. Ferreira, and L. Hui, Dynamical fric-
+tion from scalar dark matter in the relativistic regime, Phys. Rev. D 104, 103014 (2021),
+arXiv:2106.08280 [gr-qc].
+[77] R. Vicente and V. Cardoso, Dynamical friction of black holes in ultralight dark matter, Phys.
+Rev. D 105, 083008 (2022), arXiv:2201.08854 [gr-qc].
+[78] H. Bondi and F. Hoyle, On the mechanism of accretion by stars, Mon. Not. Roy. Astron. Soc.
+104, 273 (1944).
+[79] R. G. Edgar, A Review of Bondi-Hoyle-Lyttleton accretion, New Astron. Rev. 48, 843 (2004),
+arXiv:astro-ph/0406166.
+[80] N. Becker and L. Sagunski, Comparing Accretion Disks and Dark Matter Spikes in Interme-
+diate Mass Ratio Inspirals, . (2022), arXiv:2211.05145 [gr-qc].
+[81] C. Zhang and Y. Gong, Detecting electric charge with extreme mass ratio inspirals, Phys.
+Rev. D 105, 124046 (2022), arXiv:2204.08881 [gr-qc].
+[82] V. Cardoso, G. Castro, and A. Maselli, Gravitational waves in massive gravity theories: wave-
+forms, fluxes and constraints from extreme-mass-ratio mergers, Phys. Rev. Lett. 121, 251103
+(2018), arXiv:1809.00673 [gr-qc].
+

8tAyT4oBgHgl3EQfc_f6/content/tmp_files/2301.00296v1.pdf.txt

@@ -0,0 +1,579 @@
+Local Einstein relation for fractals
+L. Padilla, J. L. Iguain
+Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR) and
+Departamento de F´ısica FCEyN, Universidad Nacional de Mar del Plata,
+De´an Funes 3350, 7600 Mar del Plata, Argentina
+E-mail: [email protected]
+Abstract.
+We study single random walks and the electrical resistance for fractals
+obtained as the limit of a sequence of periodic structures. In the long-scale regime,
+power laws describe both the mean-square displacement of a random walk as a function
+of time and the electrical resistance as a function of length.
+We show that the
+corresponding power-law exponents satisfy the Einstein relation. For shorter scales,
+where these exponents depend on length, we find how the Einstein relation can be
+generalized to hold locally. All these findings were analytically derived and confirmed
+by numerical simulations.
+Keywords: Fractals, Power-law behaviours, Einstein relation.
+arXiv:2301.00296v1  [cond-mat.stat-mech]  31 Dec 2022
+
+Local Einstein relation for fractals
+2
+1. Introduction
+Fractals are characterized by quantities that exhibit power-law behaviour in space or
+time. More precisely, as scale invariance occurs for integer powers of a characteristic
+length, pure power laws are modulated by logarithmic periodic functions, that describe
+the departures from the main trend at intermediate scales. These modulations have
+been the object of recent interest and considerable effort has been devoted toward
+understanding the relation between log-periodicity and discrete-scale invariance [1–13].
+For a given fractal and some related observables, which show (modulated) power-
+law behaviours, a problem of interest is to determine whether or not the exponents
+associated with these quantities are independent. Sometimes we can expect a relation
+as a consequence of underlying physical laws.
+This is, for example, the case of the
+mass m, the electric resistance R and the mean-square-displacement (MSD) ∆r2 for
+a single random walker. On a fractal, the first two grow with length l as m(l) ∼ ldf
+and R(l) ∼ lζ, while the last one grows with time t as ∆r2(t) ∼ t2/dw. The exponents
+df, ζ and dw are known as the fractal, resistance and walk exponents, respectively, and
+these power-law behaviours hold for scales large enough to ensure self-similarity. In an
+d-dimensional euclidean space, the diffusion coefficient D and conductivity σ are related
+by the Einstein equation [14]
+σ = e2ρ
+kBT D.
+(1)
+Here, D = limt→∞ ∆r2(t)/2t, ρ and e are the density and charge of mobile particles, T
+is the temperature and kB is the Boltzmann constant. Equation (1) is one of the forms
+of the fluctuation-dissipation theorem, and can be used together with simple scaling
+heuristic arguments, to argue that the fractal, walk, and resistance exponents satisfy
+the Einstein relation [14]
+df = dw − ζ,
+(2)
+This property has been shown to hold asymptotically for some finitely ramified
+fractals [15, 16]; which has been used to analyze the periodicity of the oscillations in
+dynamic observables, in the first attempts to understand log-periodic modulation [17].
+Einstein relation was also investigated for random walks on weighted graphs [18], and,
+more recently, for karst networks structures [19].
+A deterministic fractal can be obtained as the limit of a sequence of periodic
+structures. In this procedure, the period increases at every step as Ln (n = 0, 1, 2, ...),
+where L is a basic characteristic length scale. Self-similarity is manifested in power-law
+behaviours, which occur for long enough scales. However, this does not always hold
+for shorter lengths. Thus, the local slopes of the observables as a function of time or
+length, in log-log scales, are variable quantities, which approach constant values only
+asymptotically.
+In this work we argue that the local fractal, walk, and resistance exponents are
+related through an equation that generalizes (2).
+This generalization is obtained
+
+Local Einstein relation for fractals
+3
+analytically, following the steady-state method for the calculation of the effective
+diffusion coefficients for periodic substrates [20]. To further strengthen our findings we
+perform numerical simulations for two models of fractals; which confirm the theoretical
+predictions.
+The paper is organized as follows. In Sec. 2 we relate the diffusion coefficient and the
+unit cell resistance for a periodic structure. In Sec. 3 we derive the Einstein relation for
+self-similar systems. In Sec. 4 we generalize this relation for scale-dependent exponents.
+In Sec. 5 we confirm the generalized relation by numerical simulations performed on
+models of asymptotic self-similar substrates. Finally, we give our conclusions in Sec. 6.
+2. Periodic systems
+In this section we address the problem of the diffusion coefficient for a periodic substrate.
+We follows the steady-state method developed in reference [20]. We start by introducing
+the periodic substrate with unit cell of linear dimension l, schematized in figure 1, where
+the points represent sites, and the arrows represent hopping rates. On this structure,
+a mobile particle can jump between connected sites according to the hopping rates k′s
+(for the sake of clarity only a few sites and arrows were highlighted). We focus on a
+steady-state of non-interacting particles flowing with a constant current density j.
+k
+n
+n
+(f+1)
+(f+1)
+n
+(f+1)
+n
+ns
+t
+v
+u
+k
+n
+n
+(f)
+(f)
+(f)
+n
+(f)
+n
+n(f)
+k
+s
+t
+v
+r
+u
+(f+1)
+(f+1)
+l
+krs
+st
+k u
+t
+uv
+k
+k
+k
+r
+rs
+st
+tu
+uv
+Figure 1. Two nearest-neighbor cells f and f +1, for a periodic substrate with linear
+size period l. The points represent sites, which can be occupied by mobile particles.
+The arrows represent hopping rates between pairs of sites. For clarity, only a few sites
+and hopping rates were highlighted. n(f)
+r
+corresponds to the number of particles in the
+internal site r of cell f
+As shown in [20], this steady-state consists of a set of microscopic currents
+distributed with the same periodicity as the substrate. In figure 1 two nearest-neighbor
+(NN) unit cells are depicted schematically where, for example, n(f)
+s
+represents the
+number of particles in site r (internal index) of cell f.
+Because of the mentioned
+
+Local Einstein relation for fractals
+4
+periodicity, we get that for given pair of connected sites with internal indices s and
+t,
+i(f)
+rs = i(f+1)
+rs
+,
+(3)
+where i(f)
+rs is the current from site s to site r in cell f. In addition, as hopping rates do
+not depend on the cell either but only on the internal indices, the last equation can be
+rewritten as
+ksr(n(f)
+s
+− n(f)
+r ) = ksr(n(f+1)
+s
+− n(f+1)
+r
+),
+(4)
+or
+n(f+1)
+s
+− n(f)
+s
+= n(f+1)
+r
+− n(f)
+r .
+(5)
+Therefore, in the steady-state, the difference in the occupation number for a given
+site and the equivalent site in a NN cell is the same for all sites.
+The relation of the steady-state problem with the diffusion coefficient D is provided
+by Fick’s law
+j = −D∆n
+l2 ,
+(6)
+which is valid for distances larger than l. Here ∆n corresponds to the particle number
+difference for NN cells. Note that D also determines the mean-square displacement ∆2x
+of a single random walker on the same structure, which behaves as
+∆2x(t) = 2Dt;
+(7)
+for time long enough for ∆x ≫ l.
+n
+n
+k
+a
+b
+i=(n −n )k
+b
+a
+Va
+R
+b
+Vb
+i= (V −V )/R
+a
+Figure 2. Schematics of the equivalence between Fick’s law (left) and Ohm’s law
+(right). In the mapping particles have unitary charge, while the other quantities are
+related as V = n, and R = 1/k.
+Transforming the steady-state problem into an equivalent electrical problem is
+straightforward. Indeed, for particles of unitary electric charge, a mapping between
+Fick’s law and Ohm’s law results by identifying particle number with electrostatic
+potential (Va = na) and hopping rate with conductance (k = 1/R). In figure 2 we
+represent this mapping for every pair of connected sites. Following this analogy, we see
+
+Local Einstein relation for fractals
+5
+that in the electric problem, the potential difference for a pair of equivalent sites in NN
+cells takes the constant value
+∆V = n(i+1)
+r
+− n(i)
+r ,
+(8)
+and that the difference between particle populations
+∆n =
+M
+�
+r=1
+(n(i+1)
+r
+− n(i)
+r ) = M∆V,
+(9)
+is proportional to the potential difference ∆V , where the constant of proportionality M
+corresponds to the number of sites per unit cell.
+Thus, according to equation (6), we can conclude that, given a periodic substrate
+with unit cell of linear dimension l and M sites, the diffusion coefficient and the potential
+difference between two equivalent sites in NN cells, are connected through the relation
+D = −j
+l2
+M∆V ,
+(10)
+where j is the steady-state current density.
+3. Self-similar substrates
+Deterministic fractals are usually built by a recursive procedure, that results in a
+sequence of structures called generations.
+A generation consists of a periodic array
+of sites connected by bonds. The process begins with a basic periodic structure (zeroth
+generation). At every step the unit cell is scaled by a factor L and the building rules
+ensure that self-similarity is obtained after a large number of iterations.
+Following equation (10), the diffusion coefficient Dp for the generation p and the
+potential difference ∆Vp between two equivalent points in NN unit cells are related as
+Dp = −j
+L2p
+Mp∆Vp
+,
+(11)
+where Mp is the number of sites in the unit cell, and Lp is its linear dimension. Then, for
+two consecutive generations p and p + 1, through which the same steady-state current
+flows, we obtain
+Dp
+Dp+1
+= L−2Mp+1
+Mp
+∆Vp+1
+∆Vp
+.
+(12)
+Now, since for a fractal the number of sites in a box with linear dimension l
+behaves as m(l) ∼ ldf (i. e., df is the fractal dimension defined through box-counting),
+Mp+1/Mp = (L(p+1)/Lp)df = Ldf, and the last equation can be rewritten as
+Dp
+Dp+1
+= Ldf−2∆Vp+1
+∆Vp
+,
+(13)
+
+Local Einstein relation for fractals
+6
+As previously shown [7,8], a perfect diffusive self-similar structure corresponds to
+a ratio Dp/Dp+1 which does not depend on p, i. e.,
+Dp
+Dp+1
+= 1 + λ,
+(14)
+with λ a positive constant. In this model, the mean-square displacement for a single
+random walker behaves as
+∆2x(t) = f(t)t2ν.
+(15)
+The modulation f(t) is a log-periodic function, f(tτ) = f(t), and both ν and τ can be
+analytically calculated in terms of L and λ:
+ν =
+1
+2 + log(1 + λ)
+log(L)
+(16)
+τ = L1/ν
+(17)
+The important partial conclusion in the context of this work is that, according to
+above discussion, a perfect diffusive self-similar structure implies a power-law behaviour
+for the resistance as a function of length. Indeed, equations (13) and (14) leads to
+∆Vp+1
+∆Vp
+= L1/ν−df,
+(18)
+where we have used 1 + λ = L1/ν−2, from equation (16). Thus, for a perfect diffusive
+self-similar fractal the potential difference, which corresponds to steady-state current,
+scales with length l as
+∆V ∼ lζ,
+(19)
+where the exponent ζ is given by
+ζ = 1/ν − df;
+(20)
+which is the Einstein relation (2), with dw = 1/ν.
+4. Local exponents
+We consider now a generic substrate for which diffusive self-similarity is reached only
+asymptotically. Let us assume a ratio between consecutive diffusion coefficients, that
+depends on the generation p, as
+Dp
+Dp+1
+= 1 + λp.
+(21)
+where, {λp : p = 1, 2, ...} is a sequence of non-negative real numbers, with lim
+p→∞ λp = λ.
+Because of this limit, at long enough times a single random walk on this substrate
+will show a MSD behaviour as in equation (15), and, as pointed out before, for large
+
+Local Einstein relation for fractals
+7
+enough lengths the potential difference will behave as in equation (19); with ν and ζ
+given by equations (16) and (20).
+In this section we focus on local exponents, which correspond to the slopes in log-
+log scales for finite length or time. As shown for example in [8], on a substrate on which
+diffusion coefficients for generations p and p + 1 satisfy equation (21), the MSD for a
+single random walker behaves as
+∆2x(t) ∼ t2νp,
+for
+Lp ≲ ∆x ≲ Lp+1,
+(22)
+with the local exponent νp given by
+νp =
+1
+2 + log(1 + λp)
+log(L)
+·
+(23)
+Then, after rearranging this equation as 1+λp = L1/νp−2, which corresponds to the
+left hand side of equation (13), we obtain
+∆Vp+1
+∆Vp
+= L1/νp−df.
+(24)
+Thus, we expect that the potential difference scales with length l as
+∆V (l) ∼ lζp,
+for
+Lp ≲ l ≲ Lp+1,
+(25)
+and that the local exponents satisfy the relation
+ζp = 1/νp − df.
+(26)
+Therefore, local slopes in log-log scales for the resistance as a function of length
+and for MSD of a single random walker as a function of time are related for all scales
+through equation (26); which generalizes the Einstein relation.
+5. Numerical simulations
+We study numerically the steady-state that corresponds to a unitary current on two
+models, for which diffusive self-similarity appears asymptotically.
+At finite lengths,
+the local random-walk exponent νp is not constant. Thus, we expect an also variable
+resistance exponent ζp, related to the former through equation (26).
+The first model is a substrate built on a square lattice. A random walk consists in
+a particle hopping among NN sites. If sites are connected by a bond, the hopping rate is
+k = 1/4. If the sites are not connected, the hopping rate is k = 0. A fractal is obtained
+by deleting some bonds. The characteristic scale factor is L = 3, and the unit cells for
+the first, the second and the third generations are depicted schematically in figure 3.
+For every generation the unit cell can be separated from the rest by cutting four bonds.
+As shown in a previous work, the mass on this structure shows a power-law behaviour
+with df = 2. However, the random walk exponent νp grows with time and approaches
+a value ν < 1/2 when t → ∞ [8].
+
+Local Einstein relation for fractals
+8
+We have run numerical simulations on the unit cell of the sixth generation, to reach
+the steady-state in which a unitary current flows between the left and right extremes. In
+figure 4 we plot with symbols the potential differences for lengths x = 3i (i = 0, 1, ..., 6),
+which are the unit cell linear sizes for the generations zero to six. In the same figure,
+we plot a line using the relation (26) and the numerical values for νp, which are the
+outcomes of random walk simulations reported in reference [8]. Notice that both data
+set fall on the same curve, which confirms the relation (26).
+Figure 3. Substrate in two dimensions, which results in scale-dependent walk and
+resistance exponents. The schematics correspond to the unit cells for the first, second
+and third generations. The segments represent bonds between sites.
+The second model is a generalization of the one-dimensional self-similar model
+introduced in [7]. We start with a single random walk on a one-dimensional lattice, with
+a hopping rate k0 between any pair of NN sites. This homogeneous case corresponds to
+generation zero. We introduce a natural number L to build the other generations.
+In the first generation, we reset to k1 < k0 the hopping rate for every pair of sites j
+and j +1, with mod(j, L) = 0. The other hopping rates remains as in zeroth generation.
+In the second generation, we reset to k2 < k1 the hopping rate for every pair of sites
+j and j +1, with mod(j, L2) = 0. The other hopping rates remains as in first generation.
+This recursion follows indefinitely, in such a way that generation n is obtained from
+generation n − 1 after resetting to kn < kn−1 the hopping rate for every pair of sites j
+and j + 1, with mod(j, Ln) = 0. In figure 5 we show an schematics for L = 5.
+
+Local Einstein relation for fractals
+9
+1
+10
+100
+1
+10
+100
+1000
+∆V
+x
+Figure 4. Potential difference as a function of length for a unitary current flowing
+trough the unit cell of the sixth generation substrate in figure 3.
+The symbols
+correspond to simulations of the steady-state. The line was plotted with the exponents
+ζp from equation (26) and the values of νp which result from random-walk numerical
+simulations.
+L
+L2
+k0
+k1
+k2
+Figure 5. Schematics of the one-dimensional random-walk model. We begin with
+a homogeneous lattice, and a hopping rate k0 between nearest-neighbor sites. Then,
+hopping rates are reset to kj for transitions between sites j and j + 1 for every j such
+that mod(j, Ln) = 0, and for n = 1, 2, .... In this example, L = 5.
+If we ask for perfect self-similarity for diffusion, i. e. equation (14), the hopping
+rates are found iteratively as in reference [7]. For the more general case of equation
+(21), the sequence of hopping rates is given by
+1
+ki
+=
+1
+ki−1
++ Liλi−1
+k0
+i−2
+�
+j=0
+(1 + λj),
+for i = 1, 2, 3...
+(27)
+We test the validity of the relation (26) among the local exponents for a family of
+
+Local Einstein relation for fractals
+10
+substrates given by
+λp = λ (1 − 2−p/5.).
+(28)
+At short enough lengths these substrates are nearly homogeneous (λp ≈ 0 for p ≪ 5),
+while, on the other extreme, self-similarity for diffusion is reached for lengths much larger
+than L5. The local random walk exponent (23) decreases with length and approaches
+asymptotically ν in equation (16). Thus, the variation of νp in space increases with λ
+and, because of equation (26), the same should occur with the variation of ζp. This is an
+interesting model, because the variation of the exponents with length can be adjusted
+through the parameter λ.
+100
+101
+102
+103
+104
+105
+106
+107
+108
+100
+101
+102
+103
+100
+101
+102
+103
+104
+105
+106
+100
+101
+102
+103
+∆V
+x
+∆V
+x
+Figure 6. Potential difference as a function of length for unitary current on the one-
+dimensional model with λp = λ (1−2−p/5.), and L = 2. (Main) Symbols correspond to
+data obtained with numerical simulations on a tenth-generation substrate. Lines were
+drawn using the values of theoretical exponents. From bottom to top, λ = 1 (red),
+λ = 2 (green), λ = 4 (violet), λ = 5 (blue). (Inset) More detailed structure for λ = 2.
+We have run numerical simulations for the steady-state that corresponds to a
+unitary current flowing on this model, with L = 2 and λ = 1, 2, 4, 5. All substrates
+were built until generation 10. In figure 6-main we plot with symbols the potential
+difference as a function of the length x, for x = 2j (j = 0, 1, ..., 9). The lines correspond
+to the exponents ζp obtained from equations (26) and (23). Note the excellent agreement
+between theory and simulations. The inset in the same figure shows substructure of ∆V
+for λ = 2.
+6. Conclusions
+We have studied first the connection between single random walks and steady-state
+potential difference for substrates with spatial periodicity.
+Then, by considering a
+sequence of periodic systems, a common procedure for deterministic fractal construction,
+we find that the length dependent fractal, walk and resistance exponents, for the
+
+Local Einstein relation for fractals
+11
+substrate obtained in the infinite limit of this sequence, satisfy, at every length scale,
+the relation (26). This can be considered as a local version of the Einstein relation (2).
+We have tested our predictions numerically for two models. The first model is a fractal
+in two dimensions, while the the second is a fractal in one dimension. Both models lead
+to length-dependent exponents at intermediate scales. The excellent agreement between
+the outcomes of these simulations and the theoretical predictions supports the validity
+of the mentioned relation among exponents, not only in the asymptotic self-similar limit
+but also locally, for all length scales.
+Acknowledgments
+We are grateful to H. O. M´artin for useful discussions. This research was supported
+by the Universidad Nacional de Mar del Plata, 15/E1040, and the Consejo Nacional de
+Investigaciones Cient´ıficas y T´ecnicas, PIP1748/21.
+References
+[1] Peter J. Grabner and Wolfgang Woess. Functional iterations and periodic oscillations for simple
+random walk on the sierpi?ski graph. Stochastic Processes and their Applications, 69(1):127 –
+138, 1997.
+[2] L. Acedo and S. B. Yuste. Territory covered by n random walkers on fractal media: The sierpinski
+gasket and the percolation aggregate. Phys. Rev. E, 63:011105, Dec 2000.
+[3] M. A. Bab, G. Fabricius, and E. V. Albano. On the occurrence of oscillatory modulations in the
+power law behavior of dynamic and kinetic processes in fractals. EPL (Europhysics Letters),
+81(1):10003, 2008.
+[4] M. A. Bab, G. Fabricius, and Ezequiel V. Albano. Revisiting random walks in fractal media: On
+the occurrence of time discrete scale invariance.
+The Journal of Chemical Physics, 128(4):–,
+2008.
+[5] Alberto L. Maltz, Gabriel Fabricius, Marisa A. Bab, and Ezequiel V. Albano.
+Random walks
+in fractal media:
+a theoretical evaluation of the periodicity of the oscillations in dynamic
+observables. Journal of Physics A: Mathematical and Theoretical, 41(49):495004, 2008.
+[6] Sebastian Weber, Joseph Klafter, and Alexander Blumen. Random walks on sierpinski gaskets of
+different dimensions. Phys. Rev. E, 82:051129, Nov 2010.
+[7] L. Padilla, H. O. M´artin, and J. L. Iguain. Log-periodic modulation in one-dimensional random
+walks. EPL (Europhysics Letters), 85(2):20008, January 2009.
+[8] L. Padilla, H. O. M´artin, and J. L. Iguain. Log-periodic oscillations for diffusion on self-similar
+finitely ramified structures. Phys. Rev. E, 82:011124, Jul 2010.
+[9] L. Padilla, H. M´artin, and J. Iguain.
+Anomalous diffusion with log-periodic modulation in a
+selected time interval. Physical Review E, 83(2):2–5, feb 2011.
+[10] Daniel ben Avraham and Shlomo Havlin.
+Diffusion and Reactions in Fractals and Disordered
+Systems. Cambridge University Press, 2000.
+[11] Bernhard Kr¨on and Elmar Teufl. Asymptotics of the transition probabilities of the simple random
+walk on self-similar graphs. Trans. Amer. Math. Soc., 356:393–414, 2004.
+[12] L. Padilla, H. O. M´artin, and J. L. Iguain. Anisotropic anomalous diffusion modulated by log-
+periodic oscillations. Physical Review E, 86(1):011106, jul 2012.
+[13] Frechero, M. A., Padilla, L., M´artin, H. O., and Iguain, J. L.
+Intermediate-range structure in
+ion-conducting tellurite glasses. EPL, 103(3):36002, 2013.
+[14] Amin Bunde and Shlomo Havlin (Eds.). Fractals and Disordered Systems. Springer, 1996.
+
+Local Einstein relation for fractals
+12
+[15] J A Given and B B Mandelbrot.
+Diffusion on fractal lattices and the fractal einstein relation.
+Journal of Physics A: Mathematical and General, 16(15):L565, oct 1983.
+[16] Astrid Franz, Christian Schulzky, and Karl Heinz Hoffmann.
+The Einstein relation for finitely
+ramified Sierpinski carpets. Nonlinearity, 14(5):1411, aug 2001.
+[17] Alberto L Maltz, Gabriel Fabricius, Marisa A Bab, and Ezequiel V Albano.
+Random walks
+in fractal media:
+a theoretical evaluation of the periodicity of the oscillations in dynamic
+observables. Journal of Physics A: Mathematical and Theoretical, 41(49):495004, oct 2008.
+[18] Andr´as Telcs. The Einstein Relation for Random Walks on Graphs. Journal of Statistical Physics,
+122(4):617–645, 2006.
+[19] Martin Hendrick and Philippe Renard.
+Fractal dimension, walk dimension and conductivity
+exponent of karst networks around tulum. Frontiers in Physics, 4, 2016.
+[20] C. M. Aldao, J. L. Iguain, and H. O. M´artin. Diffusion of tagged particle in an exclusion process.
+Surf. Sci., 366:483–490, Apr 1996.
+

8tAyT4oBgHgl3EQfc_f6/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf,len=309
+page_content='Local Einstein relation for fractals  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Padilla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Iguain  Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR) and  Departamento de F´ısica FCEyN, Universidad Nacional de Mar del Plata,  De´an Funes 3350, 7600 Mar del Plata, Argentina  E-mail: iguain@mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='ar  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  We study single random walks and the electrical resistance for fractals  obtained as the limit of a sequence of periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In the long-scale regime,  power laws describe both the mean-square displacement of a random walk as a function  of time and the electrical resistance as a function of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  We show that the  corresponding power-law exponents satisfy the Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' For shorter scales,  where these exponents depend on length, we find how the Einstein relation can be  generalized to hold locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' All these findings were analytically derived and confirmed  by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Keywords: Fractals, Power-law behaviours, Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='00296v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='stat-mech]  31 Dec 2022  Local Einstein relation for fractals  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Introduction  Fractals are characterized by quantities that exhibit power-law behaviour in space or  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' More precisely, as scale invariance occurs for integer powers of a characteristic  length, pure power laws are modulated by logarithmic periodic functions, that describe  the departures from the main trend at intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' These modulations have  been the object of recent interest and considerable effort has been devoted toward  understanding the relation between log-periodicity and discrete-scale invariance [1–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  For a given fractal and some related observables, which show (modulated) power-  law behaviours, a problem of interest is to determine whether or not the exponents  associated with these quantities are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Sometimes we can expect a relation  as a consequence of underlying physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  This is, for example, the case of the  mass m, the electric resistance R and the mean-square-displacement (MSD) ∆r2 for  a single random walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' On a fractal, the first two grow with length l as m(l) ∼ ldf  and R(l) ∼ lζ, while the last one grows with time t as ∆r2(t) ∼ t2/dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The exponents  df, ζ and dw are known as the fractal, resistance and walk exponents, respectively, and  these power-law behaviours hold for scales large enough to ensure self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In an  d-dimensional euclidean space, the diffusion coefficient D and conductivity σ are related  by the Einstein equation [14]  σ = e2ρ  kBT D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (1)  Here, D = limt→∞ ∆r2(t)/2t, ρ and e are the density and charge of mobile particles, T  is the temperature and kB is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Equation (1) is one of the forms  of the fluctuation-dissipation theorem, and can be used together with simple scaling  heuristic arguments, to argue that the fractal, walk, and resistance exponents satisfy  the Einstein relation [14]  df = dw − ζ,  (2)  This property has been shown to hold asymptotically for some finitely ramified  fractals [15, 16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' which has been used to analyze the periodicity of the oscillations in  dynamic observables, in the first attempts to understand log-periodic modulation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Einstein relation was also investigated for random walks on weighted graphs [18], and,  more recently, for karst networks structures [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  A deterministic fractal can be obtained as the limit of a sequence of periodic  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In this procedure, the period increases at every step as Ln (n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='),  where L is a basic characteristic length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Self-similarity is manifested in power-law  behaviours, which occur for long enough scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' However, this does not always hold  for shorter lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Thus, the local slopes of the observables as a function of time or  length, in log-log scales, are variable quantities, which approach constant values only  asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  In this work we argue that the local fractal, walk, and resistance exponents are  related through an equation that generalizes (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  This generalization is obtained  Local Einstein relation for fractals  3  analytically, following the steady-state method for the calculation of the effective  diffusion coefficients for periodic substrates [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' To further strengthen our findings we  perform numerical simulations for two models of fractals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' which confirm the theoretical  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' 2 we relate the diffusion coefficient and the  unit cell resistance for a periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' 3 we derive the Einstein relation for  self-similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' 4 we generalize this relation for scale-dependent exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' 5 we confirm the generalized relation by numerical simulations performed on  models of asymptotic self-similar substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Finally, we give our conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Periodic systems  In this section we address the problem of the diffusion coefficient for a periodic substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  We follows the steady-state method developed in reference [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' We start by introducing  the periodic substrate with unit cell of linear dimension l, schematized in figure 1, where  the points represent sites, and the arrows represent hopping rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' On this structure,  a mobile particle can jump between connected sites according to the hopping rates k′s  (for the sake of clarity only a few sites and arrows were highlighted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' We focus on a  steady-state of non-interacting particles flowing with a constant current density j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  k  n  n  (f+1)  (f+1)  n  (f+1)  n  ns  t  v  u  k  n  n  (f)  (f)  (f)  n  (f)  n  n(f)  k  s  t  v  r  u  (f+1)  (f+1)  l  krs  st  k u  t  uv  k  k  k  r  rs  st  tu  uv  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Two nearest-neighbor cells f and f +1, for a periodic substrate with linear  size period l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The points represent sites, which can be occupied by mobile particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The arrows represent hopping rates between pairs of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' For clarity, only a few sites  and hopping rates were highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' n(f)  r  corresponds to the number of particles in the  internal site r of cell f  As shown in [20], this steady-state consists of a set of microscopic currents  distributed with the same periodicity as the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In figure 1 two nearest-neighbor  (NN) unit cells are depicted schematically where, for example, n(f)  s  represents the  number of particles in site r (internal index) of cell f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Because of the mentioned  Local Einstein relation for fractals  4  periodicity, we get that for given pair of connected sites with internal indices s and  t,  i(f)  rs = i(f+1)  rs  ,  (3)  where i(f)  rs is the current from site s to site r in cell f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In addition, as hopping rates do  not depend on the cell either but only on the internal indices, the last equation can be  rewritten as  ksr(n(f)  s  − n(f)  r ) = ksr(n(f+1)  s  − n(f+1)  r  ),  (4)  or  n(f+1)  s  − n(f)  s  = n(f+1)  r  − n(f)  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (5)  Therefore, in the steady-state, the difference in the occupation number for a given  site and the equivalent site in a NN cell is the same for all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The relation of the steady-state problem with the diffusion coefficient D is provided  by Fick’s law  j = −D∆n  l2 ,  (6)  which is valid for distances larger than l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Here ∆n corresponds to the particle number  difference for NN cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Note that D also determines the mean-square displacement ∆2x  of a single random walker on the same structure, which behaves as  ∆2x(t) = 2Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (7)  for time long enough for ∆x ≫ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  n  n  k  a  b  i=(n −n )k  b  a  Va  R  b  Vb  i= (V −V )/R  a  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Schematics of the equivalence between Fick’s law (left) and Ohm’s law  (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In the mapping particles have unitary charge, while the other quantities are  related as V = n, and R = 1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Transforming the steady-state problem into an equivalent electrical problem is  straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Indeed, for particles of unitary electric charge, a mapping between  Fick’s law and Ohm’s law results by identifying particle number with electrostatic  potential (Va = na) and hopping rate with conductance (k = 1/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In figure 2 we  represent this mapping for every pair of connected sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Following this analogy, we see  Local Einstein relation for fractals  5  that in the electric problem, the potential difference for a pair of equivalent sites in NN  cells takes the constant value  ∆V = n(i+1)  r  − n(i)  r ,  (8)  and that the difference between particle populations  ∆n =  M  �  r=1  (n(i+1)  r  − n(i)  r ) = M∆V,  (9)  is proportional to the potential difference ∆V , where the constant of proportionality M  corresponds to the number of sites per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Thus, according to equation (6), we can conclude that, given a periodic substrate  with unit cell of linear dimension l and M sites, the diffusion coefficient and the potential  difference between two equivalent sites in NN cells, are connected through the relation  D = −j  l2  M∆V ,  (10)  where j is the steady-state current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Self-similar substrates  Deterministic fractals are usually built by a recursive procedure, that results in a  sequence of structures called generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  A generation consists of a periodic array  of sites connected by bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The process begins with a basic periodic structure (zeroth  generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' At every step the unit cell is scaled by a factor L and the building rules  ensure that self-similarity is obtained after a large number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Following equation (10), the diffusion coefficient Dp for the generation p and the  potential difference ∆Vp between two equivalent points in NN unit cells are related as  Dp = −j  L2p  Mp∆Vp  ,  (11)  where Mp is the number of sites in the unit cell, and Lp is its linear dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Then, for  two consecutive generations p and p + 1, through which the same steady-state current  flows, we obtain  Dp  Dp+1  = L−2Mp+1  Mp  ∆Vp+1  ∆Vp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (12)  Now, since for a fractal the number of sites in a box with linear dimension l  behaves as m(l) ∼ ldf (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', df is the fractal dimension defined through box-counting),  Mp+1/Mp = (L(p+1)/Lp)df = Ldf, and the last equation can be rewritten as  Dp  Dp+1  = Ldf−2∆Vp+1  ∆Vp  ,  (13)  Local Einstein relation for fractals  6  As previously shown [7,8], a perfect diffusive self-similar structure corresponds to  a ratio Dp/Dp+1 which does not depend on p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=',  Dp  Dp+1  = 1 + λ,  (14)  with λ a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In this model, the mean-square displacement for a single  random walker behaves as  ∆2x(t) = f(t)t2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (15)  The modulation f(t) is a log-periodic function, f(tτ) = f(t), and both ν and τ can be  analytically calculated in terms of L and λ:  ν =  1  2 + log(1 + λ)  log(L)  (16)  τ = L1/ν  (17)  The important partial conclusion in the context of this work is that, according to  above discussion, a perfect diffusive self-similar structure implies a power-law behaviour  for the resistance as a function of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Indeed, equations (13) and (14) leads to  ∆Vp+1  ∆Vp  = L1/ν−df,  (18)  where we have used 1 + λ = L1/ν−2, from equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Thus, for a perfect diffusive  self-similar fractal the potential difference, which corresponds to steady-state current,  scales with length l as  ∆V ∼ lζ,  (19)  where the exponent ζ is given by  ζ = 1/ν − df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (20)  which is the Einstein relation (2), with dw = 1/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Local exponents  We consider now a generic substrate for which diffusive self-similarity is reached only  asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Let us assume a ratio between consecutive diffusion coefficients, that  depends on the generation p, as  Dp  Dp+1  = 1 + λp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (21)  where, {λp : p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='} is a sequence of non-negative real numbers, with lim  p→∞ λp = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Because of this limit, at long enough times a single random walk on this substrate  will show a MSD behaviour as in equation (15), and, as pointed out before, for large  Local Einstein relation for fractals  7  enough lengths the potential difference will behave as in equation (19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' with ν and ζ  given by equations (16) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  In this section we focus on local exponents, which correspond to the slopes in log-  log scales for finite length or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' As shown for example in [8], on a substrate on which  diffusion coefficients for generations p and p + 1 satisfy equation (21), the MSD for a  single random walker behaves as  ∆2x(t) ∼ t2νp,  for  Lp ≲ ∆x ≲ Lp+1,  (22)  with the local exponent νp given by  νp =  1  2 + log(1 + λp)  log(L) (23)  Then, after rearranging this equation as 1+λp = L1/νp−2, which corresponds to the  left hand side of equation (13), we obtain  ∆Vp+1  ∆Vp  = L1/νp−df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (24)  Thus, we expect that the potential difference scales with length l as  ∆V (l) ∼ lζp,  for  Lp ≲ l ≲ Lp+1,  (25)  and that the local exponents satisfy the relation  ζp = 1/νp − df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (26)  Therefore, local slopes in log-log scales for the resistance as a function of length  and for MSD of a single random walker as a function of time are related for all scales  through equation (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' which generalizes the Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Numerical simulations  We study numerically the steady-state that corresponds to a unitary current on two  models, for which diffusive self-similarity appears asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  At finite lengths,  the local random-walk exponent νp is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Thus, we expect an also variable  resistance exponent ζp, related to the former through equation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The first model is a substrate built on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' A random walk consists in  a particle hopping among NN sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' If sites are connected by a bond, the hopping rate is  k = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' If the sites are not connected, the hopping rate is k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' A fractal is obtained  by deleting some bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The characteristic scale factor is L = 3, and the unit cells for  the first, the second and the third generations are depicted schematically in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  For every generation the unit cell can be separated from the rest by cutting four bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  As shown in a previous work, the mass on this structure shows a power-law behaviour  with df = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' However, the random walk exponent νp grows with time and approaches  a value ν < 1/2 when t → ∞ [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Local Einstein relation for fractals  8  We have run numerical simulations on the unit cell of the sixth generation, to reach  the steady-state in which a unitary current flows between the left and right extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In  figure 4 we plot with symbols the potential differences for lengths x = 3i (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', 6),  which are the unit cell linear sizes for the generations zero to six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In the same figure,  we plot a line using the relation (26) and the numerical values for νp, which are the  outcomes of random walk simulations reported in reference [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Notice that both data  set fall on the same curve, which confirms the relation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Substrate in two dimensions, which results in scale-dependent walk and  resistance exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The schematics correspond to the unit cells for the first, second  and third generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The segments represent bonds between sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The second model is a generalization of the one-dimensional self-similar model  introduced in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' We start with a single random walk on a one-dimensional lattice, with  a hopping rate k0 between any pair of NN sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' This homogeneous case corresponds to  generation zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' We introduce a natural number L to build the other generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  In the first generation, we reset to k1 < k0 the hopping rate for every pair of sites j  and j +1, with mod(j, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The other hopping rates remains as in zeroth generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  In the second generation, we reset to k2 < k1 the hopping rate for every pair of sites  j and j +1, with mod(j, L2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The other hopping rates remains as in first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  This recursion follows indefinitely, in such a way that generation n is obtained from  generation n − 1 after resetting to kn < kn−1 the hopping rate for every pair of sites j  and j + 1, with mod(j, Ln) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In figure 5 we show an schematics for L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Local Einstein relation for fractals  9  1  10  100  1  10  100  1000  ∆V  x  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Potential difference as a function of length for a unitary current flowing  trough the unit cell of the sixth generation substrate in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The symbols  correspond to simulations of the steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The line was plotted with the exponents  ζp from equation (26) and the values of νp which result from random-walk numerical  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  L  L2  k0  k1  k2  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Schematics of the one-dimensional random-walk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' We begin with  a homogeneous lattice, and a hopping rate k0 between nearest-neighbor sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Then,  hopping rates are reset to kj for transitions between sites j and j + 1 for every j such  that mod(j, Ln) = 0, and for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='. In this example, L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  If we ask for perfect self-similarity for diffusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' equation (14), the hopping  rates are found iteratively as in reference [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' For the more general case of equation  (21), the sequence of hopping rates is given by  1  ki  =  1  ki−1  + Liλi−1  k0  i−2  �  j=0  (1 + λj),  for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (27)  We test the validity of the relation (26) among the local exponents for a family of  Local Einstein relation for fractals  10  substrates given by  λp = λ (1 − 2−p/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  (28)  At short enough lengths these substrates are nearly homogeneous (λp ≈ 0 for p ≪ 5),  while, on the other extreme, self-similarity for diffusion is reached for lengths much larger  than L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The local random walk exponent (23) decreases with length and approaches  asymptotically ν in equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Thus, the variation of νp in space increases with λ  and, because of equation (26), the same should occur with the variation of ζp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' This is an  interesting model, because the variation of the exponents with length can be adjusted  through the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  100  101  102  103  104  105  106  107  108  100  101  102  103  100  101  102  103  104  105  106  100  101  102  103  ∆V  x  ∆V  x  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Potential difference as a function of length for unitary current on the one-  dimensional model with λp = λ (1−2−p/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' ), and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' (Main) Symbols correspond to  data obtained with numerical simulations on a tenth-generation substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Lines were  drawn using the values of theoretical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' From bottom to top, λ = 1 (red),  λ = 2 (green), λ = 4 (violet), λ = 5 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' (Inset) More detailed structure for λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  We have run numerical simulations for the steady-state that corresponds to a  unitary current flowing on this model, with L = 2 and λ = 1, 2, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' All substrates  were built until generation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' In figure 6-main we plot with symbols the potential  difference as a function of the length x, for x = 2j (j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The lines correspond  to the exponents ζp obtained from equations (26) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Note the excellent agreement  between theory and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The inset in the same figure shows substructure of ∆V  for λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Conclusions  We have studied first the connection between single random walks and steady-state  potential difference for substrates with spatial periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Then, by considering a  sequence of periodic systems, a common procedure for deterministic fractal construction,  we find that the length dependent fractal, walk and resistance exponents, for the  Local Einstein relation for fractals  11  substrate obtained in the infinite limit of this sequence, satisfy, at every length scale,  the relation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' This can be considered as a local version of the Einstein relation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  We have tested our predictions numerically for two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The first model is a fractal  in two dimensions, while the the second is a fractal in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Both models lead  to length-dependent exponents at intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The excellent agreement between  the outcomes of these simulations and the theoretical predictions supports the validity  of the mentioned relation among exponents, not only in the asymptotic self-similar limit  but also locally, for all length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Acknowledgments  We are grateful to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M´artin for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' This research was supported  by the Universidad Nacional de Mar del Plata, 15/E1040, and the Consejo Nacional de  Investigaciones Cient´ıficas y T´ecnicas, PIP1748/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  References  [1] Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Grabner and Wolfgang Woess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Functional iterations and periodic oscillations for simple  random walk on the sierpi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='ski graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Stochastic Processes and their Applications, 69(1):127 –  138, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Acedo and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Yuste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Territory covered by n random walkers on fractal media: The sierpinski  gasket and the percolation aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' E, 63:011105, Dec 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Bab, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Fabricius, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Albano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' On the occurrence of oscillatory modulations in the  power law behavior of dynamic and kinetic processes in fractals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' EPL (Europhysics Letters),  81(1):10003, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Bab, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Fabricius, and Ezequiel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Albano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Revisiting random walks in fractal media: On  the occurrence of time discrete scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The Journal of Chemical Physics, 128(4):–,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [5] Alberto L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Maltz, Gabriel Fabricius, Marisa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Bab, and Ezequiel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Albano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Random walks  in fractal media:  a theoretical evaluation of the periodicity of the oscillations in dynamic  observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Journal of Physics A: Mathematical and Theoretical, 41(49):495004, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [6] Sebastian Weber, Joseph Klafter, and Alexander Blumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Random walks on sierpinski gaskets of  different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' E, 82:051129, Nov 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Padilla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M´artin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Iguain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Log-periodic modulation in one-dimensional random  walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' EPL (Europhysics Letters), 85(2):20008, January 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Padilla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M´artin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Iguain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Log-periodic oscillations for diffusion on self-similar  finitely ramified structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' E, 82:011124, Jul 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Padilla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M´artin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Iguain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Anomalous diffusion with log-periodic modulation in a  selected time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Physical Review E, 83(2):2–5, feb 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [10] Daniel ben Avraham and Shlomo Havlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Diffusion and Reactions in Fractals and Disordered  Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Cambridge University Press, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [11] Bernhard Kr¨on and Elmar Teufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Asymptotics of the transition probabilities of the simple random  walk on self-similar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', 356:393–414, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Padilla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M´artin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Iguain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Anisotropic anomalous diffusion modulated by log-  periodic oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Physical Review E, 86(1):011106, jul 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [13] Frechero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', Padilla, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', M´artin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', and Iguain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Intermediate-range structure in  ion-conducting tellurite glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' EPL, 103(3):36002, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [14] Amin Bunde and Shlomo Havlin (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Fractals and Disordered Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Springer, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Local Einstein relation for fractals  12  [15] J A Given and B B Mandelbrot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Diffusion on fractal lattices and the fractal einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Journal of Physics A: Mathematical and General, 16(15):L565, oct 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [16] Astrid Franz, Christian Schulzky, and Karl Heinz Hoffmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  The Einstein relation for finitely  ramified Sierpinski carpets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Nonlinearity, 14(5):1411, aug 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [17] Alberto L Maltz, Gabriel Fabricius, Marisa A Bab, and Ezequiel V Albano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Random walks  in fractal media:  a theoretical evaluation of the periodicity of the oscillations in dynamic  observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Journal of Physics A: Mathematical and Theoretical, 41(49):495004, oct 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [18] Andr´as Telcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' The Einstein Relation for Random Walks on Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Journal of Statistical Physics,  122(4):617–645, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [19] Martin Hendrick and Philippe Renard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Fractal dimension, walk dimension and conductivity  exponent of karst networks around tulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Frontiers in Physics, 4, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Aldao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Iguain, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' M´artin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Diffusion of tagged particle in an exclusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content='  Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}
+page_content=', 366:483–490, Apr 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'}

9NAzT4oBgHgl3EQfFPpp/content/tmp_files/2301.01007v1.pdf.txt

@@ -0,0 +1,2688 @@
+A Bertrand duopoly game with differentiated products reconsidered
+Xiaoliang Lia and Bo Li∗b
+aSchool of Digital Economics, Dongguan City University, Dongguan 523419, China
+bSchool of Finance, Anhui University of Finance and Economics, Bengbu 233030, China
+Abstract
+In this paper, we explore a dynamic Bertrand duopoly game with differentiated products, where
+firms are boundedly rational and consumers are assumed to possess an underlying CES utility function.
+We mainly focus on two distinct degrees of product substitutability. Several tools based on symbolic
+computations such as the triangular decomposition method and the PCAD method are employed in the
+analytical investigation of the model. The uniqueness of the non-vanishing equilibrium is proved and
+rigorous conditions for the local stability of this equilibrium are established for the first time.
+Most
+importantly, we find that increasing the substitutability degree or decreasing the product differentiation
+has an effect of destabilization for our Bertrand model, which is in contrast with the relative conclusions
+for the Cournot models. This finding could be conducive to the revelation of the essential difference
+between dynamic Cournot and Bertrand oligopolies with differentiated goods.
+In the special case of
+identical marginal costs, we derive that lower degrees of product differentiation mean lower prices, higher
+supplies, lower profits, and lower social welfare. Furthermore, complex dynamics such as periodic orbits
+and chaos are reported through our numerical simulations.
+Keywords: Bertrand duopoly; differentiated product; symbolic computation; local stability
+1
+Introduction
+It is well known that Cournot [12] developed the first formal theory of oligopoly, which is a market supplied
+by only a few firms. In Cournot’s framework, firms are supposed to make decisions on their quantities of
+outputs and have perfect information on their rivals’ strategic behavior. In the strand of Cournot oligopoly
+models, the market demand function is usually supposed to be linear for simplicity by many economists
+(e.g., Fisher [16], McManus and Quandt[29]). In the real world, however, a non-linear demand is more
+likely to exist. Puu [33] investigated a Cournot duopoly game under an isoelastic market demand, where
+the price is simply the reciprocal of the total supply. Afterward, fruitful contributions including [2, 4, 7,
+9, 10, 13, 20, 21, 22, 24, 28, 31], were made in the literature on Cournot games. Related to our study,
+Zhang and Zhang [39] considered a Cournot game in which each firm produces multiple products and sells
+them in multiple markets. They obtained sufficient and necessary conditions for the local stability of the
+Cournot-Nash equilibria.
+Several decades later after Cournot’s seminal work, Bertrand [6] proposed a different framework to
+describe oligopolistic competition, where prices rather than quantities are the strategic variables of the
+competitors. Singh and Vives [35] analyzed the duality of prices and quantities, and found that Cournot
+(Bertrand) competition with substitutes is the dual of Bertrand (Cournot) competition with complements.
+L´opez and Naylor [26] compared Cournot and Bertrand equilibria in a downstream differentiated duopoly,
+and proved that the classic conclusion that profits under Cournot equilibrium exceed those under Bertrand
+competition could be reversible in the case of imperfect substitutes. Zhang et al. [40] considered a Bertrand
+model formulated under a linear inverse demand, and obtained the existence and stability of the equilibrium.
+Different from [40], Fanti et al. [15] developed a model with sound microeconomic foundations that deter-
+mine the demand for differentiated products, and showed that synchronized dynamics and intermittency
+phenomena may appear. Naimzada and Tramontana [32] also considered a Cournot-Bertrand duopoly model
+with product differentiation and emphasized the role of best response dynamics and an adaptive adjustment
+mechanism for stability. Brianzoni et al. [8] assumed quadratic costs in the study of the Bertrand duopoly
+∗Corresponding author: [email protected]
+1
+arXiv:2301.01007v1  [econ.TH]  3 Jan 2023
+
+game with horizontal product differentiation and discovered synchronized dynamics. Moreover, Ma and Guo
+[27] studied the impacts of information on the dynamical Bertrand game. They showed that there exists
+a fixed point independent of the amount of information for a triopoly, and the stable region of adjustment
+parameter increases with the amount of information for a duopoly.
+In all the aforementioned Bertrand games, the inverse demand function is supposed to be linear. Instead,
+Gori and Sodini [17] explored the local and global dynamics of a Bertrand duopoly with a nonlinear demand
+and horizontal product differentiation. Furthermore, Ahmed et al. [3] proposed a dynamic Bertrand duopoly
+game with differentiated products, where firms are boundedly rational and consumers are assumed to possess
+an underlying CES utility function. They only employed numerical simulations to investigate the dynamic
+behavior of their model because the closed form of the equilibrium is extremely difficult to compute. They
+observed that the Nash equilibrium loses its stability through a period-doubling bifurcation as the speed
+of adjustment increases.
+Motivated by [3], Agliari et al. [1] investigated a Cournot duopoly game with
+differentiated goods. We should mention that Agliari et al. [1] used the same CES utility function as [3] to
+derive the demand function of the market. They discovered that a low degree of product substitutability or
+a higher degree of product differentiation may destabilize the Cournot game. This finding is in accordance
+with that of Fanti and Gori [14], where the authors introduced a Cournot duopoly with a linear demand and
+heterogeneous players to study the influence of product differentiation on stability and found that a higher
+degree of product differentiation may destabilize the market equilibrium.
+In this paper, we re-study the Bertrand duopoly game of Ahmed et al. [3] using several tools based on
+symbolic computations such as the triangular decomposition method (see, e.g., [23]) and the PCAD method
+(see, e.g., [11]). It is worth noting that the results of symbolic computations are exact, and thus can provide
+theoretical foundations for the systematic analysis of economic models.
+We analytically investigate the
+local stability and bifurcations of the model. By using several tools based on symbolic computations, the
+uniqueness of the non-vanishing equilibrium is proved and the rigorous conditions for the local stability of this
+equilibrium are obtained for the first time. In the special case that the two companies have identical marginal
+costs, we prove that the model can lose its stability only through a period-doubling bifurcation. The most
+important finding is that increasing the substitutability degree or decreasing the product differentiation has
+an effect of destabilizing the unique non-vanishing equilibrium. A possible explanation is that a decrease in
+product differentiation may result in an increase in market competition intensity and even a price war, which
+could lead to the destabilization of the equilibrium. It should be noted that our finding is in contrast with
+the relative conclusions by Agliari et al. [1] and by Fanti and Gori [14]. This contradiction contributes to
+the literature on the connection between Cournot and Bertrand oligopolies and may help reveal the essential
+difference between them. In the special case of identical marginal costs, we derive the fact that lower degrees
+of product differentiation can lead to lower prices, higher supplies, lower profits, and lower social welfare.
+This fact is in line with our economic intuition. Complex dynamics such as periodic orbits and chaos can
+be observed through our numerical simulations, which also confirm that an increase in the substitutability
+degree leads to the emergence of instability in the considered model. Furthermore, we discover the existence
+of a Neimark-Sacker bifurcation directly on the equilibrium, which is a new finding and has not yet been
+discovered by Ahmed et al. [3]
+The rest of this paper is structured as follows. In Section 2, we revisit the construction of the Bertrand
+duopoly game investigated in our study. We analytically explore the stability and bifurcations of this model
+for two different substitutability degrees, namely α = 1/2 and α = 1/3, in Sections 3 and 4, respectively.
+The influence of the substitutability degree on the local stability of the equilibrium and related comparative
+statics are discussed in Section 5. Numerical simulations are provided in Section 6. Concluding remarks are
+given in Section 7.
+2
+Model
+In our study, we consider a market where two firms compete with each other and produce differentiated
+goods. The prices and quantities of the two goods are denoted by pi and qi, respectively, with i = 1, 2.
+Furthermore, it is assumed that the market possesses a continuum of identical consumers with a CES utility
+function of the form
+U(q1, q2) = qα
+1 + qα
+2 ,
+2
+
+where α (0 < α < 1) is called the substitutability degree between the products. Consumers choose their
+consumptions by maximizing the utility subject to the budget constraint
+p1q1 + p2q2 = 1.
+Consequently, we have the following demand functions (The reader can refer to [3] for the proof).
+q1 = pβ
+2
+p1
+1
+pβ
+1 + pβ
+2
+,
+q2 = pβ
+1
+p2
+1
+pβ
+1 + pβ
+2
+,
+where β = α/(1 − α). Thus, the inverse demands of the two goods are
+p1 =
+qα−1
+1
+qα
+1 + qα
+2
+,
+p2 =
+qα−1
+2
+qα
+1 + qα
+2
+.
+(1)
+Accordingly, a decrease in α would make the products less substitutable or more differentiated.
+In
+particular, if α = 0, the inverse demands become p1 =
+1
+2 q1 and p2 =
+1
+2 q2 , which means that the two goods
+are completely independent. If α = 1, we obtain the inverse demand p1 = p2 =
+1
+q1+q2 , which is the same as
+the famous isoelastic demand function introduced by Puu [33]. In this case, the prices of the two goods are
+equal. That is to say, the two commodities are regarded as indistinguishable or identical by consumers.
+The cost functions are assumed to be linear, i.e.,
+C1(q1) = c1q1,
+C2(q2) = c2q2,
+where c1 > 0 and c2 > 0. Then the profit of firm i (i = 1, 2) should be
+Πi(pi, p−i) = piqi − ciqi = (pi − ci)pβ
+−i
+pi
+1
+pβ
+i + pβ
+−i
+,
+(2)
+where p−i denotes the price of the commodity produced by the rival.
+Furthermore, the gradient adjustment mechanism is formulated as
+pi(t + 1) = pi(t) + ki
+∂Πi(t)
+∂pi(t) ,
+where ki > 0 controls the adjustment speed of firm i. It is known that
+∂Πi
+∂pi
+=
+−pβ
+−ip1+β
+i
+β +
+�
+p2β
+−i + pβ
+−ipβ
+i (1 + β)
+�
+ci
+p2
+i
+�
+pβ
+i + pβ
+−i
+�2
+.
+In short, the model can be described as the following iteration map.
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+p1(t + 1) = p1(t) + k1
+−pβ
+2(t)p1+β
+1
+(t)β +
+�
+p2β
+2 (t) + pβ
+2(t)pβ
+1(t) (1 + β)
+�
+c1
+p2
+1(t)
+�
+pβ
+1(t) + pβ
+2(t)
+�2
+,
+p2(t + 1) = p2(t) + k2
+−pβ
+1(t)p1+β
+2
+(t)β +
+�
+p2β
+1 (t) + pβ
+1(t)pβ
+2(t) (1 + β)
+�
+c2
+p2
+2(t)
+�
+pβ
+2(t) + pβ
+1(t)
+�2
+.
+(3)
+This game was first explored by Ahmed et al. [3] only through numerical simulations because no analytical
+expressions of the Nash equilibria are available. In this paper, we reconsider this game using methods based
+on symbolic computations and explore the influence of the substitutability degree on the local stability of
+the equilibrium. One can see that for general β, it is impossible to analyze the equilibrium point of map
+(3), because the system will have an exponential parameter. For such systems with exponential parameters,
+existing analytical tools are quite limited. Therefore, similar to [3], our study mainly focuses on two specific
+cases, namely β = 1 and β = 1/2, which are corresponding to α = 1/2 and α = 1/3, respectively.
+3
+
+3
+α = 1/2
+If α = 1/2, then β = 1. Hence, map (3) becomes
+�
+�
+�
+�
+�
+�
+�
+�
+�
+p1(t + 1) = p1(t) + k1
+−2 p2(t)p2
+1(t) +
+�
+p2
+2(t) + 2 p2(t)p1(t)
+�
+c1
+p2
+1(t) (p1(t) + p2(t))2
+,
+p2(t + 1) = p2(t) + k2
+−2 p1(t)p2
+2(t) +
+�
+p2
+1(t) + 2 p1(t)p2(t)
+�
+c2
+p2
+2(t) (p2(t) + p1(t))2
+.
+(4)
+From an economic point of view, it is important to identify the number of non-vanishing equilibria
+(p1, p2) with p1 > 0 and p2 > 0. In order to compute the equilibrium, we set p1(t + 1) = p1(t) = p1 and
+p2(t + 1) = p2(t) = p2. Then the following equations of the equilibrium are acquired.
+�
+−2p2p2
+1 +
+�
+p2
+2 + 2p2p1
+�
+c1 = 0,
+−2p1p2
+2 +
+�
+p2
+1 + 2p1p2
+�
+c2 = 0.
+(5)
+The triangular decomposition method, which can be viewed as an extension of the Gaussian elimination
+method, permits us to analyze the equilibria of non-linear economic models. Both the method of triangu-
+lar decomposition and the method of Gaussian elimination can transform a system into triangular forms.
+However, the triangular decomposition method is feasible for polynomial systems, while the Gaussian elim-
+ination method is just for linear systems. Refer to [5, 19, 23, 36, 37] for more information on triangular
+decomposition. Specifically, using the triangular decomposition method, we can decompose the solutions of
+system (5) into zeros of the following two triangular polynomial sets.
+T11 = [p1, p2] ,
+T12 =
+�
+p3
+1 − 4 c1p2
+1 + (4 c2
+1 − 2 c1c2)p1 + 3 c2
+1c2, c1p2 − p2
+1 + 2 c1p1
+�
+.
+(6)
+The zero of T11 is corresponding to the origin (0, 0). Moreover, the non-vanishing equilibria can be
+computed from T12. The first polynomial p3
+1 − 4 c1p2
+1 + (4 c2
+1 − 2 c1c2)p1 + 3 c2
+1c2 of T12 is univariate in p1 and
+the second polynomial c1p2 − p2
+1 + 2 c1p1 of T12 has degree 1 with respect to p2. Consequently, if we solve p1
+from the first polynomial, then we can substitute the solution of p1 into the second polynomial and easily
+obtain p2. As the first polynomial of T12 has degree 3 with respect to p1, we know that there are at most 3
+positive real solutions. Their analytical expressions exist but are quite complicated, though.
+This is not an easy task to identify the exact number of positive real solutions if the analytical solutions
+of T12 are complicated. However, the first author of this paper and his co-worker [25] proposed an algebraic
+algorithm to systematically identify multiplicities of equilibria in semi-algebraic economies without obtaining
+the closed-form solutions. We summarize the computational results for map (4) in Proposition 1. Interested
+readers can refer to Section 3 of [25] for additional details of the algorithm.
+Proposition 1. Let α = 1/2. The iteration map (4) possesses one unique equilibrium (p1, p2) with p1 > 0
+and p2 > 0.
+To explore the local stability of the equilibrium, the following Jacobian matrix plays an ambitious role.
+J =
+�J11
+J12
+J21
+J22
+�
+,
+where
+J11 = p6
+1 + 3 p5
+1p2 + 3 p4
+1p2
+2 +
+�
+p3
+2 + 2 k1p2
+�
+p3
+1 − 6 k1p2p2
+1c1 − 6 k1p2
+2p1c1 − 2 c1k1p3
+2
+p3
+1 (p1 + p2)3
+,
+J12 = k1 (2 c1 − p1 + p2)
+(p1 + p2)3
+,
+J21 = k2 (2 c2 + p1 − p2)
+(p1 + p2)3
+,
+J22 = p6
+2 + 3 p1p5
+2 + 3 p2
+1p4
+2 +
+�
+p3
+1 + 2 k2p1
+�
+p3
+2 − 6 k2p1p2
+2c2 − 6 k2p2
+1p2c2 − 2 c2k2p3
+1
+p3
+2 (p1 + p2)3
+.
+4
+
+Then the characteristic polynomial of J is
+CP(λ) = λ2 − Tr(J)λ + Det(J),
+where Tr(J) = J11 + J22 and Det(J) = J11J22 − J12J21 are the trace and the determinant of J, respectively.
+According to the Jury criterion [18], the conditions for the local stability include:
+1. CDJ
+1 ≡ CP(1) = 1 − Tr(J) + Det(J) > 0,
+2. CDJ
+2 ≡ CP(−1) = 1 + Tr(J) + Det(J) > 0,
+3. CDJ
+3 ≡ 1 − Det(J) > 0.
+Remark 1. Furthermore, it is known that the discrete dynamic system may undergo a fold, period-doubling,
+or Neimark-Sacker bifurcation when the equilibrium loses its stability at CDJ
+1 = 0, CDJ
+2 = 0, or CDJ
+3 = 0,
+respectively.
+3.1
+The special case of c1 = c2
+If we set c1 = c2 = c in (5), then the triangular decomposition method permits us to transform the
+equilibrium equations (5) into the following three triangular sets.
+T21 = [p1, p2],
+T22 = [p1 − 3 c, p2 − 3 c],
+T23 = [p2
+1 − cp1 − c2, p2 + p1 − c].
+The zero of T21 is simply (0, 0). From T23, we obtain two zeros1
+��√
+5
+2 + 1
+2
+�
+c,
+�
+−
+√
+5
+2 + 1
+2
+�
+c
+�
+,
+��
+−
+√
+5
+2 + 1
+2
+�
+c,
+�√
+5
+2 + 1
+2
+�
+c
+�
+,
+which are useless as the component
+�
+−
+√
+5
+2 + 1
+2
+�
+c is negative. Therefore, the only non-vanishing equilibrium
+is (3 c, 3 c), which can be obtained from T22.
+Theorem 1. Let α = 1/2 and c1 = c2 = c. The unique non-vanishing equilibrium (3 c, 3 c) is locally stable
+if
+c2 > 2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+.
+The system may undergo a period-doubling bifurcation when
+c2 = 2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+.
+Furthermore, there exist no other bifurcations of the equilibrium.
+Proof. Substituting p1 = 3 c and p2 = 3 c into J, we obtain that the Jacobian matrix at (3 c, 3 c) to be
+J(3 c, 3 c) =
+�
+27 c2−k1
+27 c2
+k1
+216 c2
+k2
+216 c2
+27 c2−k2
+27 c2
+�
+.
+Consequently,
+Tr(J) = 54 c2 − k1 − k2
+27 c2
+,
+Det(J) = 5184 c4 − 192 c2k1 − 192 c2k2 + 7 k1k2
+5184 c4
+.
+1These zeros can also be obtained from T12 in (6) by setting c1 = c2 = c.
+5
+
+One can verify that the first condition for the local stability is always fulfilled since k1, k2, c > 0 and
+CDJ
+1 ≡ 1 − Tr(J) + Det(J) = 5 k1k2
+3888 c4 .
+The second condition is
+CDJ
+2 ≡ 1 + Tr(J) + Det(J) = 15552 c4 + (−288 k1 − 288 k2) c2 + 5 k1k2
+3888 c4
+> 0,
+which means that
+15552 c4 + (−288 k1 − 288 k2) c2 + 5 k1k2 > 0,
+i.e.,
+c2 > 2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+or c2 < 2 k1 + 2 k2 −
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+.
+The third condition is
+CDJ
+3 ≡ 1 − Det(J) = (144 k1 + 144 k2) c2 − 5 k1k2
+3888 c4
+> 0,
+which implies that
+(144 k1 + 144 k2) c2 − 5 k1k2 > 0,
+i.e.,
+c2 >
+5 k1k2
+144 (k1 + k2).
+Furthermore, it can be proved that
+2 k1 + 2 k2 −
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+<
+5 k1k2
+144 (k1 + k2) < 2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+.
+Accordingly, the equilibrium is locally stable if
+c2 > 2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+.
+The rest of the proof follows immediately from Remark 1.
+Figure 1 depicts two 2-dimensional cross-sections of the stability region reported in Theorem 1. It is
+observed that an increase in the marginal cost c or a decrease in the adjustment speeds k1, k2 has an effect
+of stabilizing the unique non-vanishing equilibrium.
+(a) k2 = 1/10
+(b) c = 1/3
+Figure 1: The 2-dimensional cross-sections of the stability region of the considered model with α = 1/2 and
+c1 = c2 = c. The curves of CDJ
+2 = 0 and CDJ
+3 = 0 are marked in blue and green, respectively.
+6
+
+3.2
+The general case
+If c1 ̸= c2, then the analytical expression of the unique non-vanishing equilibrium would be quite complicated.
+Thus, the proof of Theorem 1 can not work since it is impossible to substitute the analytical expression of
+the equilibrium into the Jacobian matrix and obtain a neat result. Concerning the bifurcation analysis, we
+need to determine the conditions on the parameters that CDJ
+1 = 0, CDJ
+2 = 0, and CDJ
+3 = 0 are satisfied at
+the non-vanishing equilibrium. For this purpose, the following notation is required.
+Definition 1. Let
+A =
+m
+�
+i=0
+ai xi,
+B =
+l
+�
+j=0
+bj xj
+be two univariate polynomials in x with coefficients ai, bj, and am, bl ̸= 0. The determinant
+���������������
+am
+am−1
+· · ·
+a0
+...
+...
+...
+...
+am
+am−1
+· · ·
+a0
+bl
+bl−1
+· · ·
+b0
+...
+...
+...
+...
+bl
+bl−1
+· · ·
+b0
+���������������
+�
+�
+� l
+�
+�
+� m
+is called the Sylvester resultant (or simply resultant) of A and B with respect to x, and denoted by
+res(A, B, x).
+The following lemma reveals the main property of the resultant, which can also be found in [30].
+Lemma 1. Let A and B be two univariate polynomials in x. There exist two polynomials F and G in x
+such that
+FA + GB = res(A, B, x).
+Furthermore, A and B have common zeros in the field of complex numbers if and only if res(A, B) = 0.
+For a triangular set T = [T1(x), T2(x, y)] and a polynomial H(x, y), we define
+res(H, T ) ≡ res(res(H, T2, y), T1(x), x).
+By Lemma 1, if T1 = 0 and T2 = 0 (or simply denoted as T = 0), then one knows that H = 0 implies
+res(H, T ) = 0, which means res(H, T ) = 0 is a necessary condition for H = 0. Consequently, the following
+proposition is acquired. It should be emphasized that Proposition 2 only reports the results for the case
+of k1 = k2 because the conditions for k1 ̸= k2 are too long to list in this paper due to space limitations.
+However, readers can see that the idea of the proof also works for k1 ̸= k2 and can derive the complete
+conditions themself.
+Proposition 2. Let α = 1/2 and k1 = k2 = k. The system may undergo a period-doubling bifurcation when
+R1 = 0 and a Neimark-Sacker bifurcation when R2 = 0, where R1 and R2 are given in Appendix.
+Proof. It should be noted that the resultant is feasible only for polynomials. For CDJ
+1 , we consider its
+numerator Num(CDJ
+1 ). Then one can obtain that
+res(Num(CDJ
+1 ), T12) = 81 k6c18
+1 c6
+2 (c1 + c2)
+�
+32c2
+1 + 61c1c2 + 32c2
+2
+�
+.
+Since c1 > 0, c2 > 0, and k > 0, it is impossible that res(Num(CDJ
+1 ), T12) = 0 or CDJ
+1 = 0 provided that
+T12 = 0. Hence, the equilibrium can not lose its stability through a fold bifurcation. Furthermore, we have
+res(Num(CDJ
+2 ), T12) = −729 c32
+1 c8
+2(c1 + c2)R1,
+res(Num(CDJ
+3 ), T12) = 729 k3c32
+1 c8
+2(c1 + c2)R2,
+which will vanish only if R1 = 0 and R2 = 0, respectively.
+Consequently, the system may undergo a
+period-doubling bifurcation when R1 = 0 and a Neimark-Sacker bifurcation when R2 = 0.
+7
+
+By Proposition 1, there exists only one equilibrium (p1, p2) with p1 > 0 and p2 > 0 although its analytical
+expression is complicated. To explore the local stability, we need to determine the signs of CDJ
+1 , CDJ
+2 , and
+CDJ
+3 at this equilibrium without using its closed form. It should be noted that CDJ
+1 , CDJ
+2 , and CDJ
+3 are
+rational functions. Suppose that
+CDJ
+i = Num(CDJ
+i )
+Den(CDJ
+i ) ,
+where Num(·) and Den(·) denote the numerator and the denominator, respectively. Then the sign of CDJ
+i
+is the same as that of Num(CDJ
+i ) · Den(CDJ
+i ) if Den(CDJ
+i ) ̸= 0. One could compute that
+res(Num(CDJ
+1 ) · Den(CDJ
+1 ), T12) = −1594323 k6c50
+1 c17
+2 (c1 + c2)6(32 c2
+1 + 61 c1c2 + 32 c2
+2),
+res(Num(CDJ
+2 ) · Den(CDJ
+2 ), T12) = 129140163 c70
+1 c22
+2 (c1 + c2)6R1,
+and
+res(Num(CDJ
+3 ) · Den(CDJ
+3 ), T12) = −129140163 k3c70
+1 c22
+2 (c1 + c2)6R2.
+We should emphasize that the sign of res(Num(CDJ
+i )·Den(CDJ
+i ), T12) may not be the same as Num(CDJ
+i )·
+Den(CDJ
+i ) or CDJ
+i . However, it is known that res(Num(CDJ
+i )·Den(CDJ
+i ), T12) involves only the parameters
+and its zeros divide the parameter space into several regions. In each region, the sign of CDJ
+i is invariant.
+Consequently, we just need to select one sample point from each region and identify the sign of CDJ
+i at the
+selected sample point. The selection of sample points might be extremely complicated in general and could
+be automated using, e.g., the PCAD method [11].
+In Table 1, we list all the selected sample points and the corresponding information on whether the
+non-vanishing equilibrium is stable, i.e., whether CDJ
+1 > 0, CDJ
+2 > 0, and CDJ
+3 > 0 are simultaneously
+satisfied. Moreover, Table 1 displays the signs of R1 and R2 at these sample points. One can observe that
+the equilibrium is stable if R1 > 0 and R2 > 0, and vice versa. It should be mentioned that the calculations
+involved in Table 1 are exact and rigorous. That is, the computational results provide theoretical foundations
+for a systematic analysis of the local stability. Therefore, we acquire the following theorem.
+Theorem 2. If k1 = k2 = k, the unique non-vanishing equilibrium (p1, p2) with p1 > 0 and p2 > 0 is locally
+stable if R1 > 0 and R2 > 0, where R1 and R2 can be found in Appendix.
+Table 1: Selected Sample Points in {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} for α = 1/2
+(c1, c2, k)
+stable
+R1
+R2
+(c1, c2, k)
+stable
+R1
+R2
+(1, 1/4, 1)
+yes
++
++
+(1, 5/16, 1)
+yes
++
++
+(1, 1/4, 7)
+no
+−
++
+(1, 5/16, 10)
+no
+−
++
+(1, 1/4, 29)
+no
+−
+−
+(1, 5/16, 30)
+no
+−
+−
+(1, 1/4, 51)
+no
++
+−
+(1, 5/16, 51)
+no
++
+−
+(1, 1/2, 1)
+yes
++
++
+([1, 7/8, 1)
+yes
++
++
+(1, 1/2, 18)
+no
+−
++
+(1, 7/8, 38)
+no
+−
++
+(1, 1/2, 35)
+no
+−
+−
+(1, 7/8, 51)
+no
+−
+−
+(1, 1/2, 53)
+no
++
+−
+(1, 7/8, 65)
+no
++
+−
+(1, 9/8, 1)
+yes
++
++
+(1, 2, 1)
+yes
++
++
+(1, 9/8, 49)
+no
+−
++
+(1, 2, 70)
+no
+−
++
+(1, 9/8, 66)
+no
+−
+−
+(1, 2, 140)
+no
+−
+−
+(1, 9/8, 83)
+no
++
+−
+(1, 2, 209)
+no
++
+−
+(1, 3, 1)
+yes
++
++
+(1, 4, 1)
+yes
++
++
+(1, 3, 91)
+no
+−
++
+(1, 4, 112)
+no
+−
++
+(1, 3, 272)
+no
+−
+−
+(1, 4, 462)
+no
+−
+−
+(1, 3, 453)
+no
++
+−
+(1, 4, 811)
+no
++
+−
+8
+
+4
+α = 1/3
+If α = 1/3, then β = 1/2. We have the iteration map
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+p1(t + 1) = p1(t) + k1
+−p1(t)
+�
+p1(t)p2(t) +
+�
+2 p2(t) + 3
+�
+p1(t)p2(t)
+�
+c1
+2 p2
+1(t)
+��
+p1(t) +
+�
+p2(t)
+�2
+,
+p2(t + 1) = p2(t) + k2
+−p2(t)
+�
+p1(t)p2(t) +
+�
+2 p1(t) + 3
+�
+p1(t)p2(t)
+�
+c2
+2 p2
+2(t)
+��
+p1(t) +
+�
+p2(t)
+�2
+.
+(7)
+By setting p1(t + 1) = p1(t) = p1 and p2(t + 1) = p2(t) = p2, one can obtain the equations of the
+equilibrium
+�
+− p1
+√p1p2 + (2 p2 + 3√p1p2) c1 = 0,
+− p2
+√p1p2 + (2 p1 + 3√p1p2) c2 = 0.
+Denote √p1 = x and √p2 = y. The above equations become
+�
+− x3y + (2 y2 + 3 xy)c1 = 0,
+− y3x + (2 x2 + 3 xy)c2 = 0.
+(8)
+Using the triangular decomposition method, we decompose the solutions of system (8) into zeros of the
+following two triangular sets.
+T31 = [x, y] ,
+T32 =
+�
+x8 − 9 c1x6 + 27 c2
+1x4 + (−27 c3
+1 − 12 c2
+1c2)x2 + 20 c3
+1c2, 2 c1y − x3 + 3 c1x
+�
+.
+Evidently, T31 is corresponding to the origin (0, 0). Therefore, the identification of the number of non-
+vanishing equilibria can be transformed into the determination of the number of real solutions of the following
+semi-algebraic system.
+�
+�
+�
+�
+�
+x8 − 9 c1x6 + 27 c2
+1x4 + (−27 c3
+1 − 12 c2
+1c2)x2 + 20 c3
+1c2 = 0,
+2 c1y − x3 + 3 c1x = 0,
+x > 0, y > 0.
+Using the algebraic approach by Li and Wang [25], we know that the above system has one unique real
+solution for any parameter values of c1, c2 > 0, which implies the following proposition.
+Proposition 3. Let α = 1/3. The iteration map (7) possesses one unique equilibrium (p1, p2) with p1 > 0
+and p2 > 0.
+To investigate the local stability of the equilibrium, we consider the Jacobian matrix
+M =
+�M11
+M12
+M21
+M22
+�
+,
+where
+M11 = 12 p
+9
+2
+1
+√p2 + 4 p
+7
+2
+1 p
+3
+2
+2 − 15 c1k1p
+3
+2
+1
+√p2 − 8 c1k1p
+3
+2
+2
+√p1 + 3 k1p
+5
+2
+1
+√p2 + 4 p5
+1 + 12 p4
+1p2 − 21 c1k1p1p2 + k1p2
+1p2
+4 p
+7
+2
+1
+�√p1 + √p2
+�3
+,
+M12 =
+k1
+�√p2 p
+3
+2
+1 − p2
+1 + c1√p2√p1 + 3 p1c1
+�
+4 p2
+1
+�√p1 + √p2
+�3 √p2
+,
+M21 =
+k2
+�√p1 p
+3
+2
+2 + c2√p2√p1 + 3 p2c2 − p2
+2
+�
+4 p2
+2
+�√p1 + √p2
+�3 √p1
+,
+9
+
+M22 = 4 p
+3
+2
+1 p
+7
+2
+2 + 12 p
+9
+2
+2
+√p1 − 8 c2k2p
+3
+2
+1
+√p2 − 15 c2k2p
+3
+2
+2
+√p1 + 3 k2p
+5
+2
+2
+√p1 + 12 p1 p4
+2 + 4 p5
+2 − 21 c2k2p1p2 + k2p1p2
+2
+4p
+7
+2
+2
+�√p1 + √p2
+�3
+.
+As in Section 3, we denote
+CDM
+1 ≡ 1 − Tr(M) + Det(M),
+CDM
+2 ≡ 1 + Tr(M) + Det(M),
+CDM
+3 ≡ 1 − Det(M).
+4.1
+The special case of c1 = c2
+If we set c1 = c2 = c, then the triangular decomposition method permits us to transform the equilibrium
+equations (8) into the following triangular sets.
+T41 = [x, y],
+T42 = [x2 − c, y + x],
+T43 = [x2 − 5 c, y − x],
+T44 = [x4 − 3 c x2 + 4 c2, 2 cy − x3 + 3 cx].
+Obviously, the zeros of T41 and T42 are economically uninteresting. Moreover, all the roots of x4−3 c x2+
+4 c2 of T44, i.e.,
+�
+2
+√
+7c i + 6 c
+2
+, −
+�
+2
+√
+7c i + 6 c
+2
+,
+�
+−2
+√
+7c i + 6 c
+2
+, −
+�
+−2
+√
+7c i + 6 c
+2
+,
+are imaginary and also not of our concern.
+There exists only one non-vanishing equilibrium (p1, p2) =
+(5 c, 5 c), which is corresponding to the branch T43.
+Substituting p1 = 5 c and p2 = 5 c into M, we obtain the Jacobian matrix at the equilibrium (5 c, 5 c) to
+be
+M(5 c, 5 c) =
+�
+500 c2−3 k1
+500 c2
+k1
+1000 c2
+k2
+1000 c2
+500 c2−3 k2
+500 c2
+�
+.
+Hence,
+Tr(M) = 1000 c2 − 3 k1 − 3k2
+500 c2
+,
+Det(M) = 200000 c4 − 1200 c2k1 − 1200 c2k2 + 7 k1k2
+200000 c4
+.
+Theorem 3. Let α = 1/3 and c1 = c2 = c. The unique non-vanishing equilibrium (5 c, 5 c) is locally stable
+if
+c2 > 3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+.
+The system may undergo a period-doubling bifurcation when
+c2 = 3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+.
+Furthermore, there exist no other bifurcations of the equilibrium.
+Proof. The first condition for the local stability is always fulfilled since
+CDM
+1 ≡ 1 − Tr(M) + Det(M) =
+7 k1k2
+200000c4 .
+The second condition should be
+CDM
+2 ≡ 1 + Tr(M) + Det(M) = 800000 c4 + (−2400 k1 − 2400 k2) c2 + 7 k1k2
+200000 c4
+> 0,
+10
+
+which implies that
+800000 c4 + (−2400 k1 − 2400 k2) c2 + 7 k1k2 > 0,
+i.e.,
+c2 > 3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+or c2 < 3 k1 + 3 k2 −
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+.
+The third condition should be
+CDM
+3 ≡ 1 − Det(M) = (1200 k1 + 1200 k2) c2 − 7 k1k2
+200000 c4
+> 0,
+from which we have
+(1200 k1 + 1200 k2) c2 − 7 k1k2 > 0,
+i.e.,
+c2 >
+7 k1k2
+1200 (k1 + k2).
+It can be proved that
+3 k1 + 3 k2 −
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+<
+7 k1k2
+1200 (k1 + k2) < 3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+.
+Therefore, the equilibrium is locally stable if
+c2 > 3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+.
+The rest of the proof follows from Remark 1.
+In Figure 2, we show two 2-dimensional cross-sections of the stability region reported in Theorem 3. One
+can see that the equilibrium may lose its stability if the adjustment speeds k1, k2 are large enough or the
+marginal cost c is small enough.
+(a) k2 = 1/10
+(b) c = 1/10
+Figure 2: The 2-dimensional cross-sections of the stability region of the considered model with α = 1/3 and
+c1 = c2 = c. The curves of CDM
+2 = 0 and CDM
+3 = 0 are marked in blue and green, respectively.
+4.2
+The general case
+As in Section 3.2, we set k1 = k2 = k. We should mention that the method employed in this section also
+works for the case of k1 ̸= k2. However, the conditions for k1 ̸= k2 are tedious and not reported in this
+section due to space limitations. Interested readers can use our method to compute the complete conditions
+themself. The case of c1 = c2 has been explored in Section 4.1, hence we suppose that c1 ̸= c2 in what
+follows. The bifurcations are analyzed in the following proposition.
+11
+
+Proposition 4. Let α = 1/3, k1 = k2 = k and c1 ̸= c2. The iteration map (7) may undergo a period-
+doubling bifurcation when R3 = 0 and a Neimark-Sacker bifurcation when R4 = 0, where R3 and R4 are
+given in Appendix.
+Proof. Computing the resultant of Num(CDM
+1 ) with respect to T32, one obtains
+res(Num(CDM
+1 ), T32) = 879609302220800000 k16c51
+1 c11
+2 (c1 − c2)2 �
+2187 c2
+1 − 4031 c1c2 + 2187 c2
+2
+�2 .
+It is evident that
+2187 c2
+1 − 4031 c1c2 + 2187 c2
+2 = 2187(c1 − c2)2 + 343 c1c2 > 0.
+Therefore, res(Num(CDM
+1 ), T32) ̸= 0, which means that CDM
+1 ̸= 0 at the unique non-vanishing equilibrium.
+Hence, there exist no fold bifurcations in map (7). Furthermore, we have
+res(Num(CDJ
+2 ), T32) = 99035203142830421991929937920000000 c101
+1
+c13
+2 (c1 − c2)2 R2
+3,
+and
+res(Num(CDJ
+3 ), T32) = 99035203142830421991929937920000000 k8c101
+1
+c13
+2 (c1 − c2)10R2
+4.
+Consequently, a period-doubling bifurcation may occur when R3 = 0, while a Neimark-Sacker bifurcation
+may take place when R4 = 0.
+To investigate the local stability, we need to consider Num(CDJ
+i ) · Den(CDJ
+i ) and compute its resultant
+with respect to T32. Then it is obtained that
+res(Num(CDJ
+1 ) · Den(CDJ
+1 ), T32) =
+5708990770823839524233143877797980545530986496 · 1020
+· k16c156
+1
+c36
+2 (c1 − c2)12(2187 c2
+1 − 4031 c1c2 + 2187 c2
+2)2,
+res((Num(CDJ
+2 ) · Den(CDJ
+2 ), T32) =
+6582018229284824168619876730229402019930943462534319453394436096 · 1024
+· c218
+1
+c42
+2 (c1 − c2)10R2
+3,
+res((Num(CDJ
+3 ) · Den(CDJ
+3 ), T32) =
+6582018229284824168619876730229402019930943462534319453394436096 · 1024
+· k8c218
+1
+c42
+2 (c1 − c2)10R2
+4.
+These res(Num(CDJ
+i )·Den(CDJ
+i ), T32) involve only the parameters and their zeros divide the parameter
+set {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} into several regions.
+In each region, the signs of CDM
+1 , CDM
+2 ,
+and CDM
+3
+are fixed and can be identified by checking at a selected sample point. In Table 2, we list the
+40 selected sample points and the signs of R3, R4 at these sample points.
+Moreover, Table 2 provides
+the information on whether the non-vanishing equilibrium is stable, i.e., whether the stability conditions
+CDM
+1
+> 0, CDM
+2
+> 0 and CDM
+3
+> 0 are satisfied simultaneously.
+Interested readers may check the
+correctness of Table 2 themselves. Based on a series of computations, we acquire the following theorem.
+Theorem 4. Let k1 = k2 = k and c1 ̸= c2. The unique non-vanishing equilibrium of map (7) is locally
+stable if one of the following conditions is satisfied:
+1. R3 > 0, R4 > 0;
+2. R3 < 0, R4 > 0 and A1 > 0, A2 < 0, A3 > 0,
+where R3, R4, A1, A2, and A3 can be found in Appendix.
+Remark 2. From Table 2, we see that the equilibrium is stable if R3 > 0 and R4 > 0. Hence, R3 > 0, R4 > 0
+is a sufficient condition for the local stability. However, this condition is not necessary. For example, at the
+first sample point (1, 1/4, 1/512) listed in Table 2, the equilibrium is locally stable, but one can verify that
+R3 < 0 and R4 > 0 at this point. Thus, the second condition of Theorem 4 is needed.
+The necessity of the second condition can also be illustrated by Figure 4 (b, d, f), where the regions
+defined by the first and second conditions are marked in light grey and dark grey, respectively. By economic
+12
+
+intuition, we know that for a fixed value of the marginal cost c2, a decrease in the adjustment speed k would
+be beneficial to the local stability of the equilibrium. That is to say, the dark grey regions defined by the
+second condition would be more likely to be included in the stability regions.
+It is noted that A1, A2, and A3 involved in the second condition are contained in the so-called generalized
+discriminant list and can be picked out by repeated trials. Concerning the generalized discriminant list, the
+reader may refer to [38] for more details. The polynomials A1, A2, and A3 are needed here since the condition
+that R3 < 0 and R4 > 0 is not a sufficient condition for the local stability. For example, the model is stable
+at (1, 1/4, 1/512), where R3 < 0 and R4 > 0. But, the model is unstable at (1, 1/4, 34), where R3 < 0 and
+R4 > 0 are also satisfied. Consequently, additional polynomials are needed to constrict the region defined
+by R3 < 0 and R4 > 0 such that the complete stability conditions can be acquired.
+Table 2: Selected Sample Points in {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} for α = 1/3
+(c1, c2, k)
+stable
+R3
+R4
+(c1, c2, k)
+stable
+R3
+R4
+(1, 1/4, 1/512)
+yes
+−
++
+(1, 3/8, 1/128)
+yes
+−
++
+(1, 1/4, 1)
+yes
++
++
+(1, 3/8, 1)
+yes
++
++
+(1, 1/4, 34)
+no
+−
++
+(1, 3/8, 64)
+no
+−
++
+(1, 1/4, 153)
+no
+−
+−
+(1, 3/8, 175)
+no
+−
+−
+(1, 1/4, 273)
+no
++
+−
+(1, 3/8, 287)
+no
++
+−
+(1, 5/8, 1/32)
+yes
+−
++
+(1, 7/8, 1/128)
+yes
+−
++
+(1, 5/8, 1)
+yes
++
++
+(1, 7/8, 1)
+yes
++
++
+(1, 5/8, 145)
+no
+−
++
+(1, 7/8, 244)
+no
+−
++
+(1, 5/8, 231)
+no
+−
+−
+(1, 7/8, 302)
+no
+−
+−
+(1, 5/8, 317)
+no
++
+−
+(1, 7/8, 361)
+no
++
+−
+(1, 5/4, 1/32)
+yes
+−
++
+(1, 3/2, 1/16)
+yes
+−
++
+(1, 5/4, 1)
+yes
++
++
+(1, 3/2, 1)
+yes
++
++
+(1, 5/4, 335)
+no
+−
++
+(1, 3/2, 362)
+no
+−
++
+(1, 5/4, 436)
+no
+−
+−
+(1, 3/2, 544)
+no
+−
+−
+(1, 5/4, 538)
+no
++
+−
+(1, 3/2, 726)
+no
++
+−
+(1, 2, 1/16)
+yes
+−
++
+(1, 3, 1/16)
+yes
+−
++
+(1, 2, 1)
+yes
++
++
+(1, 3, 1)
+yes
++
++
+(1, 2, 403)
+no
+−
++
+(1, 3, 471)
+no
+−
++
+(1, 2, 804)
+no
+−
+−
+(1, 3, 1503)
+no
+−
+−
+(1, 2, 1205)
+no
++
+−
+(1, 3, 2536)
+no
++
+−
+5
+Influence of the Substitutability Degree
+Firstly, we analyze the influence of the substitutability degree α on the size of the stability region of the
+equilibrium. We start by considering the special case of c1 = c2.
+Proposition 5. Let c1 = c2. The stability region for α = 1/2 is a proper subset of that for α = 1/3.
+Proof. Recall Theorems 1 and 3. We need to prove that
+2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+> 3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+,
+which is equivalent to
+�
+2 k1 + 2 k2 +
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+216
+�2
+−
+�
+3 k1 + 3 k2 +
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2000
+�2
+> 0.
+The left-hand side of the above inequality can be simplified into
+− (4374 k1 + 4374 k2)
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2916000000
++ (250000 k1 + 250000 k2)
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+2916000000
+13
+
++ 243439 k2
+1
+1458000000 + 61771 k1k2
+2916000000 + 243439 k2
+2
+1458000000.
+It is easy to check that
+(4374 k1 + 4374 k2)
+�
+9 k2
+1 − 17 k1k2 + 9 k2
+2
+2916000000
+< (250000 k1 + 250000 k2)
+�
+4 k2
+1 − 7 k1k2 + 4 k2
+2
+2916000000
+,
+which completes the proof.
+If c1 ̸= c2, however, the conclusion of the above proposition would be incorrect. For example, if we
+assume k1 = k2 = k and take (c1, c2, k) = (261/65536, 1/2, 79/1024), then
+R1 =
+588713082686404258452596575293972215811486125608829
+6129982163463555433433388108601236734474956488734408704 > 0,
+R2 = 108130364702270905134254005155560019343
+340282366920938463463374607431768211456 > 0.
+Hence, (261/65536, 1/2, 79/1024) is in the stability region of the model for α = 1/2. But, at the same
+parameter point, namely (c1, c2, k) = (261/65536, 1/2, 79/1024), we have
+R3 = − 791461358900213183480020700044263844445257635142615074110540187
+26328072917139296674479506920917608079723773850137277813577744384 < 0,
+R4 = 526438846625624761986017962528229497389068363385599391
+374144419156711147060143317175368453031918731001856
+> 0,
+and
+A1 =
+44864955
+4294967296 > 0,
+A2 = −
+842240947483983714275440267
+81129638414606681695789005144064 < 0,
+A3 = − 63936547182666560163845458457577
+649037107316853453566312041152512 < 0.
+This means that the stability conditions of Theorem 4 for α = 1/3 are not satisfied.
+On the other hand, one can also find some points where the model is stable for α = 1/3 but unstable for
+α = 1/2. For example, at (c1, c2, k) = (3/8, 1/2, 827/64), we know
+R3 = 40079185741889580295152003015
+288230376151711744
+> 0,
+R4 = 29339436396656781
+17179869184
+> 0.
+Therefore, (3/8, 1/2, 827/64) is in the stability region for α = 1/3. However,
+R1 = −24200272602071108539
+17592186044416
+< 0,
+R2 = −96467864887
+67108864
+< 0.
+That is to say, (3/8, 1/2, 827/64) is an unstable parameter point for α = 1/2.
+Figure 3 depicts the 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3. For
+comparison purposes, we place the cross-sections for α = 1/2 on the left and those for α = 1/3 on the right.
+We set k1 = k2 = k and choose three different values of the parameter k, i.e., k = 1/2, 1, 10, to observe the
+effect of variation of k on the size of the stability regions. The curves of R1 = 0 and R3 = 0 are marked in
+blue; the curves of R2 = 0 and R3 are marked in green; the curves of A1 = 0, A2 = 0 and A3 = 0 are marked
+in red. The stability regions are colored in light grey. From Figure 3, we find that the stability region would
+shrink if the firms react or adjust their outputs faster both for α = 1/2 and α = 1/3. Similarly, in Figure
+4, we assume that k1 and k2 are identical and choose three different values of c1, i.e., c1 = 1/2, 1, 10. The
+regions of R1 > 0, R2 > 0 and those of R3 > 0, R4 > 0 are colored in light grey, while the regions defined by
+R3 < 0, R4 > 0, A1 > 0, A2 < 0, A3 > 0 are colored in dark grey. From Figure 4, we observe that increasing
+the marginal cost c1 of the first firm could result in the enlargement of the stability region for α = 1/2 and
+α = 1/3.
+As aforementioned, in the case of c1 ̸= c2 and k1 = k2, it can not be proved that the stability region
+for α = 1/3 covers that for α = 1/2. From Figures 3 and 4, however, it seems that the stability region for
+α = 1/3 is larger than that for α = 1/2. Consequently, for the Bertrand duopoly model considered in this
+paper, we may conclude that increasing the substitutability degree α has an effect of destabilizing the unique
+14
+
+non-vanishing equilibrium in some sense. In other words, product differentiation might make the considered
+model more stable, which is an important finding from an economic point of view. Shy [34] discussed the
+traditional view on the degree of product differentiation, i.e., a decrease in product differentiation may result
+in an increase in market competition intensity and even a price war among involved firms. The possible
+explanation for our finding is that a price war might destabilize the equilibrium of the Bertrand game with
+differentiated goods. It should be noted that our conclusion is in contrast with the one by Agliari et al.
+[1]. Specifically, Agliari et al. [1] investigated a Cournot duopoly model with differentiated products and
+employed the same CES utility function and the same linear cost functions as in our study.
+However,
+they discovered that a higher degree of product differentiation or a lower degree of substitutability leads to
+the destabilization of their model. This contradiction may help reveal the essential difference between the
+Bertrand and Cournot oligopolies with differentiated goods.
+From an economic point of view, the effects on economic variables such as prices and profits of changing
+the substitutability degree are interesting. In the sequel, we focus on the comparative statics in the special
+case of identical marginal costs. Let c1 = c2 = c. According to (3), the equilibrium satisfies that
+�
+− pβ
+2p1+β
+1
+β + p2β
+2 c + (pβ
+1pβ
+2)(1 + β)c = 0,
+− pβ
+1p1+β
+2
+β + p2β
+1 c + (pβ
+1pβ
+2)(1 + β)c = 0.
+(9)
+Hence,
+−pβ
+2p1+β
+1
+β + p2β
+2 c = −pβ
+1p1+β
+2
+β + p2β
+1 c,
+which implies that
+(p2β
+1 − p2β
+2 )c = (p2 − p1)pβ
+1pβ
+2β.
+Without loss of generality, we suppose that p1 ≥ p2. Since c > 0 and β > 0, we know (p2β
+1 − p2β
+2 )c ≥ 0 and
+(p2 − p1)pβ
+1pβ
+2β ≤ 0, which implies p1 = p2. Plugging p1 = p2 into the first equation of (9), one can solve
+p1 = p2 = c(2+β)
+β
+. Therefore, at the equilibrium q1 = q2 =
+β
+2 c(2+β). As β = α/(1 − α), we obtain
+∂pi
+∂α = −2 c
+α2 < 0,
+∂qi
+∂α =
+1
+(−2 + α)2 c
+> 0.
+According to (2), the profits of the two firms would be
+Π1 = Π2 =
+�c(2 + β)
+β
+− c
+�
+β
+2 c(2 + β) =
+1
+2 + β = 1 +
+1
+α − 2.
+Hence, for i = 1, 2,
+∂Πi
+∂α = −
+1
+(α − 2)2 < 0.
+Recalling the inverse demands (1), for a point (q∗
+1, q∗
+2) on the indifference curve, we define the consumer
+surplus of the first product to be
+CS1 =
+� q∗
+1
+0
+qα−1
+1
+qα
+1 + q∗α
+2
+dq1 = 1
+α
+� q∗
+1
+0
+d(qα
+1 + q∗α
+2 )
+qα
+1 + q∗α
+2
+= 1
+α ln
+�
+1 +
+�q∗
+1
+q∗
+2
+�α�
+.
+In the case of c1 = c2, the outputs of the two products are equal at the equilibrium. Therefore, we have
+that CS1 = CS2 = 1
+α ln 2. Accordingly, the social welfare is
+W = CS1 + CS2 + Π1 + Π2 = 2
+α ln 2 +
+2
+α − 2 + 2.
+Then it is known that
+∂W
+∂α = −2 ln 2
+α2
+−
+2
+(α − 2)α < 0.
+To summarize, in the special case of identical marginal costs, an increase in the substitutability degree
+α leads to a stable equilibrium with lower prices, higher supplies, lower profits, and lower welfare. In other
+words, the degree of product differentiation is positively related to the prices of the goods, the profits of the
+involved companies, and the social welfare, which is consistent with our economic intuition.
+15
+
+(a) α = 1/2, k = 1/2
+(b) α = 1/3, k = 1/2
+(c) α = 1/2, k = 1
+(d) α = 1/3, k = 1
+(e) α = 1/2, k = 10
+(f) α = 1/3, k = 10
+Figure 3: The 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3 if we set
+k1 = k2 = k and fix k = 1/2, 1, 10. The curves of R1 = 0 and R3 = 0 are marked in blue; the curves of
+R2 = 0 and R3 are marked in green; the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red. The
+stability regions are colored in light grey.
+16
+
+(a) α = 1/2, c1 = 1/2
+(b) α = 1/3, c1 = 1/2
+(c) α = 1/2, c1 = 1
+(d) α = 1/3, c1 = 1
+(e) α = 1/2, c1 = 10
+(f) α = 1/3, c1 = 10
+Figure 4: The 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3 if we set
+k1 = k2 = k and fix c1 = 1/2, 1, 10. The curves of R1 = 0 and R3 = 0 are marked in blue; the curves of
+R2 = 0 and R3 are marked in green; the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red. The
+regions of R1 > 0, R2 > 0 and those of R3 > 0, R4 > 0 are colored in light grey, while the regions defined
+by R3 < 0, R4 > 0, A1 > 0, A2 < 0, A3 > 0 are colored in dark grey.
+17
+
+6
+Numerical Simulations
+This section provides numerical simulations to illustrate the complex dynamics of the considered Bertrand
+duopoly model. The first purpose of our simulations is to confirm the main conclusion of Section 5 that
+increasing the substitutability degree α could destabilize the unique non-vanishing equilibrium. In Figure
+5, we depict the 1-dimensional bifurcation diagrams with respect to α, where we fix the other parameters
+k1 = k2 = 1, c1 = c2 = 0.2 and set the initial point to be (0.56, 1.06). The bifurcation diagrams against
+p1 and p2 are given in Figure 5 (a, c) and (b, d), respectively. It is observed that complex dynamics appear
+when α becomes large enough. Specifically, there exists one unique stable equilibrium at first, then a stable
+2-cycle orbit, and finally a chaotic set as α varies from 0.1 up to 0.7.
+To show the transition clearly,
+the 1-dimensional bifurcation diagrams are enlarged for α ∈ (0.55, 0.6) in (c, d). One can see that, when
+α = 0.553372, a branching point occurs and the unique fixed point bifurcates into a 2-cycle orbit, which,
+however, is not a period-doubling bifurcation point. This 2-cycle orbit loses its stability through a Neimark-
+Sacker bifurcation rather than a period-doubling bifurcation at α = 0.577570.
+More details can be found in Figure 6, where we plot the phase portraits for k1 = k2 = 1 and c1 = c2 = 0.2
+with the initial point (0.56, 1.06). From Figure 6 (a), we observe that, after the occurrence of a Neimark-
+Sacker bifurcation, the 2-cycle orbit (P21(0.464194, 0.607384) and P21(0.607384, 0.464194)) becomes unstable
+and bifurcates into two invariant closed orbits when α = 0.58; the unique equilibrium E1(0.492557, 0.492557)
+goes to E1new(0.489655, 0.489655) when α = 0.58.
+Furthermore, all points on the diagonal line x = y
+converge to E1new. The two invariant closed orbits marked in blue are stable and points converge to them
+from inside and outside. Figure 6 (b) depicts the phase portrait when α = 0.59 and the other parameters
+are set to be the same as (a). From (b), one can discover chaotic attractors with symmetry. The above
+observations show that an increase in the substitutability degree α leads to the emergence of instability,
+complex dynamics, and even chaos in the considered model.
+(a) against p1
+(b) against p2
+(c) against p1 and enlarged for α ∈ (0.55, 0.6)
+(d) against p2 and enlarged for α ∈ (0.55, 0.6)
+Figure 5: The 1-dimensional bifurcation diagrams with respect to α if we fix k1 = k2 = 1, c1 = c2 = 0.2 and
+set the initial point to be (0.56, 1.06).
+18
+
+4
+pi
+3
+2
+1
+0
+0.1
+0.2
+0.3
+0.4
+a0.5
+0.6
+0.77
+9
+54
+3
+2
+1
+0
+0.1
+0.2
+0.3
+0.4
+L
+a0.5
+0.6
+0.77
+9
+50.7
+pi
+0.6
+NS
+BP
+0.5
+NS
+0.4
+0.3
+0.55
+0.56
+0.57
+0.58
+a0.59
+0.61
+0.9
+0.80.7
+0.6
+NS
+BP
+0.5
+NS
+0.4
+0.3
+0.55
+0.56
+0.57
+0.58
+a0.59
+0.61
+0.9
+0.80.4
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+p1
+0.4
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+p2
+(a) α = 0.58
+0.4
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+p1
+0.4
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+p2
+(b) α = 0.59
+Figure 6: Phase portraits for k1 = k2 = 1 and c1 = c2 = 0.2 with the initial point (0.56, 1.06).
+To illustrate the influence of other parameters, several 2-dimensional bifurcation diagrams are computed
+and displayed in the sequel. Figure 7 depicts the 2-dimensional bifurcation diagram of map (4) (α = 1/2)
+with respect to k1 and k2 if we fix c1 = 0.3, c2 = 0.4 and set the initial point to be (0.5, 0.8). We detect
+periodic orbits with distinct orders and mark the corresponding parameter points in different colors in Figure
+7. It should be mentioned that the parameter points where there exist periodic orbits with orders more than
+25 are marked in light yellow as well. Two different routes from the unique stable equilibrium to complex
+dynamics can be observed. For example, if we fix k2 = 7.5 and change the value of k1 from 0.0 to 10.0, the
+dynamics of the system start from one unique stable equilibrium (the dark blue region), then transition to
+a stable 2-cycle orbit (the light blue region) and finally to invariant closed orbits as well as chaos (the light
+yellow region). This is similar to the route displayed in Figure 5, where the stable 2-cycle loses its stability
+through a Neimark-Sacker bifurcation. The other route can be discovered, e.g., if we fix k2 = 2.5 and keep
+k1 as a free parameter. Then it is observed that the unique stable equilibrium loses its stability through a
+cascade of period-doubling bifurcations.
+In Figure 8, we plot the 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to k1
+and k2 if fixing c1 = 0.1, c2 = 0.15 and setting the initial point to be (0.6, 0.9). Similar to Figure 7, the
+aforementioned two routes from local stability to complex dynamics can also be observed in Figure 8.
+The 2-dimensional bifurcation diagrams with respect to c1 and c2 for α = 1/2 and α = 1/3 are displayed
+in Figures 9 and 10, respectively. One can see that complicated dynamic phenomena take place if one of
+the cost parameters c1, c2 is small enough. Similarly, we find the above two routes to chaotic behavior, i.e.,
+through a cascade of period-doubling bifurcation and through a Neimark-Sacker bifurcation on a 2-cycle
+orbit, which have already been discovered by Ahmed et al. [3]. However, from Figure 9, we also find the
+existence of a Neimark-Sacker bifurcation directly on the unique equilibrium, which is a new result that
+has not been observed by Ahmed et al. [3] yet. Specifically, Figure 9 shows that, if we fix c1 = 0.9 and
+decrease the value of c2 from 1.0 to 0.0, the dynamics of the system directly transition from the unique
+stable equilibrium (the dark blue region) to invariant closed orbits (the light yellow region). In this case,
+the behavior of the market suddenly changes from an ordered state to a disordered state at some critical
+point, which can hardly be learned by even rational players.
+7
+Concluding Remarks
+In this paper, we investigated the local stability, bifurcations, and comparative statics of a dynamic Bertrand
+duopoly game with differentiated products. This duopoly is assumed to possess two boundedly rational
+players adopting a gradient adjustment mechanism and a continuum of identical consumers with a CES
+utility function. Moreover, the cost functions are supposed to be linear. It should be mentioned that the
+nonlinearity of the resulting demand function derived from the underlying utility permits us to extend the
+applications of Bertrand games to more realistic economies, compared to the widely used Bertrand models
+with linear demands.
+The considered game was first explored by Ahmed et al. [3], where only numerical simulations are
+19
+
+Figure 7: The 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to k1 and k2 if we fix
+c1 = 0.3, c2 = 0.4 and set the initial point to be (0.5, 0.8).
+Figure 8: The 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to k1 and k2 if we fix
+c1 = 0.1, c2 = 0.15 and set the initial point to be (0.6, 0.9).
+20
+
+10.00
+25
+24
+23
+22
+21
+20
+7.50
+19
+18
+17
+16
+15
+14
+5.00
+13
+12
+11
+10
+8
+2.50
+6
+5
+3
+2
+0.00
+0.00
+2.50
+5.00
+7.50
+10.00
+k16.00
+25 
+24
+23
+ 22 
+21
+20
+4.50
+19
+18
+17
+16
+15
+14
+3.00
+13
+12
+11
+10
+9
+1.50
+6
+5
+3
+2
+0.00
+0.00
+1.50
+3.00
+4.50
+6.00
+k1Figure 9: The 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to c1 and c2 if we fix
+k1 = 6, k2 = 12 and set the initial point to be (0.5, 0.8).
+Figure 10: The 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to c1 and c2 if we fix
+k1 = 0.3, k2 = 0.6 and set initial point to be (0.6, 0.9).
+21
+
+1.00
+2.
+25
+24
+23
+ 22 
+21
+20
+0.75.
+19
+18
+17
+16
+15
+14
+0.50
+13
+12
+11
+10
+9
+8
+0.25.
+6
+5
+4
+3
+2
+0.00
+0.00
+0.25
+0.50
+0.75
+1.00
+C10.10
+24
+25 
+ 24 
+23 
+ 22 
+21
+20
+0.08
+19
+18
+17
+16
+15
+14
+0.05
+13
+12
+11
+10
+9
+0.03
+5
+4
+3
+2
+0.00
+0.00
+0.03
+0.05
+0.08
+0.10
+C1employed to investigate the dynamic behavior and it was observed that the Nash equilibrium loses its
+stability through a period-doubling bifurcation as the speed of adjustment increases. In our study, however,
+we re-investigated this game using several tools based on symbolic computations such as the triangular
+decomposition method (refer to, e.g., [23]) and the PCAD method (refer to, e.g., [11]).
+The results of
+symbolic computations are exact, and thus provide theoretical foundations for the systematic analysis of
+economic models.
+For simplicity, our work mainly focused on two specific degrees of product substitutability, namely
+α = 1/2 and α = 1/3. In both cases, we proved the uniqueness of the non-vanishing equilibrium using the
+algebraic approach of detecting the multiplicity of equilibria proposed by the first author and his co-worker
+[25]. We introduce several tools based on symbolic computations and used them to obtain the rigorous
+conditions for the local stability of the unique non-vanishing equilibrium for the first time. In the special
+case that the two firms have identical marginal costs, we proved that the model can lose its stability only
+through a period-doubling bifurcation. From an economic point of view, the most interesting finding was
+that an increase in the substitutability degree or a decrease in the product differentiation leads to the
+destabilization of the Bertrand model. This is because a price war, which might destabilize the equilibrium,
+can take place if the substitutability degree is large enough.
+We should mention that our finding is in
+contrast with that by Agliari et al. [1] and that by Fanti and Gori [14]. This contradiction contributes to the
+literature on the connection between Cournot and Bertrand oligopolies and may help reveal the essential
+difference between them. Moreover, we conducted the comparative statics in the special case of identical
+marginal costs. The resulting conclusion was that lower degrees of product differentiation mean lower prices,
+higher supplies, lower profits, and lower social welfare, which is consistent with our economic intuition.
+Numerical simulations were provided in the end, through which complex dynamics such as periodic
+orbits and chaos can be observed. The simulations confirmed that an increase in the substitutability degree
+α leads to the emergence of instability, complex dynamics, and even chaos in the considered model. Two-
+dimensional bifurcation diagrams were also provided to show different possible routes to chaotic behavior,
+e.g., through a cascade of period-doubling bifurcation and through a Neimark-Sacker bifurcation on a 2-cycle
+orbit. Furthermore, we discovered the existence of a Neimark-Sacker bifurcation directly on the equilibrium,
+which is a new finding and has not yet been discovered by Ahmed et al. [3].
+Appendix
+R1 = 15552 c10
+1 c6
+2 + 62208 c9
+1c7
+2 + 93312 c8
+1c8
+2 + 62208 c7
+1c9
+2 + 15552 c6
+1c10
+2 + 73728 c11
+1 c3
+2k + 327168 c10
+1 c4
+2k
++ 576576 c9
+1c5
+2k + 541440 c8
+1c6
+2k + 436608 c7
+1c7
+2k + 541440 c6
+1c8
+2k + 576576 c5
+1c9
+2k + 327168 c4
+1c10
+2 k
++ 73728 c3
+1c11
+2 k + 32768 c11
+1 c2 k2 + 94208 c10
+1 c2
+2k2 + 284160 c9
+1c3
+2k2 + 1163712 c8
+1c4
+2k2 + 2855520 c7
+1c5
+2k2
++ 3825168 c6
+1c6
+2k2 + 2855520 c5
+1c7
+2k2 + 1163712 c4
+1c8
+2k2 + 284160 c3
+1c9
+2k2 + 94208 c2
+1c10
+2 k2 + 32768 c1c11
+2 k2
++ 77824 c9
+1c2 k3 + 359936 c8
+1c2
+2k3 + 644608 c7
+1c3
+2k3 + 610976 c6
+1c4
+2k3 + 494368 c5
+1c5
+2k3 + 610976 c4
+1c6
+2k3
++ 644608 c3
+1c7
+2k3 + 359936 c2
+1c8
+2k3 + 77824 c1c9
+2k3 − 4096c8
+1k4 − 12288 c7
+1c2 k4 + 4544 c6
+1c2
+2k4
++ 70360 c5
+1c3
+2k4 + 114600 c4
+1c4
+2k4 + 70360 c3
+1c5
+2k4 + 4544 c2
+1c6
+2k4 − 12288 c1c7
+2k4 − 4096 c8
+2k4
+− 1024 c5
+1c2 k5 − 3232 c4
+1c2
+2k5 − 4488 c3
+1c3
+2k5 − 3232 c2
+1c4
+2k5 − 1024 c1c5
+2k5 + 32 c3
+1c2 k6 + 61 c2
+1c2
+2k6
++ 32 c1c3
+2k6,
+R2 = 1152 c8
+1c2
+2 + 5832 c7
+1c3
+2 + 12960 c6
+1c4
+2 + 16560 c5
+1c5
+2 + 12960 c4
+1c6
+2 + 5832 c3
+1c7
+2 + 1152 c2
+1c8
+2 + 1024 c8
+1k
++ 3584 c7
+1c2 k + 5920 c6
+1c2
+2k + 6224 c5
+1c3
+2k + 5836 c4
+1c4
+2k + 6224 c3
+1c5
+2k + 5920 c2
+1c6
+2k + 3584 c1c7
+2k
++ 1024 c8
+2k + 512 c5
+1c2 k2 + 1616 c4
+1c2
+2k2 + 2244 c3
+1c3
+2k2 + 1616 c2
+1c4
+2k2 + 512 c1c5
+2k2 − 32 c3
+1c2 k3
+− 61 c2
+1c2
+2k3 − 32 c1c3
+2k3,
+R3 = − 209715200000 c12
+1 c8
+2 + 838860800000 c11
+1 c9
+2 − 1258291200000 c10
+1 c10
+2 + 838860800000 c9
+1c11
+2
+− 209715200000 c8
+1c12
+2 + 1160950579200 c13
+1 c5
+2k − 5170397184000 c12
+1 c6
+2k + 9284105011200 c11
+1 c7
+2k
+− 9178054656000 c10
+1 c8
+2k + 7806792499200 c9
+1c9
+2k − 9178054656000 c8
+1c10
+2 k + 9284105011200 c7
+1c11
+2 k
+− 5170397184000 c6
+1c12
+2 k + 1160950579200 c5
+1c13
+2 k + 626913312768 c13
+1 c3
+2k2 − 1827529703424 c12
+1 c4
+2k2
++ 6377496477696 c11
+1 c5
+2k2 − 24562717922304 c10
+1 c6
+2k2 + 56911413825536 c9
+1c7
+2k2
+22
+
+− 74841436780544 c8
+1c8
+2k2 + 56911413825536 c7
+1c9
+2k2 − 24562717922304 c6
+1c10
+2 k2
++ 6377496477696 c5
+1c11
+2 k2 − 1827529703424 c4
+1c12
+2 k2 + 626913312768 c3
+1c13
+2 k2 − 117546246144 c12
+1 c2
+2k3
++ 2268751389696 c11
+1 c3
+2k3 − 8446241806848 c10
+1 c4
+2k3 + 13848228389376 c9
+1c5
+2k3 − 12871123435008 c8
+1c6
+2k3
++ 10570707526656 c7
+1c7
+2k3 − 12871123435008 c6
+1c8
+2k3 + 13848228389376 c5
+1c9
+2k3
+− 8446241806848 c4
+1c10
+2 k3 + 2268751389696 c3
+1c11
+2 k3 − 117546246144 c2
+1c12
+2 k3 + 7346640384 c11
+1 c2 k4
++ 23872802112 c10
+1 c2
+2k4 − 79144786368 c9
+1c3
+2k4 − 389232360000 c8
+1c4
+2k4 + 1762366805056 c7
+1c5
+2k4
+− 2639431381760 c6
+1c6
+2k4 + 1762366805056 c5
+1c7
+2k4 − 389232360000 c4
+1c8
+2k4 − 79144786368 c3
+1c9
+2k4
++ 23872802112 c2
+1c10
+2 k4 + 7346640384 c1c11
+2 k4 − 153055008 c10
+1 k5 + 444048480 c9
+1c2 k5
+− 2281361760 c8
+1c2
+2k5 − 6359031360 c7
+1c3
+2k5 + 33853070112 c6
+1c4
+2k5 − 51945109632 c5
+1c5
+2k5
++ 33853070112 c4
+1c6
+2k5 − 6359031360 c3
+1c7
+2k5 − 2281361760 c2
+1c8
+2k5 + 444048480 c1c9
+2k5
+− 153055008 c10
+2 k5 + 36636624 c7
+1c2 k6 − 65578896 c6
+1c2
+2k6 + 239834412 c5
+1c3
+2k6 − 377249916 c4
+1c4
+2k6
++ 239834412 c3
+1c5
+2k6 − 65578896 c2
+1c6
+2k6 + 36636624c1c7
+2k6 − 669222 c5
+1c2 k7 + 1023534 c4
+1c2
+2k7
+− 951468 c3
+1c3
+2k7 + 1023534 c2
+1c4
+2k7 − 669222 c1c5
+2k7 + 2187 c3
+1c2 k8 − 4031 c2
+1c2
+2k8 + 2187 c1c3
+2k8,
+R4 = 17714700 c10
+1 c2
+2 − 84798900 c9
+1c3
+2 + 166819500 c8
+1c4
+2 ��� 187523100 c7
+1c5
+2 + 175575600 c6
+1c6
+2 − 187523100 c5
+1c7
+2
++ 166819500 c4
+1c8
+2 − 84798900 c3
+1c9
+2 + 17714700 c2
+1c10
+2 + 19131876 c10
+1 k − 55506060 c9
+1c2 k
++ 70441812 c8
+1c2
+2k − 70683840 c7
+1c3
+2k + 106503012 c6
+1c4
+2k − 136123200 c5
+1c5
+2k + 106503012 c4
+1c6
+2k
+− 70683840 c3
+1c7
+2k + 70441812 c2
+1c8
+2k − 55506060 c1c9
+2k + 19131876 c10
+2 k − 9159156 c7
+1c2 k2
++ 23480604 c6
+1c2
+2k2 − 24625107 c5
+1c3
+2k2 + 19286271 c4
+1c4
+2k2 − 24625107 c3
+1c5
+2k2 + 23480604 c2
+1c6
+2k2
+− 9159156 c1c7
+2k2 + 334611 c5
+1c2 k3 − 511767 c4
+1c2
+2k3 + 475734 c3
+1c3
+2k3 − 511767 c2
+1c4
+2k3 + 334611 c1c5
+2k3
+− 2187 c3
+1c2 k4 + 4031 c2
+1c2
+2k4 − 2187 c1c3
+2k4,
+A1 = 243 c2
+1 + 352 c1c2 − 9 k,
+A2 = − 8000 c5
+1c3
+2 + 19683 c6
+1k − 17496 c5
+1c2 k + 3024 c4
+1c2
+2k + 1728 c3
+1c3
+2k − 2187 c4
+1k2 + 3564 c3
+1c2 k2
+− 432 c2
+1c2
+2k2 + 81 c2
+1k3 + 36 c1c2 k3 − k4,
+A3 = 12754584 c7
+1 − 12171384 c6
+1c2 + 3708504 c5
+1c2
+2 + 84096 c4
+1c3
+2 + 2519424 c3
+1c4
+2 − 72171 c5
+1k − 3576744 c4
+1c2 k
++ 5126856 c3
+1c2
+2k − 629856 c2
+1c3
+2k − 25272 c3
+1k2 + 98966 c2
+1c2 k2 + 52488 c1c2
+2k2 + 387 c1 k3 − 1458 c2 k3.
+Acknowledgements
+The authors wish to thank Dr. Li Su for the beneficial discussions. The authors are grateful to the anonymous
+referees for their helpful comments.
+This work has been supported by Philosophy and Social Science Foundation of Guangdong (Grant No.
+GD21CLJ01), Natural Science Foundation of Anhui Province (Grant No. 2008085QA09), University Natural
+Science Research Project of Anhui Province (Grant No. KJ2021A0482), Major Research and Cultivation
+Project of Dongguan City University (Grant No. 2021YZDYB04Z).
+References
+[1] A. Agliari, A. Naimzada, and N. Pecora. Nonlinear dynamics of a Cournot duopoly game with differ-
+entiated products. Applied Mathematics and Computation, 281:1–15, 2016.
+[2] E. Ahmed, H. Agiza, and S. Hassan. On modifications of Puu’s dynamical duopoly. Chaos, Solitons &
+Fractals, 11(7):1025–1028, 2000.
+[3] E. Ahmed, A. Elsadany, and T. Puu. On Bertrand duopoly game with differentiated goods. Applied
+Mathematics and Computation, 251:169–179, 2015.
+[4] S. Askar. The rise of complex phenomena in Cournot duopoly games due to demand functions without
+inflection points. Communications in Nonlinear Science and Numerical Simulation, 19(6):1918–1925,
+2014.
+23
+
+[5] P. Aubry and M. Moreno Maza. Triangular sets for solving polynomial systems: a comparative imple-
+mentation of four methods. Journal of Symbolic Computation, 28(1–2):125–154, 1999.
+[6] J. Bertrand. Review of ‘Th´eorie Math´ematique de la Richesse Sociale’ and ‘Recherches sur les Principes
+Math´ematiques de la Richesse’. Journal des Savants, pages 499–508, 1883.
+[7] G. I. Bischi, A. Naimzada, and L. Sbragia. Oligopoly games with local monopolistic approximation.
+Journal of Economic Behavior & Organization, 62(3):371–388, 2007.
+[8] S. Brianzoni, L. Gori, and E. Michetti. Dynamics of a Bertrand duopoly with differentiated products
+and nonlinear costs: analysis, comparisons and new evidences. Chaos, Solitons & Fractals, 79:191–203,
+2015.
+[9] J. S. C´anovas and M. Mu˜noz-Guillermo. On the dynamics of Kopel’s Cournot duopoly model. Applied
+Mathematics and Computation, 330:292–306, 2018.
+[10] F. Cavalli, A. Naimzada, and F. Tramontana. Nonlinear dynamics and global analysis of a heterogeneous
+Cournot duopoly with a local monopolistic approach versus a gradient rule with endogenous reactivity.
+Communications in Nonlinear Science and Numerical Simulation, 23(1-3):245–262, 2015.
+[11] G. E. Collins and H. Hong.
+Partial cylindrical algebraic decomposition for quantifier elimination.
+Journal of Symbolic Computation, 12(3):299–328, 1991.
+[12] A. A. Cournot. Recherches sur les Principes Math´ematiques de la Th´eorie des Richesses. L. Hachette,
+Paris, 1838.
+[13] A. A. Elsadany. Dynamics of a Cournot duopoly game with bounded rationality based on relative profit
+maximization. Applied Mathematics and Computation, 294:253–263, 2017.
+[14] L. Fanti and L. Gori. The dynamics of a differentiated duopoly with quantity competition. Economic
+Modelling, 29(2):421–427, 2012.
+[15] L. Fanti, L. Gori, C. Mammana, and E. Michetti. The dynamics of a Bertrand duopoly with differ-
+entiated products: synchronization, intermittency and global dynamics. Chaos, Solitons & Fractals,
+52:73–86, 2013.
+[16] F. M. Fisher. The stability of the Cournot oligopoly solution: the effects of speeds of adjustment and
+increasing marginal costs. The Review of Economic Studies, 28(2):125, 1961.
+[17] L. Gori and M. Sodini. Price competition in a nonlinear differentiated duopoly. Chaos, Solitons &
+Fractals, 104:557–567, 2017.
+[18] E. Jury, L. Stark, and V. Krishnan. Inners and stability of dynamic systems. IEEE Transactions on
+Systems, Man, and Cybernetics, (10):724–725, 1976.
+[19] M. Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic
+varieties. Journal of Symbolic Computation, 15(2):143–167, 1993.
+[20] M. Kopel. Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons &
+Fractals, 7(12):2031–2048, 1996.
+[21] B. Li, Q. He, and R. Chen. Neimark-Sacker bifurcation and the generate cases of Kopel oligopoly model
+with different adjustment speed. Advances in Difference Equations, 2020(1):1–18, 2020.
+[22] B. Li, H. Liang, L. Shi, and Q. He. Complex dynamics of Kopel model with nonsymmetric response
+between oligopolists. Chaos, Solitons & Fractals, 156:111860, 2022.
+[23] X. Li, C. Mou, and D. Wang. Decomposing polynomial sets into simple sets over finite fields: the
+zero-dimensional case. Computers & Mathematics with Applications, 60(11):2983–2997, 2010.
+[24] X. Li and L. Su. A heterogeneous duopoly game under an isoelastic demand and diseconomies of scale.
+Fractal and Fractional, 6(8):459, 2022.
+24
+
+[25] X. Li and D. Wang. Computing equilibria of semi-algebraic economies using triangular decomposition
+and real solution classification. Journal of Mathematical Economics, 54:48–58, 2014.
+[26] M. C. L´opez and R. A. Naylor. The Cournot–Bertrand profit differential: a reversal result in a differ-
+entiated duopoly with wage bargaining. European Economic Review, 48(3):681–696, 2004.
+[27] J. Ma and Z. Guo. The influence of information on the stability of a dynamic Bertrand game. Com-
+munications in Nonlinear Science and Numerical Simulation, 30(1-3):32–44, 2016.
+[28] A. Matsumoto, Y. Nonaka, and F. Szidarovszky. Nonlinear dynamics and adjunct profits in two bound-
+edly rational models of monopoly. Communications in Nonlinear Science and Numerical Simulation,
+116:106868, 2022.
+[29] M. McManus and R. E. Quandt. Comments on the stability of the Cournot oligipoly model. The
+Review of Economic Studies, 28(2):136–139, 1961.
+[30] B. Mishra. Algorithmic Algebra. Springer-Verlag, New York, 1993.
+[31] A. Naimzada and F. Tramontana.
+Controlling chaos through local knowledge.
+Chaos, Solitons &
+Fractals, 42(4):2439–2449, 2009.
+[32] A. Naimzada and F. Tramontana.
+Dynamic properties of a Cournot–Bertrand duopoly game with
+differentiated products. Economic Modelling, 29(4):1436–1439, 2012.
+[33] T. Puu. Chaos in duopoly pricing. Chaos, Solitons & Fractals, 1(6):573–581, 1991.
+[34] O. Shy. Industrial Organization: Theory and Applications. MIT Press, Cambridge, 1995.
+[35] N. Singh and X. Vives. Price and Quantity Competition in a Differentiated Duopoly. The RAND
+Journal of Economics, 15(4):546–554, 1984.
+[36] D. Wang.
+Computing triangular systems and regular systems.
+Journal of Symbolic Computation,
+30(2):221–236, 2000.
+[37] W.-T. Wu.
+Basic principles of mechanical theorem proving in elementary geometries.
+Journal of
+Automated Reasoning, 2(3):221–252, 1986.
+[38] L. Yang, X. Hou, and B. Xia. A complete algorithm for automated discovering of a class of inequality-
+type theorems. Science in China Series F, 44:33–49, 2001.
+[39] A. Zhang and Y. Zhang. Stability of a Cournot-Nash equilibrium: the multiproduct case. Journal of
+Mathematical Economics, 26(4):441–462, 1996.
+[40] J. Zhang, Q. Da, and Y. Wang. The dynamics of Bertrand model with bounded rationality. Chaos,
+Solitons & Fractals, 39(5):2048–2055, 2009.
+25
+

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+arXiv:2301.11984v1  [eess.SY]  27 Jan 2023
+1
+Dual Control of Exploration and Exploitation for
+Self-Optimisation Control in Uncertain
+Environments
+Zhongguo Li, Member, IEEE, Wen-Hua Chen, Fellow, IEEE, Jun Yang, Fellow, IEEE
+Yunda Yan, Member, IEEE
+Abstract—This paper develops a dual control framework for
+exploration and exploitation (DCEE) to solve a self-optimisation
+problem in unknown and uncertain environment. In general,
+there is a fundamental conflict between tracking an unknown
+optimal operational condition and parameter identification. Dif-
+ferent from existing adaptive control methods, the proposed
+DCEE does not need to introduce additional perturbation signals,
+since it naturally embraces an exploration effect to actively
+probe the uncertain environment to reduce belief uncertainty. An
+ensemble based multi-estimator approach is developed to learn
+the environmental parameters and in the meanwhile quantify the
+estimation uncertainty in real time. The control action is devised
+with dual effects, which not only minimises the tracking error
+between the current state and the believed unknown optimal
+operational condition but also reduces belief uncertainty by
+actively exploring the environment. Formal properties of the
+proposed DCEE framework like convergence are established. A
+numerical example is used to validate the effectiveness of the
+proposed DCEE. Simulation results for maximum power point
+tracking are provided to further demonstrate the potential of
+this new framework in real world applications.
+Index Terms—Dual control, self-optimisation control, active
+learning, exploration and exploitation, adaptation and control.
+I. INTRODUCTION
+Traditionally, adaptive control algorithms are mostly de-
+signed for either regulation problems with known setpoints or
+tracking problems with known reference trajectories. In many
+applications, setpoints or references are usually dependent
+on unknown or changing environment parameters, and thus
+cannot be pre-specified in advance. Operating a system at
+optimal condition is strongly desirable for best profit, pro-
+ductivity or efficiency, but it can be particularly challenging
+in an unknown or changing environment due to the presence
+of uncertainties, disturbances and noises. Typical examples
+include anti-lock braking systems to maintain maximal friction
+under various unknown road surfaces and vehicle conditions
+This work was supported by the UK Engineering and Physical Sciences
+Research Council (EPSRC) Established Career Fellowship “Goal-Oriented
+Control Systems: Disturbance, Uncertainty and Constraints” under the grant
+number EP/T005734/1.
+Z. Li is with Department of Computer Science, University College London,
+London, WC1E 6BT, U.K. (email: [email protected]).
+W.-H. Chen and J. Yang are with Department of Aeronautical and Automo-
+tive Engineering, Loughborough University, Loughborough, LE11 3TU, U.K.
+(emails: [email protected]; [email protected]).
+Y.
+Yan
+is
+with
+School
+of
+Engineering
+and
+Sustainable
+Devel-
+opment,
+De
+Montfort
+University,
+Leicester,
+LE1
+9BH,
+U.K.
+(email:
+[email protected]).
+[1], maximum power point tracking to continuously deliver the
+highest possible power to the load in presence of variations in
+environments [2], [3].
+As a classic control problem with a wide range of applica-
+tions, early solution for static optimal operation can be traced
+as far back as 1922 [4], [5]. It was popular in 1950s and 1960s,
+and regained significant attention since 2000s due to a solid
+theoretical foundation established for the stability and per-
+formance in [6], [7]. Several approaches have been proposed
+under different names including self-optimisation control [8],
+extremum seeking control [6], [9] and hill-climbing systems
+[10]. The goal of self-optimisation control is to keep the
+system operating at a setpoint that optimises a performance
+function dependent upon unknown or changing environment
+parameters, despite uncertainties, disturbances and noises.
+Since the optimal operation is unknown and possibly changes
+during the operation, a control system must be able to adapt
+to unknown or changing environments, for example, by means
+of learning, adaptation and action through limited interactions
+between the system and its operational environment. Then,
+the control system devises possible strategies to track the
+estimated setpoints or references based on its perceived en-
+vironment knowledge and the level of confidence.
+Generally speaking, there are dual objectives in a self-
+optimisation control problem in an unknown and uncertain
+environment: parameter identification and optimality tracking.
+Quite often, the dual objectives are conflicting in the sense
+that new observations do not provide sufficient information
+for identifying the unknown parameters when the system state
+settles to some local optimal solutions. This phenomenon
+widely exists in adaptive extremum seeking when an extreme
+searching algorithm converges to its local optimal solution, the
+identifiability will naturally loss due to the lack of persistent
+excitation (PE). As a trade-off, dither perturbations are intro-
+duced on purpose to sustain the identifiability, but such dithers
+inevitably deteriorate the tracking performance. Various ap-
+proaches have been proposed to design the dither signals, e.g.,
+sinusoidal perturbations [6], [11], stochastic perturbations [12],
+[13] and decaying perturbations [14]. However, they are usu-
+ally pre-specified, and thereby cannot make online adjustments
+according to real-time inference performance. In other words,
+active learning cannot be embedded, that is, actively generate
+data for the purpose of learning.
+This paper proposes a new approach to self-optimisation
+control by embedding active learning from a new perspective:
+
+2
+dual control of exploration and exploitation (DCEE). DCEE
+was originally proposed in [15] for autonomous search of
+sources of atmospheric release where the source location
+and other environmental factors are unknown. To realise
+autonomous search, it proposes each move of the robotic
+agent shall have dual effects: driving the agent towards the
+believed location of the source (exploitation) and probing the
+environment to reduce the level of uncertainty of the current
+belief (exploration). An optimal autonomous search strategy is
+realised by optimally trading-off these two effects. We argue
+in this paper that DCEE is actually applicable to a much wider
+range of systems that operate in an unknown or uncertain
+environment without well-defined control specifications (e.g.
+the reward or cost functions are unknown). We present a new
+self-optimisation control framework by extending DCEE from
+a specific autonomous search application to a general design
+approach for achieving or maintaining optimal operation in an
+unknown environment.
+The contribution of this paper is of twofold. On one side,
+for self-optimisation control problems, we propose a new
+and systematic framework which is able to actively probe
+the environment to reduce the level of uncertainty through
+active learning. There is no need to artificially introduce
+perturbation as in the current extremum seeking control. It
+also provides an optimal transition from any initial operation
+condition to acquire the unknown optimal operation condition
+in terms of a reformulated objective conditional upon current
+knowledge and future predicted information. By formulating
+the self-optimisation control in this framework, it enables to
+establish proven properties by getting access to a wide range of
+theoretic tools in control theory such as parameter adaptation
+and optimal control. On the other side, we generalise and
+extend the DCEE concept from a specific application, where
+specific system dynamics, reward function and properties are
+considered, to a general control system problem. A systematic
+design procedure for general descriptions of the system and
+control objectives is presented. We show that DCEE provides a
+powerful and promising framework to design control systems
+operating in an uncertain environment, which is an important
+feature of autonomous systems.
+Compared
+with
+all
+the
+existing
+schemes
+for
+self-
+optimisation control, our approach is most related to the work
+where the model based approach is adopted and the uncertainty
+of the objective or system dynamics are parameterised by
+uncertain parameters [6], [8], [9], [16], [17]. There are three
+main features in the new DCEE based self-optimisation control
+framework, detailed as follows.
+1) It is developed through an optimal control approach,
+which is able to achieve best transition from any admis-
+sible initial operation condition to the optimal operation
+condition in terms of a reformulated objective.
+2) It embeds an active learning effect allowing the system
+to actively explore the unknown environment to reduce
+the level of uncertainty. Instead of using computationally
+expensive particle filters in information-driven methods,
+this paper develops an efficient multi-estimator based en-
+semble approach to quantify the estimation uncertainty
+online, based on which the controller effectively trades
+off between exploration and exploitation to balance the
+dual objectives of identification and tracking.
+3) Different from all the existing schemes where probing
+effect is artificially introduced or inserted (usually by
+means of dithers and perturbations), the probing effect
+naturally occurs depending on the confidence of the es-
+timation by assembling the outcomes of these individual
+estimators.
+The rest of this paper is organised as follows. In Sec-
+tion II, we formulate the self-optimisation control problem and
+demonstrate the dual effects embedded in the new formulation.
+In Section III, an active learning based ensemble approach is
+developed for unknown environment acquisition and then a
+dual controller for exploration and exploitation is designed
+to achieve optimal trade-off between parameter identification
+and optimality tracking for a special single integrator system.
+Section IV extends DCEE to general linear systems and formal
+properties of the proposed self-optimisation control method are
+established. Section V demonstrates the effectiveness of the
+proposed algorithm using a numerical example. Section VI
+formulates maximum power point tracking (MPPT) problem
+as a self-optimisation control problem and compares the pro-
+posed algorithm with other existing approaches. Section VII
+concludes this paper.
+II. PROBLEM STATEMENT
+In this section, we elaborate the dual effects embedded
+in the reformulated self-optimisation control problem. Then,
+an ensemble active learning based approach is introduced to
+realise efficient parameter adaptation and assess the estimation
+performance.
+A. Dual Control Reformulation
+Consider a reward function for a system operating in an
+unknown environment
+J(θ∗, y) = φT(y)θ∗
+(1)
+where θ∗ ⊂ Rm is unknown, depending on the operational
+environment, y ∈ Rq is the system output, and φ(y) ∈ Rm
+is the basis function of the reward function. In other words,
+the reward function is parameterised by unknown θ∗. Without
+loss of generality, it is assumed the the optimal condition is
+achieved at the maximum of J. A self-optimisation control
+is designed to automatically drive the system to the unknown
+operational condition, maintain there despite disturbances and
+automatically adjust the optimal operation condition accord-
+ingly when the operational environment changes.
+The system dynamics under concern are described by
+x(k + 1) = Ax(k) + Bu(k)
+y(k) = Cx(k)
+(2)
+where x(k) ∈ Rn, u(k) ∈ Rp and y(k) ∈ Rq are system state,
+control input and output, respectively, and A ∈ Rn×n, B ∈
+Rn×p, C ∈ Rq×n are constant matrices. Suppose that at each
+time, the system output and the reward J(k) can be measured
+or derived subject to measurement noise v(k). We have
+z(k) = [x(k); y(k); J(k) + v(k)]
+(3)
+
+3
+and the information state is denoted as
+Ik = [u(k − 1); z(k)]
+(4)
+All the measurement up to the current time k is given by
+Ik = [I0, I1, . . . , Ik]
+(5)
+with I0 = [z(0)].
+There are two ways to formulate this problem using the dual
+control for exploration and exploitation (DCEE) concept. The
+first approach is similar to extremum seeking control [9], [16]
+aiming to select the control such that the reward function is
+maximised with all the information up to now including the
+prior and all the measurements
+max
+u(k)∈Rp Eθ,Ik+1|k{J(θ, y(k + 1|k))|Ik+1|k}
+(6)
+subject to the system dynamics (2), where Ik+1|k
+=
+[Ik, Ik+1|k] with Ik+1|k = [u(k), z(k + 1|k)]. z(k + 1|k)
+consists of the predicted output y(k + 1|k) and the predicted
+reward function under the control u(k).
+Another approach is to drive the system output to the
+unknown optimal condition directly, which is closer to the
+classic self-optimisation control [8]. Since the optimal oper-
+ation condition is unknown, the best one can do is to drive
+the system to the best estimation of the optimal operation
+condition with all the information up to now. This can be
+formulated as
+min
+u(k)∈Rp E{(y(k + 1|k) − r∗)T(y(k + 1|k) − r∗)|Ik+1|k} (7)
+where r∗ = l(θ∗) denotes the predicted optimal operational
+condition conditional upon Ik+1|k, which is a function of
+the environment parameter θ∗. In realm of self-optimisation
+control, it is often required that the mapping l(θ) is a smooth
+function of θ and r∗ = l(θ∗) is a unique optimum of the
+objective function [6].
+These two problems have been solved previously in au-
+tonomous search [15], [18]. The research question is how to
+extend these results from this specific application to general
+self-optimisation control problems. In this paper, we will focus
+our attention on the latter formulation in (7), which is related to
+the operational condition determined by unknown environment
+parameters.
+Before proceeding further, we demonstrate that the control
+input u(k) obtained by minimising (7) naturally carries dual
+effects corresponding to exploration and exploitation, respec-
+tively. Intuitively, the control input u(k) will influence the
+future system state y(k+1|k) via the system dynamics (2), and
+at the same time affect the future information to be collected
+Ik+1|k via the reward function in (1) from the environment
+subject to uncertainties.
+We define the predicted nominal operational condition as
+¯r(k + 1|k) = E
+�
+r∗(k + 1|k)|Ik+1|k
+�
+(8)
+based on which the prediction error conditional on Ik+1|k can
+be written as
+˜r(k + 1|k) = r∗(k + 1|k) − ¯r(k + 1|k).
+(9)
+Expanding (7) and substituting (8) and (9) into (7), we have
+E
+�
+∥y(k + 1|k) − ¯r(k + 1|k) − ˜r(k + 1|k)∥2|Ik+1|k
+�
+= E
+�
+∥y(k + 1|k) − ¯r(k + 1|k)∥2|Ik+1|k
+�
+− 2 E
+�
+(y(k + 1|k) − ¯r(k + 1|k))T˜r(k + 1|k)|Ik+1|k
+�
++ E
+�
+∥˜r(k + 1|k)∥2|Ik+1|k
+�
+.
+(10)
+It follows from the definition of ˜r(k + 1|k) in (9) that
+E
+�
+˜r(k + 1|k)|Ik+1|k
+�
+= 0. Thus, by further noting that
+y(k + 1|k) and ¯r(k + 1|k) are deterministic, the cross term in
+(10) equals to zero, yielding
+D(u(k)) := E
+�
+∥y(k + 1|k) − ¯r(k + 1|k)∥2|Ik+1|k
+�
++ E
+�
+∥˜r(k + 1|k)∥2|Ik+1|k
+�
+.
+(11)
+Remark 1: The objective function in (11) exhibits dual
+effects. Minimising the first term in (11) drives the system
+state to estimated nominal value, which corresponds to the
+exploitation effect. In control terminology, it can be understood
+as tracking a nominal reference, thus also referred to as
+optimality tracking. The second term characterises the level
+of uncertainty (variance) associated with the predicted optimal
+operational condition, which is related to the exploration
+effect. According to the classic dual control concept [19],
+[20], a control input is said to have dual effects if it can
+affect at least one rth-order central moment of a state variable
+(r > 1), in addition to its effect on the state. In fact,
+the dual control framework developed in this paper is a
+generalisation of the classic one [19] in the sense that our
+formulation deals with not only system uncertainty but also
+environment uncertainty (the operational condition r∗ = l(θ∗)
+is determined by the environment parameters θ∗). This subtle
+difference endows the system with capability of exploring the
+operational environment and in the meanwhile exploiting its
+current belief. Recently, DCEE has demonstrated superior and
+promising performance in autonomous search [15], [21].
+Remark 2: According to [22], the level of autonomy can be
+measured in terms of the set of goals that the system is able to
+accomplish subject to a set of uncertainties. As a result, it is
+required that the system can exploit its available knowledge to
+accomplish the goals, and at the same time it should be able to
+actively explore the operational environment to reduce knowl-
+edge uncertainty. Effective trading-off between exploration and
+exploitation has been a long standing issue, particularly in
+artificial intelligence, control and decision-making in complex
+and uncertain environment. In control society, some recent
+works explicitly introduce trade-off coefficients to incorporate
+the exploration terms into model predictive control problems,
+e.g., [17], [23]. This inevitably incurs tedious efforts in tuning
+the coefficients to balance exploration and exploitation. In
+view of the derivation of (11), it is clear that the dual effects
+in DCEE are naturally embedded, since they are derived from
+a physically meaningful value function in (7).
+B. Ensemble based Active Learning
+Efficient gradient descent algorithms can be used to es-
+timate the unknown parameters. The performance of single
+estimator based optimisation algorithm is quite poor, due to
+
+4
+noisy measurement and nonlinear modelling (see examples in
+autonomous search [18], [24]). Recently, the ensemble-based
+approximation in machine learning community has demon-
+strated great success with tractable computational load [25],
+[26]. In this paper, we develop a multi-estimator based learning
+method for parameter adaptation, which shows comparable
+performance as particle filter using much less computational
+resources in autonomous search application [18].
+Considering an ensemble of N estimators, the dual formu-
+lation in (11) becomes
+min
+u(k)∈Rp D(u) = ∥y(k + 1|k) − ¯r(k + 1|k)∥2 + P(k + 1|k)
+subject to x(k + 1|k) = Ax(k) + Bu(k)
+y(k + 1|k) = Cx(k + 1|k)
+(12)
+where the nominal estimate and variance of the estimated
+optimal condition are drawn from the ensemble, i.e.,
+¯r(k + 1|k) = 1
+N
+N
+�
+i=1
+ri(k + 1|k) = 1
+N
+N
+�
+i=1
+l(θi(k + 1|k))
+(13)
+P(k + 1|k) = 1
+N
+N
+�
+i=1
+(ri(k + 1|k) − ¯ri(k + 1|k))T
+× (ri(k + 1|k) − ¯ri(k + 1|k))
+(14)
+where the subscript i ∈ N denotes the index of the estimators,
+with N representing the set of the ensemble. Note that
+the relationship between the predicted optimal condition and
+the unknown parameter, i.e., ri
+k+1|k = l(θi
+k+1|k), is usually
+known. For example, in autonomous search application, θ∗
+is composed of the unknown source location and other envi-
+ronment parameters, like wind direction and wind speed. The
+optimal operation condition r∗ in autonomous search is the
+source location, i.e., part of θ∗, which serves as a tracking
+reference for the search agent.
+In order to estimate the unknown parameter θ∗, we apply a
+gradient-descent regression method [27], designed as
+θi(k) =θi(k − 1) − ηiφ(y(k − 1))
+×
+�
+φ(y(k − 1))Tθi(k − 1) − J(k − 1)
+�
+, ∀i ∈ N.
+(15)
+where ηi > 0 is the learning rate of the ith estimator; J(k−1)
+denotes the observed reward with measurement noise in (3) at
+y(k − 1); and θ(k) denotes the estimate of unknown reward
+parameter θ∗. The estimators are randomly initialised or they
+can be initialised according to a priori pdfs of the unknown
+parameters if available. Denote the estimation error as ˜θi(k) =
+θi(k) − θ∗. Then, by noting J(k − 1) = φ(y(k − 1))Tθ∗ +
+v(k − 1), we have
+˜θi(k) =
+���
+Im − ηiφ(y(k − 1))φ(y(k − 1))T� ˜θi(k − 1)
+− ηiφ(y(k − 1))v(k), ∀i ∈ N.
+(16)
+Denoting
+the
+extended
+parameter
+error
+as
+˜Θ(k)
+=
+col{˜θ1(k), . . . , ˜θN(k)}, where col{·} denotes a column vector
+formed by stacking the elements on top of each other, (16)
+can be written in a compact form as
+˜Θ(k) =
+�
+IN ⊗
+�
+Im − ηiφ(y(k − 1))φ(y(k − 1))T� �˜Θ(k − 1)
+−
+�
+IN ⊗ ηiφ(y(k − 1))
+�
+(1N ⊗ v(k − 1)).
+(17)
+In an ensemble-based adaptation, we take their average
+as the current estimation of the unknown parameters. Thus,
+averaging (16), we have
+˜Θav(k) = 1
+N (1T
+N ⊗ Im)˜Θ(k)
+= 1
+N (1T
+N ⊗ Im)
+�
+IN ⊗ (Im − ηφφT)
+� ˜Θ(k − 1)
+− 1
+N (1T
+N ⊗ Im)
+�
+IN ⊗ ηφ
+�
+(1N ⊗ v(k − 1))
+= 1
+N (1T
+N ⊗ Im)˜Θ(k − 1) − 1
+N (1T
+N ⊗ ηφφT)˜Θ(k − 1)
+− ηφv(k − 1).
+(18)
+Remark 3: An important observation is that even though we
+have the same regressor φ at one time instant, its excitation im-
+pact to each estimator will be different since φφT˜θi ̸= φφT˜θj,
+∀i ̸= j, almost surely. Due to the introduction of parameter
+extension by multiple estimators, at any time instant the aver-
+age estimation can always be excited when there are sufficient
+estimators. In addition, by introducing a group of estimators,
+it is possible to evaluate and make full use of the estimation
+uncertainty by sampling the outcomes of the ensemble in an
+online manner, which is proved to be crucial in DCEE [15]
+as we will discuss in the sequel. Another desirable feature of
+the ensemble approach is its resilience to measurement noises.
+In view of the last term in (18), instantaneous noises will be
+averaged out under multiple estimators such that the overall
+performance of the ensemble can be improved.
+III. DCEE FOR SINGLE INTEGRATOR
+A. Algorithm Development
+In high-level decision-making, system behaviours are usu-
+ally simplified as single integrators by ignoring low-level
+dynamics. In this paper, we begin with DCEE for this special
+case
+y(k + 1) = y(k) + u(k).
+(19)
+For general linear systems, we will use this as an internal
+reference generator, as will be shown later in Section IV.
+With the estimated environment parameter in (15), the dual
+controller can be designed as
+y(k + 1) = y(k) + u(k)
+u(k) = −δk
+�
+∇yC(k + 1|k) + ∇yP(k + 1|k)
+�
+(20)
+where C(k + 1|k) = ∥y(k) − ¯r(k + 1|k)∥2 denotes the
+exploitation term, and P(k+1|k) is the exploration term in the
+dual objective (12). To obtain the future mean and covariance,
+we utilise the classic principles in extended Kalman filter.
+According to the gradient-descent regression in (15), the
+
+5
+predicted mean of the N ensemble θi(k + 1|k), denoted as
+¯θ(k + 1|k), is given by
+¯θ(k + 1|k) = 1
+N
+N
+�
+i=1
+θi(k + 1|k)
+= 1
+N
+N
+�
+i=1
+(1m − ηiFi(k + 1|k))Tθi(k)
+(21)
+where
+Fi(k + 1|k) =[J(θi(k), y) − J(k + 1|k)]φ(y)
+(22)
+with J(k + 1|k) being the predicted future reward based
+on current belief {θi(k), ∀i ∈ N}. Note that the predicted
+future reward is noise-free as there is no influence from
+sensory devices in prediction. In this paper, we use the average
+of θi(k), ∀i ∈ N to evaluate the predicted future reward.
+Similarly, the predicted variance of the ensemble is given by
+P(k + 1|k) = trace(F (k + 1|k)TP(k|k)F (k + 1|k))
+(23)
+where
+F (k + 1|k) = col{F1(k + 1|k), . . . , FN(k + 1|k)}
+P(k|k) = cov{θi(k), ∀i ∈ N}
+= diag{(θ1(k) − ¯θ(k))(θ1(k) − ¯θ(k))T, . . . ,
+(θN(k) − ¯θ(k))(θN(k) − ¯θ(k))T}
+(24)
+where cov{·} is a covariance operator evaluating the covari-
+ance matrix of the ensemble, and diag{·} denotes a block-
+diagonal matrix by putting the elements on its main diagonal.
+Using the predicted mean ¯θi(k + 1|k) and the predicted co-
+variance P(k+1|k) of the unknown environmental parameter,
+the dual control terms in (2) can be obtained by using the
+mapping between the operational condition and the unknown
+environmental parameter, i.e., r = l(θ).
+B. Convergence Analysis
+In this section, we will examine the convergence of the
+proposed dual control algorithm, by leveraging parameter
+adaptation and optimisation techniques. To this end, we in-
+troduce some fundamental assumptions that will be used to
+facilitate the convergence analysis of the proposed dual control
+algorithm.
+Assumption 1: There exist positive constants T ∈ Z+ and
+β > 0 such that
+t+T
+�
+k=t
+[φ(y(k))][φ(y(k))]T ≥ βIm > 0, ∀t > 0.
+(25)
+Assumption 2: The measurement noise v(k) is independent
+and identically distributed with bounded variance, i.e.,
+E [v(k)] = 0
+E
+�
+∥v(k)∥2�
+≤ ̺2.
+(26)
+Assumption 3: The reward function J(θ, y) is twice dif-
+ferentiable and strictly concave on y for any θ ∈ Rm, that
+is,
+∂2J(θ, y)
+∂y2
+> 0.
+(27)
+Remark 4: Assumption 1 is a standard persistent excitation
+(PE) condition to ensure the identifiability of the unknown
+environmental parameter θ. Extensive techniques on parameter
+adaptation have been reported in the past few decades aiming
+at relaxing or fulfilling the conditions of PE [9], [27]. If we
+introduce a memory-based regressor extension to the param-
+eter adaptation algorithm in (15), the PE condition can be
+relaxed to interval excitation [27]. Assumption 2 implies that
+the noises imposed on sensory information are unbiased with
+bounded variances. Assumption 3 guarantees the existence
+and uniqueness of the optimal operational condition, i.e.,
+r∗ = l(θ∗), which is widely used in adaptive self-optimisation
+and extremum seeking control [9], [28]. Note that the mapping
+between the optimal operational condition and parameter θ can
+be obtained by solving ∂J(θ,y)
+∂y
+= 0.
+First, we examine the convergence of the gradient-descent
+regression method in (15).
+Theorem 1: Under Assumptions 1 and 2, there exists a
+constant η∗ > 0 such that, for any 0 < ηi < η∗, the estimates,
+ˆθi(k), ∀i ∈ N, converge to a bounded neighbourhood of the
+true environmental parameter θ∗. Moreover, the mean-square-
+error of the estimator is convergent and bounded by
+E ∥˜θi(k)∥2 ≤
+η2
+i L2̺2
+1 − maxj∈{1,...,k−1} ∥Ai(j)∥
+(28)
+where Ai(j) = Im − ηi[φ(y(j))][φ(y(j))]T and L denotes the
+bound of the regressor φ. Moreover, in absence of measure-
+ment noises, limk→∞ E ∥˜θi(k)∥2 = 0.
+Proof: In view of (16) and Assumption 2, the expectation of
+the estimate is given by
+E[˜θi(k)] =
+�
+Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�
+E[˜θi(k − 1)]
+∀i ∈ N.
+(29)
+According to Assumption 1, there exists a constant η∗ such
+that, for any 0 < ηi < η∗, 0 < ηi[φ(y(k−1))][φ(y(k−1))]T <
+Im. Consequently, for any 0 < ηi < η∗, we have
+0 < Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T < Im.
+(30)
+It follows from (29) that
+∥ E[˜θi(k)]∥ ≤
+��Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T��
+× ∥ E[˜θi(k − 1)]∥, ∀i ∈ N.
+(31)
+Therefore,
+∥ E[˜θi(k)]∥ ≤
+k
+�
+j=1
+��Im − ηi[φ(y(j − 1))][φ(y(j − 1))]T��
+× ∥ E[˜θi(0)]∥, ∀i ∈ N.
+(32)
+For any bounded error ˜θi(0), the expectation of the estimator
+converge to zero.
+
+6
+Moreover, the variance of the estimators can be bounded
+under Assumption 2. Taking the squared Euclidean norm of
+(16) yields
+∥˜θi(k)∥2 =
+���
+Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�˜θi(k − 1)
+��2
++ ∥ηiφ(y(k − 1))v(k)∥2
+− 2
+�
+[Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T]
+× ˜θi(k − 1)
+�
+[ηiφ(y(k − 1))v(k)] , ∀i ∈ N.
+(33)
+Applying expectation operation to (33) leads to
+E ∥˜θi(k)∥2 = E
+�� �
+Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�
+× ˜θi(k − 1)
+��2
++ E ∥ηiφ(y(k − 1))v(k)∥2, ∀i ∈ N.
+(34)
+where E [v(k)] = 0 has been used to eliminate the cross term.
+Denoting Ai(k−1) = Im −ηi[φ(y(k−1))][φ(y(k−1))]T and
+applying the variance bound in (26), we have
+E ∥˜θi(k)∥2 ≤ E
+��˜θi(k − 1)
+��2
+Ai(k−1) + η2
+i L2̺2.
+(35)
+For any 0 < ηi < η∗, the mean-square-error of the estimator
+is convergent and bounded by
+E ∥˜θi(k)∥2 ≤
+η2
+i L2̺2
+1 − maxj∈{1,...,k−1} ∥Ai(j)∥.
+(36)
+In absence of measurement noise v(k)
+=
+0, limk→∞
+E ∥˜θi(k)∥2 = 0. This completes the proof.
+■
+Remark 5: Theorem 1 establishes the convergence of the
+estimators under mild assumptions on the measurement noises
+and persistent excitation. The parameter adaptation algorithm
+together with its convergence analysis under measurement
+noises forms a new feature of this paper since existing studies
+mainly focus on noise-free scenarios [27], [29]. As having
+been discussed in Remark 4, PE is a standard and commonly-
+used condition to guarantee the convergence of parameter
+estimators. Despite significant research efforts have been dedi-
+cated to explore weak/alternative assumptions, very few result
+has been obtained (see recent survey in [27]). In the proposed
+dual controller (20), a probing effort is inherently embedded
+aiming to reduce the estimation uncertainty. Such an explo-
+ration effect from active learning is beneficial to environment
+acquisition, which has been validated in autonomous search
+application [15], [18].
+Remark 6: The proposed multi-estimator assisted ensemble
+method for environment adaptation is a hybrid approach that
+combines both model-based and model-free techniques. The
+model-based estimators are trained according to the model
+structures of the reward function in (1). A model-free ensemble
+approximation is used to estimate the mean and variance of the
+unknown environmental parameters in an online manner. It is
+widely perceived in machine learning community that model-
+based approach benefits from high learning efficiency due
+to the utilisation of model knowledge but inevitably inherits
+model biased errors; on the other hand, model-free approach
+provides a reliable way to quantify the level of estimation
+uncertainty but may incur additional computational burden.
+Recently, the hybrid method has demonstrated superior perfor-
+mance in simulation and experiment in machine learning due
+to its combined strength from both model-based and model-
+free learning [25], [26]. Theoretical guarantee on convergence
+and performance of the hybrid approach has not been well-
+established but mainly verified by extensive simulation and
+experimental results. Inspired by its recent success, we develop
+a concurrent active learning based ensemble algorithm and
+establish its formal properties in this paper.
+Denote the tracking error between current state and un-
+known optimal condition r∗ as ˜y(k) = y(k) − r∗. Then, it
+follows from (20) that
+˜y(k + 1) = ˜y(k) − δk
+�
+∇yC(k + 1|k) + ∇yP(k + 1|k)
+�
+.
+(37)
+Now, we analyse the convergence to the optimal operational
+condition.
+Theorem 2: Under Assumptions 1-3, for any 0 < ηi < η∗,
+y converges to a bounded neighbourhood of the optimal
+operational condition r∗ = l(θ∗) if there exists a step size
+δk such that 0 < 2∥[In − δkL(k)]∥2 < 1 with L(k) =
+� 1
+0 ∇2
+yC(r∗ + τ ˜y(k))dτ.
+Proof: To relate the gradient term ∇yC(k + 1|k) with ˜y(k),
+we recall the mean value theorem [30], that is, for a twice-
+differentiable function h(y) : Rm → R,
+∇h(y1) =∇h(y2) +
+�� 1
+0
+∇2h[y2 + τ(y1 − y2)]dτ
+�
+(y1 − y2),
+∀y1, y2 ∈ Rm .
+(38)
+Thus, we have
+∇yC(y(k)) =∇yC(r∗) +
+� � 1
+0
+∇2
+yC(r∗ + τ ˜y(k))dτ
+�
+˜y(k)
+(39)
+where the time stamps in C(k+1|k) have been dropped for no-
+tational convenience. Denoting L(k) =
+� 1
+0 ∇2
+yC(r∗+τ ˜y(k))dτ
+and applying ∇yC(r∗) = 0, we have
+∇yC(y(k)) = L(k)˜y(k).
+(40)
+Applying (40) to (37) results in
+˜y(k + 1) = [In − δkL(k)]˜y(k) − δk∇yP(k + 1|k).
+(41)
+To examine the boundedness of the tracking error, we take the
+Euclidean norm for both sides of (41), yielding
+∥˜y(k + 1)∥2 =∥[In − δkL(k)]˜y(k)∥2 + ∥δk∇yP(k + 1|k)∥2
+− 2δk[(In − δkL(k))˜y(k)]T∇T
+yP(k + 1|k).
+(42)
+Taking the expectation of (42) leads to
+E ∥˜y(k + 1)∥2 ≤∥[In − δkL(k)]∥2 E ∥˜y(k)∥2
++ E ∥δk∇yP(k + 1|k)∥2
++ E[−2δk∇T
+yP(k + 1|k)[(In − δkL(k))˜y(k)]].
+(43)
+The last term in (43) can be written as
+E[−2δk∇T
+yP(k + 1|k)[(In − δkL(k))˜y(k)]]
+≤ ∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + E ∥δk∇yP(k + 1|k)∥2.
+(44)
+
+7
+Therefore, substituting (44) into (43) results in
+E ∥˜y(k + 1)∥2 ≤2∥[In − δkL(k)]∥2 E ∥˜y(k)∥2
++ 2 E ∥δk∇yP(k + 1|k)∥2.
+(45)
+From Theorem 1, the estimation errors are bounded within
+E ∥˜θi(k)∥2 ≤ max
+�
+∥˜θi(0)∥2,
+η2
+i L2̺2
+1 − maxj∈{1,...,k−1} ρ(Ai(j))
+�
+.
+(46)
+As a result, 0 ≤ E ∥δk∇yP(k +1|k)∥2 ≤ µ is upper bounded,
+since it is a measure of covariance of the bounded estimators.
+Consequently, we have
+E ∥˜y(k + 1)∥2 ≤2∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + µ.
+(47)
+If there exists a step size δk such that 0 < 2∥[In−δkL(k)]∥2 <
+1, then the expected mean square of the tracking error is
+convergent. Recursively iterating (47) gives
+E ∥˜y(k + 1)∥2 ≤ ¯αk E ∥˜y(0)∥2 +
+k−1
+�
+j=0
+¯αjµ
+(48)
+where ¯α := maxj∈{1,...,k} αj with 0 < αk := 2∥[In −
+δjL(j)]∥2 < 1. Since limk→∞ ¯αk E ∥˜y(0)∥2 → 0, we have
+lim
+k→∞ E ∥y(k) − r∗∥2 ≤
+µ
+1 − ¯α.
+(49)
+This completes the proof.
+■
+Remark 7: In general, traditional adaptive control can
+be regarded as passive learning [9], [17] where parameter
+estimators are updated by accidentally collected data sam-
+ples. For example, MPC in autonomous search is targeted at
+navigating the agent to the source position, whereas during
+this pure exploitation process the estimators are updated pas-
+sively by accidentally collected concentration measurements
+from the environment [15], [31]. Recently, there are a wide
+range of engineering problems involved in balancing between
+exploration and exploitation, e.g., machine learning, control
+and decision-making in uncertain environment [20], [32]–
+[34]. In control society, related works are usually focused on
+stochastic model predictive control with active learning [17]. A
+similar concept is referred to as active reinforcement learning
+in artificial intelligence [34], [35]. Nevertheless, there is a
+critical distinction between previous works and the proposed
+DCEE framework for self-optimisation control. In existing
+dual control formulation, the probing effect is introduced to
+learn the system states or parameters (e.g. MPC with active
+learning [36] and active adaptive control [37], [38]), while
+in our formulation the probing effect is used to actively
+explore the operational environment. We believe that future
+autonomous control should be able to deal with not only
+system uncertainty but also environment uncertainty [15], [22].
+IV. DCEE FOR LINEAR SYSTEMS
+In this section, we deal with general linear systems. As
+the environment estimators are designed by information mea-
+surements, the parameter adaptation algorithm in (15) can be
+used and Theorem 1 remains valid. Now, we design a dual
+controller that regulates the system output y(k) to minimise
+the reformulated objective function defined in (12).
+The dual controller is proposed as
+u(k) = −Kx(k) + (G + KΨ)ξ(k)
+(50)
+where the optimal reference ξ(k) is generated by
+ξ(k) = ξ(k − 1) + ψ(k)
+ψ(k) = −δk
+�
+∇ξC(k + 1|k) + ∇ξP(k + 1|k)
+�
+(51)
+where G and Ψ are gain matrices obtained by solving
+(A − I)Ψ + BG = 0
+CΨ − I = 0.
+(52)
+and K is chosen such that A − BK is Schur stable as (A, B)
+is controllable. Note that ψ(k) is exactly the dual gradient
+term used in the integrator dynamics in Section III. For
+linear systems, the control input u(k) not only needs to have
+dual effects for exploration and exploitation but additionally
+requires control effort to stabilise the system dynamics as in
+(50).
+Assumption 4: The pair (A, B) is controllable, and
+rank
+�
+A − I
+B
+C
+0
+�
+= n + q.
+(53)
+Remark 8: The dual control design in (50)-(52) is partly
+inspired by conventional internal model approaches [39]. The
+solvability of (52) is guaranteed by (53), which is widely
+known as regulation equations [39]. The existence of Ψ
+ensures the existence of optimal state x∗ = Ψr∗ such that
+Cx∗ = r∗.
+Define state transformations xs(k) = Ψξ(k), us(k) =
+Gξ(k). Let ¯x(k) = x(k) − xs(k) and ¯u(k) = u(k) − us(k).
+Applying the transformation to the system dynamics (2) leads
+to
+¯x(k + 1) = x(k + 1) − xs(k + 1)
+= Ax(k) + Bu(k) − Ψ(ξ(k) + ψ(k))
+= A¯x(k) + B¯u(k) − Ψψ(k)
+e(k) = C¯x(k)
+(54)
+where (52) has been used to derive above dynamics. Applying
+the control input (50), we have the closed loop dynamics
+¯x(k + 1) = (A − BK)¯x(k) − Ψψ(k)
+e(k) = C¯x(k).
+(55)
+The following lemma can be regarded as input-to-output
+stability of the transformed dynamics (55) by viewing ψ(k)
+and e(k) as the input and output, respectively.
+Lemma 1: Let Assumptions 1–4 hold. Suppose the condi-
+tions specified in Theorems 1–2 hold. If the gain matrices G
+and Ψ are designed according to (52) and K is chosen such
+that (A − BK) is Schur stable, then
+lim sup
+k→∞
+∥e(k)∥ ≤
+1
+1 − ∥A − BK∥ lim sup
+k→∞
+∥ψ(k)∥
+(56)
+Furthermore, if lim supk→∞ ψ(k) = 0, then lim supk→∞ e(k)
+= 0.
+Proof: Putting (52) into a matrix form leads to
+� A − I
+B
+C
+0
+� � Ψ
+G
+�
+=
+� 0
+I
+�
+(57)
+
+8
+of which the solvability is guaranteed under (53) in Assump-
+tion 4 by transforming the matrix equation (57) to standard
+linear algebraic equations. For notational convenience, we
+denote Ac = A − BK and Bc = −Ψ. Then, we have
+¯x(k + 1) = Ac¯x(k) + Bcψ(k).
+(58)
+Recursively iterating (58) results in
+¯x(k) = Ak
+c ¯x(0) +
+k−1
+�
+j=0
+Ak−j−1
+c
+Bcψ(j).
+(59)
+Hence, we have
+e(k) = C¯x(k) = CAk
+c ¯x(0) −
+k−1
+�
+j=0
+Ak−j−1
+c
+ψ(j)
+(60)
+where CΨ − I = 0 has been used. Because Ac is Schur, we
+have limk→∞ CAk
+c ¯x(0) = 0.
+The convergence of reference generator (51) has been
+established in Theorem 2, and thereby ψ(k), i.e., the gradient
+of the dual controller, is bounded and converges to zero as
+k → ∞. Denoting ̟ := lim supk→∞ ∥ψ(k)∥, it can be
+obtained that, for any small constant ǫ > 0, there exists a
+positive time index ζ > 0 such that
+∥ψ(k)∥ < ̟ + ǫ, ∀k > ζ.
+(61)
+Now, the second term in (60) can be separated into two
+parts, written as
+k−1
+�
+j=0
+Ak−j−1
+c
+ψ(j) =
+ζ
+�
+j=0
+Ak−j−1
+c
+ψ(j) +
+k−1
+�
+j=ζ+1
+Ak−j−1
+c
+ψ(j).
+(62)
+Taking the Euclidean norm of (62) and invoking (61), we
+obtain
+����
+k−1
+�
+j=0
+Ak−j−1
+c
+ψ(j)
+���� =
+����
+ζ
+�
+j=0
+Ak−j−1
+c
+ψ(j)
++
+k−1
+�
+j=ζ+1
+Ak−j−1
+c
+ψ(j)
+����
+≤
+��Ak−ζ−1
+c
+��
+����
+ζ
+�
+j=0
+Aζ−j
+c
+ψ(j)
+���� + (̟ + ǫ)
+����
+k−1
+�
+j=ζ+1
+Ak−j−1
+c
+����.
+(63)
+Therefore, combining (60) and (63) leads to
+lim sup
+k→∞
+∥e(k)∥ ≤
+1
+1 − ∥Ac∥ (̟ + ε)
+(64)
+by noting that
+t−1
+�
+j=ζ+1
+∥Ac∥t−1−j = 1 − ∥Ac∥t−ζ
+1 − ∥Ac∥
+<
+1
+1 − ∥Ac∥
+(65)
+and that
+lim
+k→∞
+��Ak−ζ−1
+c
+�� = 0
+(66)
+since Ac is Schur stable. As ǫ can be set arbitrarily small, it
+follows from (64) that
+lim sup
+k→∞
+∥e(k)∥ ≤
+1
+1 − ∥Ac∥ lim sup
+k→∞
+∥ψ(k)∥.
+(67)
+This completes the proof.
+■
+Now, combining the results in Theorems 1–2 and Lemma 1,
+we are ready to establish the convergence of the self-
+optimisation control for linear systems.
+Theorem 3: Let Assumptions 1–4 hold. Suppose the
+conditions specified in Theorems 1–2 and Lemma 1 hold.
+The output y(k) of the linear system (2) converges to the
+neighbourhood of the optimum r∗, using control input (50) to-
+gether with reference generator (51). Moreover, in the absence
+of measurement noises, y(k) converges to the true optimal
+solution r∗.
+Proof: Denoting ˜x(k) = x(k) − Ψr∗, we have
+˜x(k + 1) = Ax(k) + B[−Kx(k) + (G + KΨ)ξ(k)] − Ψr∗
+= (A − BK)˜x(k) + B(G + KΨ)(ξ(k) − r∗)
+(68)
+It follows from Theorems 1–2 that ξ(k) converges to the
+neighbourhood of r∗ with bounded error. Thus, the result can
+be concluded by treating B(G + KΨ)(ξ(k) − r∗) as ψ(k) in
+Lemma 1.
+■
+Remark 9: The self-optimisation control in this paper is
+similar to the classic formulation of reinforcement learning
+in the sense that both of them are targeted to operate a
+system in an unknown and uncertain environment. There are
+two bottlenecks in widely applying reinforcement learning,
+particularly deep RL: one is a large number of trials are
+required to achieve a satisfactory performance (big data) and
+the other is its performance could significantly degrade if
+the real operational environment is different from the training
+environment (poor adaptiveness) [40]. DCEE establishes a new
+control framework to provide a promising and complementary
+method to reinforcement learning in control and robotics
+society. In fact, active learning for exploration and exploitation
+in machine intelligence can find strong evidence in human
+intelligence, which is supported by the biological principles
+in functional integration in the human brain and neuronal in-
+teractions (known as free-energy principle and active inference
+in neuroscience [41]). Interested readers are referred to [40]
+for detailed discussions.
+V. NUMERICAL EXAMPLE
+In this section, we verify the effectiveness of the proposed
+algorithm using a dedicate numerical example. Consider a
+linear system (2) with
+A =
+�
+0
+1
+2
+1
+�
+, B =
+�
+1
+1
+�
+, C =
+� 0
+1 �
+.
+(69)
+The reward function is given by
+J(θ∗, y) = 2y − θ∗y2 =
+� 2y
+−y2 � � 1
+θ∗
+�
+(70)
+where θ∗ is affected by the unknown environment. The true
+value is θ∗ = 1 but unavailable a priori. The optimal oper-
+ational condition r∗ is determined by θ∗, i.e., r∗ = l(θ∗) =
+1/θ∗ = 1.
+We assume the measurements are subject to Gaussian noise
+v(k) ∼ N(0, 2), which implies that the observations from
+environment are J(k) = J(θ∗, y(k))+ v(k). Decision-making
+
+9
+under uncertain environment with noisy measurements is of
+significant importance to promote the system intelligence.
+In order to explore the uncertain environment, the first step
+is to quantify the level of uncertainty. An ensemble based
+multi-estimator approach has been developed in previous
+sections. Now, the size of the estimator ensemble is chosen
+as N
+= 100, and each of them is randomly initialised
+according to a uniform distribution between 0 and 20, i.e.,
+θi(0) ∼ U(0, 20), ∀i = 1, 2, . . . , 100. The step sizes are set
+as ηi = 0.005 and δk = 0.5. The system is controllable
+and regulation condition in (53) is satisfied such that the gain
+matrices can be obtained as Ψ = [ 1
+3, 1]T and G = − 2
+3. The
+gain matrix K = [−1.24, 1.14] is chosen by placing the poles
+of (A − BK) at [0.4; 0.7].
+Fig. 1 shows the estimated environmental parameters. Ini-
+tially, the mean and standard deviation of the ensemble
+{θi, i = 1, . . . , 100} are 10.87 and 5.57, respectively, ran-
+domly initialised using a uniform distribution. The mean of
+the estimators converges to the true environment parameter
+θ∗ = 1, and the standard deviation among the estimators
+shrinks quickly, indicating that the estimation uncertainty
+reduces (quantified by the variance among the estimators in
+the ensemble). Despite increasing the iteration k significantly,
+the estimated parameters remain fluctuating within a small
+neighbourhood of the true value due to the presence of noisy
+measurements. Fig. 2 displays the observed rewards from the
+environment. Even though we have imposed quite significant
+noises to the measurements, the performance of the estimators
+is fairly satisfactory, which manifests the ensemble based
+active learning provides superior robustness against noises.
+Implementing the dual control in (50) not only contributes to
+enhanced parameter adaptation performance but also drives the
+system output to the optimal operational condition, as shown in
+Fig. 3. The system output approaches the optimal operational
+point r∗ = 1 as shown in Fig. 3, and the system states are
+displayed in Fig. 4. It can be verified that x∗ = Ψr∗ = [ 1
+3, 1]T.
+The tracking error is determined by the estimation error. In this
+process, there is no need to tune the weights of exploration
+and exploitation. As a principled approach, the dual controller
+in (50) is derived from a physically meaningful objective
+function, which naturally embeds balanced dual effects for
+active environment learning and optimality tracking.
+VI. APPLICATION FOR MPPT
+DCEE was originally developed to solve autonomous search
+problem in [15], which demonstrates outstanding performance
+compared with other existing approaches. In this section, we
+take the optimal control for photovoltaic (PV) systems as an
+example to illustrate that DCEE can be implemented to solve
+a much wider class of self-optimisation control problems in
+real-world applications. Extracting maximum power is a long-
+lasting pursuit in operating PV systems. Despite significant
+research efforts made over the past few decades [42]–[44],
+the energy conversion efficiency of PV systems remains very
+poor due to high environment uncertainties in temperature,
+irradiance level, partial shading and other atmospheric con-
+ditions. The primary goal in PV operation is simply to
+Fig. 1: Mean and standard deviation of estimated θ(k) using
+ensemble based estimators.
+Fig. 2: Observed reward J(k) from unknown and uncertain
+environment with measurement noises v(t).
+extract solar energy as much as possible despite changing
+operational environment, termed as maximum power point
+tracking (MPPT). There have been a wide variety of methods
+targeting to solve this problem, which can be roughly classified
+into three categories: offline methods, online methods, and
+other methods. Detailed comparisons and classifications can
+be found in comprehensive survey papers, e.g., [42], [43].
+In this section, the proposed DCEE is implemented as an
+alternative approach to achieve MPPT, and two representa-
+tive approaches, hill climbing method (HC) and incremental
+conductance method (IC), are deployed for comparison. It is
+Fig. 3: System output y(k) using DCEE.
+
+10
+Fig. 4: System state x(k).
+Fig. 5: Time-varying solar irradiance profile.
+worth noting that all the three algorithms can be classified
+as online methods. It has been widely perceived that online
+methods usually outperform offline counterparts in terms of
+conversion efficiency due to their inherent adaptiveness to
+changing environment. According to the curve-fitting based
+MPPT [42], the power and voltage (P-V ) characteristics can
+be modelled by
+P = φT(V )θ
+(71)
+where φ(V ) is the polynomial regressor [1, V, V 2, . . . , V n]T
+and θ ∈ Rn+1 is the polynomial coefficient. To solve the
+maximum problem of (71), we need to estimate the unknown
+parameters θ and then maximise the power output by regulat-
+ing the voltage V according to
+V (k + 1) = V (k) + u(k).
+(72)
+We use solar panel A10Green Technology model number
+A10J-S72-175 for this simulation [45]. To mimic the real
+operational environment of PV systems, a time-varying solar
+irradiance profile is stimulated as shown in Fig. 5, and the
+temperature is initially set as 25°C and then jumps to 35°C
+at t = 1s. It should be noted that the unknown environment
+parameter θ changes as the operational condition varies. Al-
+though the proposed algorithm is theoretically analysed for
+static parameters identification, the use of constant learning
+rate ηi renders the adaptation algorithm in (15) with the
+capability of tracking drifting parameters.
+Simulation results using different algorithms (DCEE, HC
+and IC) are shown in Fig. 6, 7 and 8. To illustrate more de-
+tailed features of different algorithms, enlarged sub-figures are
+displayed for the time intervals t ∈ [0, 0.1], and t ∈ [0.3, 0.4].
+The power losses, as displayed in Fig. 9, are calculated by
+integrating the differences between the maximum power point
+and real power outputs stimulated using different algorithms.
+Convergence speed, sensed signals, algorithm complexity and
+conversion efficiency are four commonly-used criteria to as-
+sess the characteristics of MPPT techniques. According to the
+simulation results, we summarise and compare the features of
+different approaches in Table I. Conversion efficiency directly
+influences the energy extracted from the PV systems, which
+is ratio between real generated energy and maximum energy
+(accumulated over the simulation time interval [0, 2]). DCEE
+produces quite high efficiency (99.1%). Due to the use of
+perturbed signals in hill climbing method, there are very
+large voltage and current fluctuations in steady state. This
+undesirable property not only causes low conversion efficiency
+but also leads to fast degradation in low level electronic
+devices. The oscillations are partially solved by incremental
+conductance method, which measures incremental current and
+voltage changes to predict the effect of voltage change.
+Different from HC, incremental inductance method is able
+to maintain at MPP without oscillations when there is no
+change in operational environment. From the simulation re-
+sults using HC and IC, there is a trade-off between transient
+convergence speed and steady-state oscillations. The steady-
+state oscillation of IC is reduced at the cost of slow tracking
+performance, leading to larger power loss with a conversion
+efficiency 97.2%. It is argued that DCEE as a balanced ap-
+proach is able to optimally trade-off between exploitation and
+exploration: when there is large uncertainty in estimated MPP,
+it will explore quickly to gain information to construct more
+accurate estimate of MPP; and when there is less change in
+operational environment, it will maintain at the current belief
+of MPP without causing large oscillations. All three algorithms
+need to measure voltage and current: DCEE requires voltage
+and power (calculated by the product of current and voltage) to
+construct P-V curve in (71) (i.e., reward-state mapping), while
+HC and IC use incremental power to deicide the direction of
+voltage regulation. As mature MPPT techniques, both HC and
+IC are simple to implement using dedicated hardware devices.
+Since efficient ensemble approximation and gradient based
+control are developed in this new approach, DCEE is ready to
+be implemented in real PV platforms without incurring heavy
+computational load.
+VII. CONCLUSION
+In this paper, a general framework of dual control for ex-
+ploration and exploitation has been developed to solve a wide
+range of self-optimisation control problems in an uncertain
+environment. A real-time ensemble based estimation approach
+is proposed for efficient environment acquisition, which con-
+sequently provides a measure of knowledge uncertainty to
+the unknown environment. The proposed DCEE algorithm
+optimally balances between exploration and exploitation to
+
+11
+TABLE I: Features of different MPPT techniques.
+Methods
+Convergence speed
+Sensed variables
+Algorithm complexity
+Conversion efficiency
+1
+DCEE
+Fast
+Voltage and current
+Medium
+99.1%
+2
+Hill climbing
+Fast
+Voltage and current
+Simple
+98.3%
+3
+Incremental conductance
+Medium
+Voltage and current
+Simple
+97.2%
+Fig. 6: Power profile using different algorithms.
+Fig. 7: Voltage profile using different algorithms.
+Fig. 8: Current profile using different algorithms.
+Fig. 9: Power losses using different algorithms.
+handle the intrinsic conflict between parameter identifiability
+and optimality tracking. Guaranteed convergence and perfor-
+mance are established in relation to the reward function and
+the noise characteristics. A numerical example and a classic
+application of MPPT are provided to validate the effectiveness
+and potential of DCEE.
+REFERENCES
+[1] C. Zhang and R. Ordonez, “Numerical optimization-based extremum
+seeking control with application to ABS design,” IEEE Transactions on
+Automatic Control, vol. 52, no. 3, pp. 454–467, 2007.
+[2] R. Leyva, C. Alonso, I. Queinnec, A. Cid-Pastor, D. Lagrange, and
+L. Martinez-Salamero, “MPPT of photovoltaic systems using extremum-
+seeking control,” IEEE Transactions on Aerospace and Electronic Sys-
+tems, vol. 42, no. 1, pp. 249–258, 2006.
+[3] Z.-D. Zhong, H.-B. Huo, X.-J. Zhu, G.-Y. Cao, and Y. Ren, “Adaptive
+maximum power point tracking control of fuel cell power plants,”
+Journal of Power Sources, vol. 176, no. 1, pp. 259–269, 2008.
+[4] Y. Tan, W. H. Moase, C. Manzie, D. Neˇsi´c, and I. M. Mareels,
+“Extremum seeking from 1922 to 2010,” in Proceedings of the 29th
+Chinese Control Conference.
+IEEE, 2010, pp. 14–26.
+[5] M. Leblanc, “Sur l’electrification des chemins de fer au moyen de
+courants alternatifs de frequence elevee,” Revue g´en´erale de l’´electricit´e,
+vol. 12, no. 8, pp. 275–277, 1922.
+[6] M. Krsti´c and H.-H. Wang, “Stability of extremum seeking feedback
+for general nonlinear dynamic systems,” Automatica, vol. 36, no. 4, pp.
+595–601, 2000.
+[7] M. Krsti´c, “Performance improvement and limitations in extremum
+seeking control,” Systems & Control Letters, vol. 39, no. 5, pp. 313–326,
+2000.
+[8] S. Skogestad, “Plantwide control: The search for the self-optimizing
+control structure,” Journal of Process Control, vol. 10, no. 5, pp. 487–
+507, 2000.
+[9] M. Guay and T. Zhang, “Adaptive extremum seeking control of nonlinear
+dynamic systems with parametric uncertainties,” Automatica, vol. 39,
+no. 7, pp. 1283–1293, 2003.
+[10] K. A. Sullivan and S. H. Jacobson, “A convergence analysis of general-
+ized hill climbing algorithms,” IEEE Transactions on Automatic Control,
+vol. 46, no. 8, pp. 1288–1293, 2001.
+[11] Y. Tan, D. Neˇsi´c, and I. Mareels, “On non-local stability properties of
+extremum seeking control,” Automatica, vol. 42, no. 6, pp. 889–903,
+2006.
+[12] C. Manzie and M. Krstic, “Extremum seeking with stochastic pertur-
+bations,” IEEE Transactions on Automatic Control, vol. 54, no. 3, pp.
+580–585, 2009.
+
+8
+6 
+5
+loss
+power
+3
+2
+hill climbing
+1
+incremental conductance
+DCEE
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+t [s]12
+[13] S.-J. Liu and M. Krstic, “Stochastic source seeking for nonholonomic
+unicycle,” Automatica, vol. 46, no. 9, pp. 1443–1453, 2010.
+[14] S. Xie and L. Y. Wang, “Adaptive optimization with decaying periodic
+dither signals,” IEEE Transactions on Automatic Control, 2022.
+[15] W.-H. Chen, C. Rhodes, and C. Liu, “Dual control for exploitation and
+exploration (DCEE) in autonomous search,” Automatica, vol. 133, no.
+109851, 2021.
+[16] M. Guay and D. J. Burns, “A proportional integral extremum-seeking
+control approach for discrete-time nonlinear systems,” International
+Journal of Control, vol. 90, no. 8, pp. 1543–1554, 2017.
+[17] A. Mesbah, “Stochastic model predictive control with active uncertainty
+learning: A survey on dual control,” Annual Reviews in Control, vol. 45,
+pp. 107–117, 2018.
+[18] Z. Li, W.-H. Chen, and J. Yang, “Concurrent active learning in au-
+tonomous airborne source search: Dual control for exploration and
+exploitation,” IEEE Transactions on Automatic Control, 2022.
+[19] A. A. Feldbaum, “Dual control theory I,” Avtomatika i Telemekhanika,
+vol. 21, no. 9, pp. 1240–1249, 1960.
+[20] Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence, and sep-
+aration in stochastic control,” IEEE Transactions on Automatic Control,
+vol. 19, no. 5, pp. 494–500, 1974.
+[21] C. Rhodes, C. Liu, and W.-H. Chen, “Autonomous source term estima-
+tion in unknown environments: From a dual control concept to UAV
+deployment,” IEEE Robotics and Automation Letters, vol. 7, no. 2, pp.
+2274–2281, 2021.
+[22] P. Antsaklis, “Autonomy and metrics of autonomy,” Annual Reviews in
+Control, vol. 49, pp. 15–26, 2020.
+[23] T. A. N. Heirung, B. E. Ydstie, and B. Foss, “Dual adaptive model
+predictive control,” Automatica, vol. 80, pp. 340–348, 2017.
+[24] M. Hutchinson, H. Oh, and W.-H. Chen, “A review of source term
+estimation methods for atmospheric dispersion events using static or
+mobile sensors,” Information Fusion, vol. 36, pp. 130–148, 2017.
+[25] K. Chua, R. Calandra, R. McAllister, and S. Levine, “Deep rein-
+forcement learning in a handful of trials using probabilistic dynamics
+models,” Advances in Neural Information Processing Systems (NIPS
+2018), vol. 31, 2018.
+[26] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable
+predictive uncertainty estimation using deep ensembles,” Advances in
+Neural Information Processing Systems, vol. 30, 2017.
+[27] R. Ortega, V. Nikiforov, and D. Gerasimov, “On modified parameter
+estimators for identification and adaptive control. a unified framework
+and some new schemes,” Annual Reviews in Control, vol. 50, pp. 278–
+293, 2020.
+[28] V. Adetola and M. Guay, “Parameter convergence in adaptive extremum-
+seeking control,” Automatica, vol. 43, no. 1, pp. 105–110, 2007.
+[29] F. Ding and T. Chen, “Performance analysis of multi-innovation gradient
+type identification methods,” Automatica, vol. 43, no. 1, pp. 1–14, 2007.
+[30] W. Rudin, Principles of Mathematical Analysis, 3rd ed.
+New York,
+NY, USA: McGraw-hill, 1976.
+[31] Z. Li, W.-H. Chen, and J. Yang, “A dual control perspective for
+exploration and exploitation in autonomous search,” in European Control
+Conference, 2022, Conference Proceedings.
+[32] E. Tse and Y. Bar-Shalom, “An actively adaptive control for linear
+systems with random parameters via the dual control approach,” IEEE
+Transactions on Automatic Control, vol. 18, no. 2, pp. 109–117, 1973.
+[33] A. I. Cowen-Rivers, D. Palenicek, V. Moens, M. Abdullah, A. Sootla,
+J. Wang, and H. Ammar, “Samba: Safe model-based & active reinforce-
+ment learning,” arXiv preprint arXiv:2006.09436, 2020.
+[34] M. Ghavamzadeh, S. Mannor, J. Pineau, and A. Tamar, “Bayesian
+reinforcement learning: A survey,” Foundations and Trends® in Machine
+Learning, vol. 8, no. 5-6, pp. 359–483, 2015.
+[35] H. Jeong, B. Schlotfeldt, H. Hassani, M. Morari, D. D. Lee, and G. J.
+Pappas, “Learning q-network for active information acquisition,” arXiv
+preprint arXiv:1910.10754, 2019.
+[36] A. Mesbah, “Stochastic model predictive control: An overview and
+perspectives for future research,” IEEE Control Systems Magazine,
+vol. 36, no. 6, pp. 30–44, 2016.
+[37] M. K. Bugeja, S. G. Fabri, and L. Camilleri, “Dual adaptive dynamic
+control of mobile robots using neural networks,” IEEE Transactions on
+Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 39, no. 1, pp.
+129–141, 2008.
+[38] T. Alpcan and I. Shames, “An information-based learning approach to
+dual control,” IEEE transactions on Neural Networks and Learning
+Systems, vol. 26, no. 11, pp. 2736–2748, 2015.
+[39] J. Huang, Nonlinear Output Regulation: Theory and Applications.
+SIAM, 2004.
+[40] W.-H. Chen, “Perspective view of autonomous control in unknown envi-
+ronment: Dual control for exploitation and exploration vs reinforcement
+learning,” Neurocomputing, 2022.
+[41] K. Friston, “The free-energy principle: a unified brain theory?” Nature
+Reviews Neuroscience, vol. 11, no. 2, pp. 127–138, 2010.
+[42] P. Bhatnagar and R. Nema, “Maximum power point tracking control
+techniques: State-of-the-art in photovoltaic applications,” Renewable and
+Sustainable Energy Reviews, vol. 23, pp. 224–241, 2013.
+[43] A. R. Reisi, M. H. Moradi, and S. Jamasb, “Classification and com-
+parison of maximum power point tracking techniques for photovoltaic
+system: A review,” Renewable and Sustainable Energy Reviews, vol. 19,
+pp. 433–443, 2013.
+[44] T. Esram and P. L. Chapman, “Comparison of photovoltaic array
+maximum power point tracking techniques,” IEEE Transactions on
+Energy Conversion, vol. 22, no. 2, pp. 439–449, 2007.
+[45] I. Shams, S. Mekhilef, and K. S. Tey, “Maximum power point tracking
+using modified butterfly optimization algorithm for partial shading,
+uniform shading, and fast varying load conditions,” IEEE Transactions
+on Power Electronics, vol. 36, no. 5, pp. 5569–5581, 2020.
+

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+There are 2-tough 4-regular graphs with claws
+W. Goddard, Clemson University
+Chv´atal [1] defined the toughness of a graph G to be the minimum value of
+|S|/k(G−S) where k(G−S) denotes the number of components of G−S and the
+minimum is taken over all cut-sets S ⊆ V (G). It is immediate that the toughness
+is at most half the connectivity. Matthews and Sumner [5] showed that there is
+equality if the graph is claw-free.
+For cubic graphs, Jackson and Katerinis [4] showed that being claw-free is
+also necessary for the graph to have toughness 3
+2. In [2] we conjectured that the
+analogous result holds for all r-regular graphs, and in [3] we expressed the belief
+that the analogous result does not hold for all r, thus ensuring that we have to
+be correct at least once.
+We note here that it is the latter belief that is true.
+The graph below is
+4-regular and has claws and its toughness is 2. It is one of the two of smallest
+order.
+References
+[1] V. Chv´atal. Tough graphs and Hamiltonian circuits. Discrete Math. 5 (1973),
+215–28.
+[2] W. Goddard and H.C. Swart. On the toughness of a graph. Quaestiones Math.
+13 (1990), 217–232.
+1
+arXiv:2301.13632v1  [math.CO]  27 Jan 2023
+
+[3] W. Goddard. The toughness of cubic graphs. Graphs Combin. 12 (1996), 17–
+22.
+[4] B. Jackson and P. Katerinis. A characterization of 3
+2-tough cubic graphs. Ars
+Combin. 38 (1994), 145–148.
+[5] M.M. Matthews and D.P. Sumner. Hamiltonian results in K1,3-free graphs.
+J. Graph Theory 8 (1984), 139–146.
+2
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+What Makes Good Examples for Visual In-Context Learning?
+Yuanhan Zhang 1 Kaiyang Zhou 1 Ziwei Liu 1
+Abstract
+Large-scale models trained on broad data have
+recently become the mainstream architecture in
+computer vision due to their strong generalization
+performance. In this paper, the main focus is
+on an emergent ability in large vision models,
+known as in-context learning, which allows
+inference on unseen tasks by conditioning on
+in-context examples (a.k.a. prompt) without
+updating the model parameters. This concept has
+been well-known in natural language processing
+but has only been studied very recently for large
+vision models. We for the first time provide a
+comprehensive investigation on the impact of
+in-context examples in computer vision, and find
+that the performance is highly sensitive to the
+choice of in-context examples.
+To overcome
+the problem, we propose a prompt retrieval
+framework to automate the selection of in-context
+examples. Specifically, we present (1) an unsuper-
+vised prompt retrieval method based on nearest
+example search using an off-the-shelf model,
+and (2) a supervised prompt retrieval method,
+which trains a neural network to choose examples
+that
+directly
+maximize
+in-context
+learning
+performance. The results demonstrate that our
+methods can bring non-trivial improvements
+to visual in-context learning in comparison to
+the commonly-used random selection.
+The
+code and models are available at https:
+//github.com/ZhangYuanhan-AI/
+visual_prompt_retrieval.
+1. Introduction
+In recent years, large-scale models have emerged in com-
+puter vision: they have enormous parameter size and are
+pre-trained on broad data to gain wide-ranging knowledge.
+These models have demonstrated remarkable generaliza-
+tion performance and have great potential for numerous
+1S-Lab, Nanyang Technological University, Singapore. Corre-
+spondence to: Ziwei Liu <[email protected]>.
+Preliminary work. Do not distribute.
+downstream applications (Bommasani et al., 2021). How-
+ever, due to the large model size and the potentially pro-
+prietary data used for training, entities able to develop
+large-scale models typically only provide users with APIs,
+known as Model-as-a-Service (Maas). Representative exam-
+ples include the prominent text-to-image generation models,
+DALL·E (Ramesh et al., 2021) and Imagen (Saharia et al.,
+2022), and OpenAI’s powerful language models like GPT-
+3/ChatGPT (Radford et al., 2021). As a result, users are
+unable to apply full fine-tuning or some parameter-efficient
+tuning techniques, such as prompt learning (Li & Liang,
+2021; Lester et al., 2021; Zhou et al., 2022c;b; Zhang et al.,
+2022; Pan et al., 2022), for model adaptation, largely limit-
+ing downstream performance.
+In-context learning, which is a “hidden” capability origi-
+nally found in large autoregressive language models (Rad-
+ford et al., 2021), has recently been investigated for large
+vision models (Bar et al., 2022), and more importantly, has
+the potential to become the mainstream approach for MaaS
+applications in the near future. Without the need to update
+any parameter for previously unseen tasks, in-context learn-
+ing simply prepends some domain-specific input-output
+pairs, called in-context examples or prompt,1 to a test ex-
+ample, which together guide the model to produce an ideal
+result. For instance, in natural language processing one
+could prepend a French-English sentence pair to a French
+sentence, and the model would produce an English transla-
+tion of the French sentence. In computer vision, Bar et al.
+(2022) pre-trained a neural network to fill missing patches
+in grid-like images, which allows the model to perform in-
+context learning for unseen tasks like image segmentation
+(see the grid images in Fig. 1(a) bottom).
+In this work, we focus on visual in-context learning, a rel-
+atively new concept with little existing research regarding
+how to better apply it in practice. We for the first time
+conduct a comprehensive investigation on the impact of in-
+context examples for large vision models, and identify a
+critical issue: downstream performance is highly sensitive
+to the choice of in-context examples. This is evidenced by
+the large variances observed for a variety of test examples
+shown in Fig. 1(a) top. By visualizing the results in Fig. 1(a)
+bottom, it seems to suggest that the closer the in-context
+1These two terms are used interchangeably in this paper.
+arXiv:2301.13670v1  [cs.CV]  31 Jan 2023
+
+What Makes Good Examples for Visual In-Context Learning?
+0
+20
+40
+60
+80
+100
+1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
+Foreground segmentation (mIoU)
+Average
+Standard deviation
+Best prompt
+Worst prompt
+…
+…
+Example
+Annotation
+Query
+Output
+Database (train)
+Query (test)
+Large-scale
+vision model
+(b) Prompt retrieval for visual in-context learning
+Score
+function
+Retrieved
+image
+(a)Visual in-context learning is sensitive to prompt selection
+Query image index
+Figure 1: (a) Different choices of in-context examples (outlined in green) often lead to significantly different results. Here
+we show 30 random query images (x-axis) from Pascal-5i (Shaban et al., 2017) split 0, and measure the performance range
+using 50 different in-context examples. (b) We propose a prompt retrieval framework aiming to automate the selection of
+in-context examples. We provide two implementations of the idea: one is unsupervised while the other is supervised, both
+outperforming random selection by a clear margin.
+example to the query, the better the result. For example, the
+best prompt image is closer to the query as they are similar
+in object pose and background; on the other hand, the worst
+prompt image has a drastically different style than the query
+image, which might explain why the predicted mask focuses
+on the wrong region, i.e., the white pillar instead of the cat.
+Clearly, designing a proper prompt containing the optimal
+in-context example(s) by hand would be extremely difficult.
+To overcome the problem, we propose a prompt retrieval
+framework where the core component is a score function,
+which aims to give each source instance a score to indicate
+the level of suitability for being included in the prompt.
+Once the scoring process is done, we can simply pick one
+or multiple examples with the highest score(s) to construct
+a prompt. An overview of our framework is depicted in
+Fig. 1(b).
+We provide two implementations for the prompt retrieval
+framework, both interpreting the score as the cosine distance
+measuring similarity between a query and a source exam-
+ple. The first is an unsupervised method based on nearest
+example search using an off-the-shelf model. The second
+is a supervised method, which learns a neural network to
+choose examples that directly maximize in-context learning
+performance. Since there is no ground-truth score to be
+used as the supervisory signal, we resort to a contrastive
+learning paradigm: source examples that result in better (or
+worse) in-context learning performance should get closer
+(or farther) to the query in feature space.
+Our contributions and the main findings are summarized
+as follows. (1) We present the first comprehensive study
+concerning how to select good examples for the emerg-
+ing visual in-context learning, and reveal a critical issue
+that the choice of in-context examples has a huge impact
+on performance. (2) From the technical perspective, we
+present a prompt retrieval framework that can automate the
+prompt selection process, and provide two simple implemen-
+tations: an unsupervised method and a supervised method.
+(3) By conducting extensive experiments on three visual
+in-context learning tasks (which have not been seen dur-
+ing pre-training), namely foreground segmentation, single
+object detection and image colorization, we share valuable
+insights with the community on how to find good visual
+in-context examples, e.g., the supervised method performs
+the best and often finds examples that are both semantically
+close and spatially similar to a query.
+2. Methods
+2.1. Visual In-Context Learning
+In-context learning is a new paradigm that originally
+emerged from large autoregressive language models pre-
+trained on broad data, such as GPT-3 (Brown et al., 2020).
+Unlike traditional learning methods, in-context learning
+
+What Makes Good Examples for Visual In-Context Learning?
+Feature
+space
+Learnable
+feature
+extractor
+…
+…
+IoU:60.2%
+IoU:30.1% 
+Query
+🔥
+🧊
+Database (train)
+Positive
+Negative
+Large-scale
+vision model
+Query (test)
+high
+low
+🔥
+Figure 2: Overview of the supervised prompt retrieval method. The main idea is to compute the in-context learning result for
+each source example, and pick those with the highest/lowest results to form a positive/negative set for contrastive learning.
+does not require any parameter update and instead condi-
+tions prediction on some in-context examples in the form
+of input-output pairs. For example, in natural language pro-
+cessing one might give a French-English sentence pair and
+a test French sentence as input to the model, which then
+produces the English version of the sentence. In computer
+vision, such a paradigm has only been studied very recently.
+For example, Bar et al. (2022) trained a neural network to
+fill missing patches in grid-like images, which in turn allows
+the model to perform in-context learning on unseen tasks.
+Formally, given a dataset D = {(xn, yn)}N
+n=1 containing
+N image-label pairs (e.g., an image and its segmentation
+mask), a query example xq, and a model gτ, in-context
+learning can be formulated as:
+yq = gτ(P, xq),
+(1)
+where P is called a prompt, which consists of K input-
+output pairs, P = {xc1, yc1, ..., xcK, ycK} ⊂ D. In partic-
+ular, the prompt P provides some context for guiding the
+model to produce the ideal yq for xq without updating the
+large model’s parameters τ.
+Problem. The most common approach for designing the
+prompt P in the vision domain is (within-class) random
+selection proposed by Bar et al. (2022): one or multiple
+image-label pairs (with the same label as the test example)
+are randomly chosen from the training dataset. As illus-
+trated in Fig. 1(a), the performance is highly sensitive to the
+selection of in-context examples—the gap between the best
+and worst prompt could reach over 70% mIoU. Below we
+propose two automatic prompt selection methods to tackle
+this problem.
+2.2. Prompt Retrieval
+Our goal is to automatically select the most suitable exam-
+ple(s) from the training dataset for a query xq. To this end,
+we propose a prompt retrieval framework in the following
+form,
+x∗ = arg max
+xn∈D fθ(xn, xq),
+(2)
+where fθ is a function parameterized by θ, aiming to produce
+a score for a pair of xn and xq. When K = 1, we choose
+the optimal example pair as the prompt, P = {x∗, y∗}.
+When K > 1, we rank the training examples by their scores
+and choose the top-K example pairs. An overview of our
+methods is provided in Fig. 1(b).
+In this work, we implement fθ as a combination of a neural
+network for feature extraction and the cosine distance func-
+tion for measuring similarity between two feature vectors.
+2.2.1. UNSUPERVISED PROMPT RETRIEVAL
+Our first method is unsupervised prompt retrieval where
+the key idea is to use an off-the-shelf feature extractor for
+extracting image features so that we can compare the cosine
+distance between the query xq and each training example
+xn ∈ D. In this case, the parameters θ for the score function
+fθ correspond to the off-the-shelf feature extractor, which
+are kept fixed.
+2.2.2. SUPERVISED PROMPT RETRIEVAL
+The unsupervised method discussed above is not explicitly
+optimized for in-context learning; instead, it relies on how
+the feature extractor was pre-trained and the objective (func-
+tion) used in pre-training may well not align with that of
+in-context learning. We propose a second method based on
+
+AIR
+CANADAWhat Makes Good Examples for Visual In-Context Learning?
+supervised prompt retrieval where we assume the source
+data contains labels. The goal is to directly optimize the
+score function fθ such that the chosen in-context example(s)
+can maximize the log-likelihood,
+max
+P
+log p(yq|P, xq).
+(3)
+In this work, we present a simple implementation for the
+supervised method, which simply turns the unsupervised
+method into a supervised one by making the feature extrac-
+tor learnable. In other words, we directly optimize Eq. 3
+with respect to the feature extractor. Below we explain in
+detail how we train the feature extractor (see Fig. 2 for an
+overview).
+Data. Recall that we interpret the score fθ(·, ·) as the co-
+sine distance between two images in feature space. We
+would like to learn a space such that an image pair (xn, xq)
+with high in-context learning performance is close to each
+other, or far away from each other if the performance is
+low. Since there is no label defining how close a distance
+should be, we resort to contrastive learning for training the
+feature extractor. The goal is then to find a positive and
+a negative set for each training example xn ∈ D treated
+as a query. Specifically, for each example xn we compute
+the prediction ˆyn = gτ((xm, ym), xn) where gτ is the large
+vision model defined in Sec. 2.1 and xm ∈ D but xm ̸= xn.
+Since we have the ground truth yn for xn, we can measure
+the performance by comparing the prediction ˆyn with the
+ground truth yn. Then, for each xn we choose the top-5
+examples with the highest/lowest performance to form a
+positive/negative set.
+Training. Let zn denote the features of xn extracted by the
+neural network we aim to optimize. At each iteraction, we
+sample a mini-batch B from the training dataset. Then, for
+each example in B, we sample one example from the top-5
+positive and negative sets, respectively. The contrastive loss
+is computed as
+ℓ = − 1
+|B|
+�
+xn∼B
+log
+ecos(zn,z+
+n )
+ecos(zn,z+
+n ) +
+�
+z−
+n ∈N
+ecos(zn,z−
+n ) , (4)
+where cos(·, ·) is the cosine distance function, z+
+n denotes
+the feature representation of a positive example, and z−
+n
+denotes the feature representation of a negative example. It
+is worth noting that for mini-batch training, the negative
+set N contains a negative example of xn sampled from
+the top-5 negative set and other examples within the same
+mini-batch.
+3. Experiments
+In this section we conduct a comprehensive evaluation using
+different prompt selection methods (Sec. 3.1) and compare
+their robustness to distribution shifts (Sec. 3.2). We also
+provide extensive quantitative and qualitative analyses in
+Sec. 3.3 to help understand why our methods work and how
+to better apply them in practice. Source code will be released
+to the community for reproducing the full experiments.
+Methods. All experiments are based on the image inpaint-
+ing model pre-trained by Bar et al. (2022) on a dataset
+consisting of academic figures.2 We mainly compare the fol-
+lowing methods: (1) Random, the baseline method that ran-
+domly samples in-context examples from the source training
+dataset; (2) Unsupervised prompt retrieval (UnsupPR), our
+first proposed method that uses off-the-shelf features for
+nearest example search. The main experiments are based
+on CLIP’s vision encoder (Radford et al., 2021), which was
+pre-trained using multimodal contrastive learning; (3) Su-
+pervised prompt retrieval (SupPR), our second proposed
+method that fine-tunes CLIP’s vision encoder by directly
+optimizing in-context learning performance on downstream
+datasets. A variety of backbones are evaluated in Sec. 3.3.
+Training details for the supervised model. The super-
+vised model is trained for 200 epochs using SGD. The initial
+learning rate is set to 0.005, decayed by the cosine annealing
+rule.
+3.1. Main Results
+Setup. Following Bar et al. (2022), we evaluate our meth-
+ods on three computer vision tasks, which have not been
+seen during the training of the image inpainting model. We
+provide the details about the datasets used for these tasks
+as follows. (1) Foreground segmentation: We use Pascal-
+5i (Shaban et al., 2017), which has four non-overlapping
+splits each containing five categories. The results are aver-
+aged over all splits. (2) Single object detection: The experi-
+ments are done on Pascal VOC (Everingham et al., 2015).
+(3) Colorization: We use ImageNet-2012 (Russakovsky
+et al., 2015), where the original validation set containing
+50,000 images is used as our test set. The training data used
+to learn our supervised prompt retrieval model is created by
+randomly sampling 50,000 images from ImageNet’s 1.2M
+training set. For all experiments, in-context examples come
+from the training set.
+Results. Table 1 shows the results on the three benchmarks
+covering foreground segmentation, single object detection,
+and colorization. We summarize our findings as follows.
+First, prompt retrieval clearly outperforms random selection.
+In particular, the improvements of prompt retrieval over
+random selection are significant in foreground segmentation
+and single object detection: more than 6% on the former
+2https://github.com/amirbar/visual_
+prompting
+
+What Makes Good Examples for Visual In-Context Learning?
+Table 1: Main results. The two prompt retrieval methods outperform random selection, and the supervised method achieves
+the best performance.
+Seg. (mIoU) ↑
+Det. (mIoU) ↑
+Color. (mse) ↓
+Split-0
+Split-1
+Split-2
+Split-3
+Avg
+Random
+28.66
+30.21
+27.81
+23.55
+27.56
+25.45
+0.67
+UnsupPR
+34.75
+35.92
+32.41
+31.16
+33.56
+26.84
+0.63
+SupPR
+37.08
+38.43
+34.40
+32.32
+35.56
+28.22
+0.63
+Table 2: Results on distribution shifts (from Pascal to
+MSCOCO). Despite being a learning-based approach,
+SupPR shows stronger robustness than UnsupPR and Ran-
+dom, which do not require any training.
+Seg. (mIoU) ↑
+Split-0
+Split-1
+Split-2
+Split-3
+Avg
+Random
+12.17
+18.47
+20.55
+15.94
+16.78
+UnsupPR
+12.67
+19.62
+21.33
+18.44
+18.02
+SupPR
+13.62
+21.25
+24.46
+20.44
+19.95
+and 1% on the latter. However, the gains on colorization
+are only marginal (0.63 vs. 0.67), suggesting that the image
+inpainting model is probably weak at image colorization.
+Second, the supervised prompt retrieval method performs
+the best. This is not surprising as the supervised method
+optimizes in-context learning performance concerning the
+prompt selection module. In contrast, the unsupervised
+method relies more on the off-the-shelf feature extractor.
+Overall, the results well justify the design of the prompt
+retrieval framework, which can serve as a strong baseline
+for future research.
+3.2. Experiments on Distribution Shifts
+Setup. Distribution shifts are commonly seen in real-world
+applications, and therefore AI models need to be robust to
+distribution shifts (Zhou et al., 2022a). To test this ability in
+visual in-context learning, we create a new protocol focusing
+on foreground segmentation where the source dataset is
+Pascal while the target dataset is MSCOCO (Lin et al., 2014).
+Specifically, we follow the design of Pascal-5i and create
+MSCOCO-5i, which also has four splits, each having the
+same set of categories as in the corresponding split in Pascal-
+5i. Note that such a shift mainly affects the supervised
+prompt retrieval method that requires training but not the
+unsupervised UnsupPR and Random.
+Results. The results are shown in Table 2. First of all,
+the unsupervised prompt retrieval method beats the random
+selection method by a clear margin. By comparing the
+two prompt retrieval methods, we find that the supervised
+method again performs better than the unsupervised one
+despite being a learning-based approach—this is an exciting
+Table 3: Comparison between different backbones pre-
+trained using different methods: multimodal contrastive
+learning for CLIP, self-supervised learning for EVA, and
+supervised learning for ViT. Overall, the performance is
+insensitive to the choice of different backbones.
+Seg. (mIoU) ↑
+Split-0
+Split-1
+Split-2
+Split-3
+Avg
+UnsupPR
+CLIP
+34.75
+35.92
+32.41
+31.16
+33.56
+EVA
+34.75
+36.09
+32.11
+31.61
+33.64
+ViT
+35.10
+37.37
+32.05
+30.80
+33.83
+SupPR
+CLIP
+37.08
+38.43
+34.40
+32.32
+35.56
+EVA
+36.11
+39.14
+34.31
+33.30
+35.71
+ViT
+36.80
+39.70
+34.71
+33.25
+36.12
+finding as it means the supervised method does not have
+the overfitting problem here. Nonetheless, we observe that
+the gains achieved by the prompt retrieval methods here
+are generally smaller than the gains achieved on the stan-
+dard foreground segmentation benchmark: here SupPR is
+only around 3% better on average than Random (19.95%
+vs. 16.78%) while the improvement in Table 1 reaches 8%
+(35.56% vs. 27.56%). One potential solution to reduce the
+gap might be to improve the image inpainting model, which
+is beyond the scope of this paper.
+3.3. Further Analysis
+What are good in-context examples? To answer this ques-
+tion, we visualize the in-context examples found by Un-
+supPR and SupPR in Fig. 3. We focus on foreground seg-
+mentation and choose two categories from Pascal (person
+and cow).3 In each grid, the first row corresponds to the re-
+trieved in-context example (i.e., an input-output pair) while
+the second row contains the query and model prediction. By
+comparing the in-context examples picked by UnsupPR and
+those picked by SupPR, we find the reason why SupPR per-
+forms better than UnsupPR: the examples found by SupPR
+are more similar to the queries in terms of semantics (e.g.,
+Fig. 3(e)), background (e.g., Fig. 3(a)), object pose (e.g.,
+Fig. 3(b), object appearance (e.g., Fig. 3(i)), viewpoint (e.g.,
+3The results of the remaining categories of Pascal and the
+results on other tasks are provided in the supplementary.
+
+What Makes Good Examples for Visual In-Context Learning?
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+IoU: 37.85
+IoU: 47.48
+IoU: 42.36
+IoU: 69.46
+IoU: 26.47
+IoU: 27.78
+IoU: 59.34
+IoU: 46.74
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+IoU: 49.12
+IoU: 23.21
+IoU: 66.93
+IoU: 61.25
+IoU: 29.34
+IoU: 63.38
+IoU: 8.45
+IoU: 36.67
+IoU: 86.44
+IoU: 86.64
+IoU: 92.32
+IoU: 80.14
+IoU: 63.14
+IoU: 79.22
+IoU: 57.48
+IoU: 49.87
+vvv
+v v
+vv
+Figure 3: In-context examples retrieved by UnsupPR and SupPR. In each grid, the first row contains the prompt while the
+second row contains the query and prediction. The in-context examples found by SupPR are more similar than those found
+by UnsupPR to the queries in a numer of ways: semantics (e.g., (e)), background (e.g., (a)), object pose (e.g., (b), object
+appearance (e.g., (i)), viewpoint (e.g., (k)), etc. More examples can be found in the supplementary.
+Fig. 3(k)), and so on. We also observe similar patterns in
+other categories/tasks (please refer to the supplementary).
+Backbone. To understand if using a different backbone than
+CLIP would make a big difference, we further evaluate our
+prompt retrieval methods, UnsupPR and SupPR, on the fore-
+ground segmentation benchmark using two other backbones:
+EVA (Fang et al., 2022) pre-trained using self-supervised
+learning (i.e., masked image modeling) and ViT (Dosovit-
+skiy et al., 2020) pre-trained using supervised learning. The
+results are reported in Table 3. Although these three back-
+bones perform differently on image recognition under the
+fine-tuning setting—EVA performed the best—the gap be-
+tween them for both UnsupPR and SupPR is less than 1%.
+Therefore, we can conclude that the backbone for visual
+in-context learning does not matter much.
+Size of retrieval set. Recall that in-context examples are
+sampled from the training dataset, namely the retrieval set.
+We are interested to know whether the size has any impact on
+performance, especially for the supervised prompt retrieval
+method. To this end, we build seven subsets for each split
+in Pascal-5i, which cover a wide range of sizes (see the
+x-axis in Fig. 4 left). The results are plotted in Fig. 4 left.
+For random selection, the size does not matter at all. In
+contrast, the two prompt retrieval methods clearly benefit
+from a bigger size. But their performance plateaus when the
+size reaches a certain level. It is worth noting that for the
+supervised method, 20% of the total data is sufficient for
+achieving a decent performance.
+
+What Makes Good Examples for Visual In-Context Learning?
+27.58
+27.58
+27.66
+27.63
+27.83
+27.42
+27.56
+29.81
+32.01
+32.46
+32.85
+33.09
+33.50
+33.56
+31.30
+34.62
+35.42
+35.41
+35.55
+35.64
+35.56
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+0.01
+0.1
+0.2
+0.4
+0.6
+0.8
+1
+mIou
+Size of retrieval set (% of full set)
+Random
+UnsupPR
+SupPR
+33.56
+35.56
+34.15
+33.70
+35.56
+34.03
+33.71
+35.57
+34.05
+33
+34
+35
+36
+UnsupPR
+SupPR
+AVG
+mIOU
+Similarity metric selection
+Cosine
+Euclidean
+Manhattan
+Figure 4: (Left) Impact of the size of retrieval set. (Right) Ablation study on distance metric used to compute the score
+function in Eq. 2. It can be observed that different metrics perform similarly.
+Table 4: Impact of the order of in-context examples.
+Seg. (mIoU) ↑
+Split-0
+Split-1
+Split-2
+Split-3
+Avg
+Random
+17.93 ± 0.20
+25.48 ± 0.27
+21.34 ± 0.73
+21.12 ± 0.53
+21.46 ± 0.43
+UnsupPR
+20.22 ± 0.31
+27.58 ± 0.40
+22.42 ± 0.38
+23.36 ± 0.42
+23.39 ± 0.37
+SupPR
+20.74 ± 0.40
+28.19 ± 0.37
+23.09 ± 0.34
+24.22 ± 0.48
+24.06 ± 0.40
+Number of in-context examples. We follow Bar et al.
+(2022) and create a large grid enough to fit 8 examples
+at maximum (as shown in Fig. 5 right). By varying the
+number of in-context examples from 1 to 7, we obtain a
+set of results and plot them in Fig. 5 left. Clearly, more
+in-context examples lead to better performance for all three
+methods, including SupPR, UnsupPR, and Random. This
+is probably because in-context examples can be viewed
+as “training data”, and having more training data typically
+benefits performance—in visual in-context learning, more
+training data gives a more comprehensive “context.” We
+show a few example cases in Fig. 5 right to explain this
+observation.
+Order of in-context examples. To understand if changing
+the order of in-context examples makes a difference, we
+fix the number of in-context examples to 3, evaluate all
+possible combinations, and compute the mean and standard
+deviation. As shown in Table 4, the standard deviation is
+generally small, so the order is not a concern as long as
+good examples are chosen.
+Distance metric. We use the cosine distance by default to
+compute the score function in Eq. 2. Here we evaluate other
+design choices including Euclidean distance and Manhattan
+distance. As shown in Fig. 4 right, the results are very
+similar for different distance metrics.
+4. Related Work
+4.1. In-Context Learning
+In-context learning is a novel paradigm that emerged in large
+language models, such as GPT-3 (Brown et al., 2020). It al-
+lows an autoregressive language model to perform inference
+on unseen tasks by conditioning the input on some target-
+specific input-output pairs serving as “context.” Such a pow-
+erful paradigm allows users to customize a model’s output
+according to their downstream datasets without changing
+the internal model parameters, which are often inaccessi-
+ble. Recent research in natural language processing has
+shown that in-context learning can be applied to numerous
+language tasks, such as machine translation (Garcia & Firat,
+2022), sentiment analysis (Min et al., 2021), and question
+answering (Press et al., 2022).
+In computer vision, in-context learning is still a relatively
+new concept. One of the earliest works tackling in-context
+learning is Flamingo (Alayrac et al., 2022), a large visual
+language model taking language as instruction and allowing
+the processing of both images and videos. More relevant
+to our work is a pure vision model developed by Bar et al.
+(2022), which was pre-trained to fill missing patches in
+images made of academic figures and infographics. Bar
+et al. (2022) found that such an image inpainting model
+can solve problems unseen during training, like foreground
+segmentation and image colorization.
+Our work follows Bar et al. (2022) but studies visual in-
+context learning from a different dimension: how to find
+
+What Makes Good Examples for Visual In-Context Learning?
+IoU: 20.98
+IoU: 15.68
+IoU: 32.50
+IoU:29.09
+(a)
+Num. of examples = 1
+(b)
+Num. of examples = 7
+18.56
+21.90
+22.87
+25.39
+21.22
+23.41
+24.39
+26.51
+21.90
+24.43
+25.87
+27.99
+18
+20
+22
+24
+26
+28
+1
+3
+5
+7
+mIoU
+Number of in-context
+examples
+Random
+UnsupPR
+SupPR
+(c)
+IoU:34.19
+IoU:56.46
+Figure 5: (Left) Impact of the number of in-context examples. (Right) More in-context examples can lead to better
+performance. The query in each grid is shown in the bottom right.
+good visual in-context examples that benefit downstream
+performance.
+4.2. Prompt Retrieval in NLP
+The natural language processing community has found that
+the choice of in-context examples has a huge impact on per-
+formance (Agrawal et al., 2022; Liu et al., 2021). Moreover,
+the way how in-context examples, also called prompts, are
+constructed can also affect performance, e.g., prompt length
+and the order of in-context examples, as reported in the lit-
+erature (Agrawal et al., 2022). These findings prompted the
+community to study how to find good in-context examples
+for large language models, which has inspired our research.
+Liu et al. (2021) assumed that good in-context examples
+should be semantically close to query sentences, based on
+which they proposed to select nearest neighbors in the train-
+ing set measured by a sentence encoder like RoBERTa (Liu
+et al., 2019). Rubin et al. (2021) first used an unsupervised
+method to retrieve some candidates, among which top ex-
+amples were chosen using a supervised prompt retriever to
+maximize downstream performance.
+5. Discussion and Conclusion
+Our research presents a timely study on an emergent abil-
+ity termed in-context learning for large vision models. We
+systematically investigate how the choice of in-context ex-
+amples impacts downstream performance, exposing a crit-
+ical issue that different in-context examples could lead to
+drastically different results. We then propose an effective
+prompt retrieval framework for visual in-context learning,
+with two simple implementations provided: one based on
+unsupervised learning and the other based on supervised
+learning. Our methods obtain significant improvements over
+random selection under various problem settings, showing
+the potential of using prompt retrieval in vision applications
+with a Model-as-a-Service (MaaS) business structure.
+Our research also unveils some intriguing phenomena. For
+instance, we show that a good in-context example should
+be semantically similar to the query and closer in context,
+e.g., viewpoint, background, and appearance. As such, state-
+of-the-art vision models like CLIP would not be sufficient
+because these models often emphasize semantics but not
+other elements critical to finding good visual in-context
+examples. A model that can better balance spatial and se-
+mantic closedness in feature space would be more ideal for
+visual in-context learning. We hope the insights presented in
+this work could pave the way for developing more effective
+prompt retrieval methods.
+Our experiments show that our methods are not strong
+enough to cope with distribution shifts. Though our meth-
+ods outperform random selection under distribution shifts,
+the gap is much smaller than that on a standard benchmark,
+suggesting huge room for improvement.
+
+What Makes Good Examples for Visual In-Context Learning?
+References
+Agrawal, S., Zhou, C., Lewis, M., Zettlemoyer, L., and
+Ghazvininejad, M. In-context examples selection for
+machine translation. arXiv preprint arXiv:2212.02437,
+2022. 8
+Alayrac, J.-B., Donahue, J., Luc, P., Miech, A., Barr, I.,
+Hasson, Y., Lenc, K., Mensch, A., Millican, K., Reynolds,
+M., et al. Flamingo: a visual language model for few-shot
+learning. arXiv preprint arXiv:2204.14198, 2022. 7
+Bar, A., Gandelsman, Y., Darrell, T., Globerson, A., and
+Efros, A. A. Visual prompting via image inpainting. arXiv
+preprint arXiv:2209.00647, 2022. 1, 3, 4, 7
+Bommasani, R., Hudson, D. A., Adeli, E., Altman, R.,
+Arora, S., von Arx, S., Bernstein, M. S., Bohg, J., Bosse-
+lut, A., Brunskill, E., et al. On the opportunities and risks
+of foundation models. arXiv preprint arXiv:2108.07258,
+2021. 1
+Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D.,
+Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G.,
+Askell, A., et al. Language models are few-shot learn-
+ers. Advances in neural information processing systems
+(NeurIPS), 33:1877–1901, 2020. 2, 7
+Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn,
+D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M.,
+Heigold, G., Gelly, S., et al. An image is worth 16x16
+words: Transformers for image recognition at scale. arXiv
+preprint arXiv:2010.11929, 2020. 6
+Everingham, M., Eslami, S., Van Gool, L., Williams, C. K.,
+Winn, J., and Zisserman, A. The pascal visual object
+classes challenge: A retrospective. International journal
+of computer vision (ICCV), 111(1):98–136, 2015. 4
+Fang, Y., Wang, W., Xie, B., Sun, Q., Wu, L., Wang, X.,
+Huang, T., Wang, X., and Cao, Y. Eva: Exploring the
+limits of masked visual representation learning at scale.
+arXiv preprint arXiv:2211.07636, 2022. 6
+Garcia, X. and Firat, O. Using natural language prompts for
+machine translation. arXiv preprint arXiv:2202.11822,
+2022. 7
+Lester, B., Al-Rfou, R., and Constant, N. The power of scale
+for parameter-efficient prompt tuning. arXiv preprint
+arXiv:2104.08691, 2021. 1
+Li, X. L. and Liang, P. Prefix-tuning: Optimizing continuous
+prompts for generation. arXiv preprint arXiv:2101.00190,
+2021. 1
+Lin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ra-
+manan, D., Doll´ar, P., and Zitnick, C. L. Microsoft coco:
+Common objects in context. In European conference on
+computer vision (ECCV), pp. 740–755. Springer, 2014. 5
+Liu, J., Shen, D., Zhang, Y., Dolan, B., Carin, L., and Chen,
+W. What makes good in-context examples for gpt-3?
+arXiv preprint arXiv:2101.06804, 2021. 8
+Liu, Y., Ott, M., Goyal, N., Du, J., Joshi, M., Chen, D.,
+Levy, O., Lewis, M., Zettlemoyer, L., and Stoyanov, V.
+Roberta: A robustly optimized bert pretraining approach.
+arXiv preprint arXiv:1907.11692, 2019. 8
+Min, S., Lewis, M., Zettlemoyer, L., and Hajishirzi, H.
+Metaicl: Learning to learn in context. arXiv preprint
+arXiv:2110.15943, 2021. 7
+Pan, J., Lin, Z., Zhu, X., Shao, J., and Li, H. St-adapter:
+Parameter-efficient image-to-video transfer learning for
+action recognition. arXiv preprint arXiv:2206.13559,
+2022. 1
+Press, O., Zhang, M., Min, S., Schmidt, L., Smith, N. A.,
+and Lewis, M.
+Measuring and narrowing the com-
+positionality gap in language models. arXiv preprint
+arXiv:2210.03350, 2022. 7
+Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G.,
+Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J.,
+et al. Learning transferable visual models from natural
+language supervision. In International Conference on
+Machine Learning (ICML), pp. 8748–8763. PMLR, 2021.
+1, 4
+Ramesh, A., Pavlov, M., Goh, G., Gray, S., Voss, C., Rad-
+ford, A., Chen, M., and Sutskever, I. Zero-shot text-to-
+image generation. In International Conference on Ma-
+chine Learning (ICML), pp. 8821–8831. PMLR, 2021.
+1
+Rubin, O., Herzig, J., and Berant, J.
+Learning to re-
+trieve prompts for in-context learning. arXiv preprint
+arXiv:2112.08633, 2021. 8
+Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S.,
+Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein,
+M., Berg, A. C., and Fei-Fei, L. ImageNet Large Scale
+Visual Recognition Challenge. International Journal of
+Computer Vision (IJCV), 115(3):211–252, 2015. doi:
+10.1007/s11263-015-0816-y. 4
+Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton,
+E., Ghasemipour, S. K. S., Ayan, B. K., Mahdavi, S. S.,
+Lopes, R. G., et al. Photorealistic text-to-image diffusion
+models with deep language understanding. arXiv preprint
+arXiv:2205.11487, 2022. 1
+Shaban, A., Bansal, S., Liu, Z., Essa, I., and Boots, B. One-
+shot learning for semantic segmentation. british machine
+vision conference (BMVC), 2017. 2, 4
+
+What Makes Good Examples for Visual In-Context Learning?
+Zhang, Y., Zhou, K., and Liu, Z. Neural prompt search.
+arXiv preprint arXiv:2206.04673, 2022. 1
+Zhou, K., Liu, Z., Qiao, Y., Xiang, T., and Loy, C. C. Do-
+main generalization: A survey. IEEE Transactions on Pat-
+tern Analysis and Machine Intelligence (TPAMI), 2022a.
+5
+Zhou, K., Yang, J., Loy, C. C., and Liu, Z. Conditional
+prompt learning for vision-language models. In Con-
+ference on Computer Vision and Pattern Recognition
+(CVPR), 2022b. 1
+Zhou, K., Yang, J., Loy, C. C., and Liu, Z. Learning to
+prompt for vision-language models. International Jour-
+nal of Computer Vision (IJCV), 2022c. 1
+
+What Makes Good Examples for Visual In-Context Learning?
+A. Illustration of In-context Examples
+In the supplementary material, we illustrate more in-context
+learning results of foreground segmentation, single object
+detection, and colorization tasks.
+A.1. Foreground Segmentation
+The main paper presents the in-context examples from the
+person and cow categories. In the supplementary, as shown
+in Fig. 6-11, we present examples from the remained 18
+categories in Pascal-5i.
+A.2. Single Object Detection
+As shown in Fig. 12-13, we illustrate the in-context ex-
+amples from the single object detection task. By compar-
+ing the in-context examples picked by UnsupPR and those
+picked by SupPR, we find the examples found by SupPR
+are more similar to the queries in terms of object pose (e.g.,
+Fig. 12(f)), viewpoint (e.g., Fig. 12(r).
+A.3. Coloralization
+As shown in Fig. 14-15, we illustrate the in-context exam-
+ples from the colorization task. This task aims to map a
+gray-scale image to a color image. By comparing the in-
+context examples picked by UnsupPR and those picked by
+SupPR, we find the ground truth images of examples found
+by SupPR are more similar to that of the queries in terms of
+image style, e.g. the background color (e.g., Fig. 14(g)(h)).
+
+What Makes Good Examples for Visual In-Context Learning?
+(d)
+IoU: 85.00
+IoU: 47.70
+IoU: 90.00
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(a)
+(b)
+(c)
+(e)
+(f)
+IoU: 13.23
+IoU: 12.32
+IoU: 37.01
+IoU: 56.60
+IoU: 30.35
+IoU: 24.85
+IoU: 25.80
+IoU: 60.85
+IoU: 74.10
+IoU: 73.68
+IoU: 36.65
+IoU: 74.71
+IoU: 51.86
+IoU: 53.91
+IoU: 80.97
+IoU: 63.18
+IoU: 31.63
+IoU: 65.34
+IoU: 13.23
+IoU: 65,34
+IoU: 52.47
+IoU: 27.78
+IoU: 63.30
+IoU: 66.49
+IoU: 51.86
+IoU: 75.23
+IoU: 70.98
+IoU: 65.87
+IoU: 82.55
+IoU: 80.37
+IoU: 27.79
+IoU: 30.71
+IoU: 48.08
+IoU: 53.17
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+Figure 6: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These
+grids show examples from the train, tv, and bus categories.
+
+What Makes Good Examples for Visual In-Context Learning?
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+IoU: 67.02
+IoU: 71.39
+IoU: 38.79
+IoU: 47.04
+IoU: 5.50
+•
+IoU: 28.45
+IoU: 13.89
+IoU: 52.95
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+IoU: 29.58
+IoU: 7.06
+IoU: 34.12
+IoU: 43.23
+IoU: 74.64
+IoU: 78.78
+IoU: 4.09
+IoU: 43.13
+IoU: 40.52
+IoU: 35.16
+IoU: 38.64
+IoU: 16.38
+IoU: 21.61
+IoU: 20.45
+IoU: 66.50
+IoU: 55.35
+IoU: 66.08
+IoU: 57.10
+IoU: 30.44
+IoU: 45.98
+IoU: 65.49
+IoU: 54.25
+IoU: 77.60
+IoU: 81.50
+IoU: 35.28
+IoU: 66.75
+IoU: 61.31
+IoU: 34.47
+Figure 7: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These
+grids show examples from the bottle, sheep, and bird categories.
+
+What Makes Good Examples for Visual In-Context Learning?
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+IoU: 6.695
+IoU: 21.72
+IoU: 8.06
+IoU: 26.55
+IoU: 5.50
+•
+IoU: 6.96
+IoU: 13.89
+IoU: 25.93
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+IoU: 5.24
+IoU: 11.12
+IoU: 1.96
+IoU: 61.63
+IoU: 43.06
+IoU: 52.62
+IoU: 62.24
+IoU: 58.01
+IoU: 28.07
+IoU: 56.08
+IoU: 17.95
+IoU: 57.14
+IoU: 38.60
+IoU: 7.50
+IoU: 34.96
+IoU: 71.44
+IoU: 29.17
+IoU: 70.87
+IoU: 74.21
+IoU: 14.59
+IoU: 41.31
+IoU: 64.74
+IoU: 52.98
+IoU: 63.29
+IoU: 68.04
+IoU: 72.49
+IoU: 8.52
+IoU: 15.43
+Figure 8: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These
+grids show examples from the boat, airplane, and bicycle categories.
+
+What Makes Good Examples for Visual In-Context Learning?
+(d)
+IoU: 59.41
+•
+IoU: 67.96
+IoU: 76.11
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+IoU: 0.00
+IoU: 40.91
+IoU: 10.53
+IoU: 23.59
+IoU: 5.49
+IoU: 28.56
+IoU: 28.38
+IoU: 42.44
+(a)
+(b)
+(c)
+(e)
+(f)
+IoU: 23.46
+IoU: 0.25
+IoU: 33.00
+IoU: 28.87
+IoU: 34.29
+IoU: 62.26
+IoU: 9.00
+IoU: 50.00
+IoU: 0.00
+IoU: 0.00
+IoU: 63.77
+IoU: 63.17
+IoU: 26.03
+IoU: 5.00
+IoU: 84.00
+IoU: 75.39
+IoU: 79.47
+IoU: 51.00
+IoU: 48.62
+IoU: 51,92
+IoU: 72.58
+IoU: 92.00
+IoU: 57.00
+IoU: 26.25
+IoU: 36.67
+Figure 9: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These
+grids show examples from the car, cat, and chair categories.
+
+What Makes Good Examples for Visual In-Context Learning?
+(d)
+IoU: 22.37
+IoU: 65.46
+IoU: 65.50
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+IoU: 0.00
+IoU: 54.00
+IoU: 0.00
+IoU: 68.04
+IoU: 25.92
+IoU: 21.99
+IoU: 70.16
+IoU: 38.34
+(a)
+(b)
+(c)
+(e)
+(f)
+IoU: 48.93
+IoU: 73.70
+IoU: 41.00
+IoU: 54.02
+IoU: 41.22
+IoU: 60.00
+IoU: 42.66
+IoU: 26.76
+IoU: 39.71
+IoU: 67.09
+IoU: 17.91
+IoU: 22.38
+IoU: 59.27
+IoU: 71.61
+IoU: 85.94
+IoU: 49.44
+IoU: 71.77
+IoU: 68.91
+IoU: 65.63
+IoU: 60.82
+IoU: 70.87
+IoU: 72.97
+IoU: 71.77
+IoU: 51.60
+IoU: 77.78
+Figure 10: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These
+grids show examples from the dog, horse, and motorbike categories.
+
+14What Makes Good Examples for Visual In-Context Learning?
+(d)
+IoU: 0.00
+IoU: 63.09
+IoU: 6.45
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+IoU: 31.88
+IoU: 45.71
+IoU: 34.58
+IoU: 0.00
+IoU: 60.62
+IoU: 27.09
+(a)
+(b)
+(c)
+(e)
+(f)
+IoU: 0.74
+IoU: 8.34
+IoU: 23.25
+IoU: 3/00
+IoU: 43.73
+IoU: 27.15
+IoU: 29.16
+IoU: 2.44
+IoU: 56.38
+IoU: 0.00
+IoU: 2.23
+IoU: 28.82
+IoU: 62.40
+IoU: 41.35
+IoU: 93.05
+IoU: 39.02
+IoU: 45.35
+IoU: 65.69
+IoU: 18.00
+IoU: 46.55
+IoU: 33.91
+IoU: 35.21
+IoU: 33.75
+IoU: 47.07
+IoU: 41.56
+IoU: 7.08
+IoU: 38.61
+Figure 11: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These
+grids show examples from the table, plant, and sofa categories.
+
+What Makes Good Examples for Visual In-Context Learning?
+(d)
+IoU: 10.59
+IoU: 33.28
+IoU: 40.88
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(a)
+(b)
+(c)
+(e)
+(f)
+IoU: 3.23
+IoU: 18.61
+IoU: 0.00
+IoU: 26.82
+IoU: 25.88
+IoU: 44.57
+IoU: 33.40
+IoU: 61.47
+IoU: 51.28
+IoU: 59.97
+IoU: 68.02
+IoU: 66.10
+IoU: 32.34
+IoU: 30.13
+IoU: 54.34
+IoU: 27.73
+IoU: 0.00
+IoU: 20.43
+IoU: 21.28
+IoU: 0.00
+IoU: 1.73
+IoU: 11.90
+IoU: 36.78
+IoU: 71.30
+IoU: 51.64
+IoU: 27.65
+IoU: 61.60
+IoU: 32.60
+IoU: 58.09
+IoU: 0.00
+IoU: 22.71
+IoU: 23.47
+IoU: 68.81
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+Figure 12: In-context examples, which are from the single object detection task, retrieved by UnsupPR and SupPR. We find
+the examples found by SupPR are more similar to the queries in terms of object pose (e.g., (f)), viewpoint (e.g., (r))
+
+What Makes Good Examples for Visual In-Context Learning?
+(d)
+IoU: 31.79
+IoU: 32.98
+IoU: 63.17
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(a)
+(b)
+(c)
+(e)
+(f)
+IoU: 3.23
+IoU: 36.21
+IoU: 13.14
+IoU: 15.34
+IoU: 6.89
+IoU: 39.50
+IoU: 14.15
+IoU: 73.74
+IoU: 29.59
+IoU: 67.14
+IoU: 67.26
+IoU: 69.75
+IoU: 32.34
+IoU: 53.40
+IoU: 71.97
+IoU: 52.23
+IoU: 26.33
+IoU: 20.43
+IoU: 0.00
+IoU: 20.72
+IoU: 41.23
+IoU: 4.51
+IoU: 8.05
+IoU: 71.30
+IoU: 29.61
+IoU: 42.64
+IoU: 7.53
+IoU: 7.26
+IoU: 45.97
+IoU: 0.00
+IoU: 45.88
+IoU: 24.76
+IoU: 58.6
+(m)
+(n)
+(o)
+(p)
+(r)
+(s)
+c
+Figure 13: In-context examples, which are from the single object detection task, retrieved by UnsupPR and SupPR. We find
+the examples found by SupPR are more similar to the queries in terms of object pose (e.g., (l)), viewpoint (e.g., (m))
+
+BAGGAGEWhat Makes Good Examples for Visual In-Context Learning?
+mse : 0.79
+mse : 0.88
+mse : 1.13
+mse: 0.91
+mse : 0.85
+mse : 0.63
+(d)
+mse : 0.49
+(a)
+(b)
+(c)
+(e)
+(f)
+mse : 0.53
+mse : 0.67
+mse : 0.52
+mse : 0.78
+mse : 0.51
+mse : 1.15
+mse : 0.68
+mse : 0.76
+mse : 2.16
+mse : 0.99
+mse : 0.38
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+mse : 0.88
+mse : 0.73
+mse : 0.23
+mse : 0.17
+mse : 0.40
+mse : 0.48
+Figure 14: In-context examples, which are from the colorization task, retrieved by UnsupPR and SupPR. We also show the
+ground truth of the query image. The query image is the gray-scale version of its ground truth. The ground truth images
+of the in-context examples found by SupPR are more similar than those found by UnsupPR to the ground truth images of
+queries in terms of image style, e.g. the background color (g).
+
+What Makes Good Examples for Visual In-Context Learning?
+mse : 0.66
+mse : 1.15
+mse : 1.51
+mse: 1.01
+mse : 0.78
+mse : 0.79
+(d)
+mse : 0.44
+(a)
+(b)
+(c)
+(e)
+(f)
+mse : 0.80
+mse : 0.64
+mse : 0.80
+mse : 0.55
+mse : 0.51
+mse : 3.48
+mse : 1.19
+mse : 0.79
+mse : 0.88
+mse : 0.80
+mse : 0.99
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+mse : 0.36
+mse : 0.52
+mse : 0.85
+mse : 0.73
+mse : 0.49
+mse : 0.34
+Figure 15: In-context examples, which are from the colorization task, retrieved by UnsupPR and SupPR. We also show the
+ground truth of the query image. The query image is the gray-scale version of its ground truth. The ground truth images
+of the in-context examples found by SupPR are more similar than those found by UnsupPR to the ground truth images of
+queries in terms of image style, e.g. the background color (h).
+
+odidasacnosadidas
+adidas
+adidos