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+# Model Adaptation: Historical Contrastive Learning for Unsupervised Domain Adaptation without Source Data
+
+Jiaxing Huang, Dayan Guan, Aoran Xiao, Shijian Lu∗ School of Computer Science Engineering, Nanyang Technological University {Jiaxing.Huang, Dayan.Guan, Aoran.Xiao, Shijian.Lu}@ntu.edu.sg
+
+# Abstract
+
+Unsupervised domain adaptation aims to align a labeled source domain and an unlabeled target domain, but it requires to access the source data which often raises concerns in data privacy, data portability and data transmission efficiency. We study unsupervised model adaptation (UMA), or called Unsupervised Domain Adaptation without Source Data, an alternative setting that aims to adapt source-trained models towards target distributions without accessing source data. To this end, we design an innovative historical contrastive learning (HCL) technique that exploits historical source hypothesis to make up for the absence of source data in UMA. HCL addresses the UMA challenge from two perspectives. First, it introduces historical contrastive instance discrimination (HCID) that learns from target samples by contrasting their embeddings which are generated by the currently adapted model and the historical models. With the historical models, HCID encourages UMA to learn instance-discriminative target representations while preserving the source hypothesis. Second, it introduces historical contrastive category discrimination (HCCD) that pseudo-labels target samples to learn category-discriminative target representations. Specifically, HCCD re-weights pseudo labels according to their prediction consistency across the current and historical models. Extensive experiments show that HCL outperforms and state-of-the-art methods consistently across a variety of visual tasks and setups.
+
+# 1 Introduction
+
+Deep neural networks (DNNs) [28, 73, 23] have achieved great success in various computer vision tasks [8, 55, 60, 59, 28, 73, 23] but often generalize poorly to new domains due to the inter-domain discrepancy [1]. Unsupervised domain adaptation (UDA) [78, 51, 76, 64, 66, 79, 77, 103, 102, 25, 71, 44, 26, 86] addresses the inter-domain discrepancy by aligning the source and target data distributions, but it requires to access the source-domain data which often raises concerns in data privacy, data portability, and data transmission efficiency.
+
+In this work, we study unsupervised model adaptation (UMA), an alternative setting that aims to adapt source-trained models to fit target data distribution without accessing the source-domain data. Under the UMA setting, the only information carried forward is a portable source-trained model which is usually much smaller than the source-domain data and can be transmitted more efficiently [45, 42, 43, 72, 48] as illustrated in Table 1. Beyond that, the UMA setting also alleviates the concern of data privacy and intellectual property effectively. On the other hand, the absence of the labeled source-domain data makes domain adaptation much more challenging and susceptible to collapse.
+
+Table 1: Source data have much larger sizes than source-trained models.   
+
+<table><tr><td rowspan="2">Storage size (MB)</td><td colspan="2">Semantic segmentation</td><td>Object detection</td><td>Image classification</td></tr><tr><td>GTA5</td><td>SYNTHIA</td><td>Cityscapes</td><td>VisDA17</td></tr><tr><td>Source dataset</td><td>62,873.6</td><td>22,323.2</td><td>12,697.6</td><td>7,884.8</td></tr><tr><td>Source-trained model</td><td>179.1</td><td>179.1</td><td>553.4</td><td>172.6</td></tr></table>
+
+To this end, we develop historical contrastive learning (HCL) that aims to make up for the absence of source data by adapting the source-trained model to fit target data distribution without forgetting source hypothesis, as illustrated in Fig. 1. HCL addresses the UMA challenge from two perspectives. First, it introduces historical contrastive instance discrimination (HCID) that learns target samples by comparing their embeddings generated by the current model (as queries) and those generated by historical models (as keys): a query is pulled close to its positive keys while pushed apart from its negative keys. HCID can thus be viewed as a new type of instance contrastive learning for the task of UMA with historical models, which learns instance-discriminative target representations without forgetting source-domain hypothesis. Second, it introduces historical contrastive category discrimination (HCCD) that pseudo-labels target samples for learning category-discriminative target representations. Specifically, HCCD re-weights the pseudo labels according to their consistency across the current and historical models.
+
+The proposed HCL tackles UMA with three desirable features: 1) It introduces historical contrast and achieves UMA without forgetting source hypothesis; 2) The HCID works at instance level, which encourages to learn instance-discriminative target representations that generalize well to unseen data [98]; 3) The HCCD works at category level (i.e., output space) which encourages to learn category-discriminative target representation that is well aligned with the objective of down-stream tasks.
+
+The contributions of this work can be summarized in three aspects. First, we investigate memorybased learning for unsupervised model adaptation that learns discriminative representations for unlabeled target data without forgetting source hypothesis. To the best of our knowledge, this is the first work that explores memory-based learning for the task of UMA. Second, we design historical contrastive learning which introduces historical contrastive instance discrimination and category discrimination, the latter is naturally aligned with the objective of UMA. Third, extensive experiments show that the proposed historical contrastive learning outperforms state-of-the-art methods consistently across a variety of visual tasks and setups.
+
+# 2 Related Works
+
+Our work is closely related to several branches of research in unsupervised model adaptation, domain adaptation, memory-based learning and contrastive learning.
+
+Unsupervised model adaptation aims to adapt a source-trained model to fit target data distributions without accessing source-domain data. This problem has attracted increasing attention recently with a few pioneer studies each of which focuses on a specific visual task. For example, [45, 46] freezes the classifier of source-trained model and performs information maximization on target data for classification model adaptation. [42] tackles classification model adaptation with a conditional GANs that generates training images with target-alike styles and source-alike semantics. [43] presents a self-entropy descent algorithm to improve model adaptation for object detection. [72] reduces the uncertainty of target predictions (by source-trained model) for segmentation model adaptation. [48] introduces data-free knowledge distillation to transfer source-domain knowledge for segmentation model adaptation. Despite the different designs for different tasks, the common motivation of these studies is to make up for the absence of source data in domain adaptation. [40] and [88] tackle source-free domain adaptation from a generative manner by generating samples from the source classes and generating reference distributions, respectively.
+
+We tackle the absence of source data by a memory mechanism that encourages to memorize source hypothesis during model adaptation. Specifically, we design historical contrastive learning that learns target representations by contrasting historical and currently evolved models. To the best of our knowledge, this is the first work that explores memory mechanism for UMA.
+
+${ x _ { s r c } } / { x _ { t g t } }$ : Source/target data $G ^ { t } / G ^ { t - m }$ : Current/historical model ????????/???????? : Source/target feature $f ^ { t } / f ^ { t - m }$ Current/historical feature:
+
+![](images/a7b73cae93218e46b16f87ef85b82645c03ffed18ea4ededb7c1b6796d7b9609.jpg)  
+Figure 1: Illustration of unsupervised domain adaptation, unsupervised model adaptation and the proposed historical contrastive learning which exploits historical source hypothesis (or memorized knowledge) to make up for the absence of source supervision in the process of UMA. Here the historical source hypothesis could be the original source hypothesis $G ^ { 0 }$ (i.e. $\scriptstyle { \mathrm { t } } = { \mathrm { m } }$ , trained using the labeled source data only), the adapted source hypothesis $\bar { G } ^ { t - m }$ (i.e. $m < t$ , trained in the last $m$ epoch), or other types of previous models.
+
+![](images/5f47b4088424c1bf835a66b73a5cf9927777552921bdaa0df571f46f0f833254.jpg)
+
+Domain adaptation is related to UMA but it requires to access labeled source data in training. Most existing work handles UDA from three typical approaches. The first exploits adversarial training to align source and target distributions in the feature, output or latent space [78, 51, 76, 14, 96, 64, 66, 79, 34, 77, 35, 21, 94, 29, 10, 9, 80, 20]. The second employs self-training to generate pseudo labels to learn from unlabeled target data iteratively [103, 69, 100, 102, 31, 19, 90, 91]. The third leverages image translation to modify image styles to reduce domain gaps [25, 71, 44, 95, 26, 86, 32, 33, 93, 92]. In addition, [30] proposes a categorical contrastive learning for domain adaptation.
+
+Memory-based learning has been studied extensively. Memory networks [81] as one of early efforts explores to use external modules to store memory for supervised learning. Temporal ensemble [41], as well as a few following works [74, 12] extend the memory mechanism to semi-supervised learning. It employs historical hypothesis/models to regularize the current model and produces stable and competitive predictions. Mean Teacher [74] leverages moving-average models as the memory model to regularize the training, and similar idea was extended for UDA [16, 99, 4, 52]. Mutual learning [97] has also been proposed for learning among multiple peer student models.
+
+Most aforementioned methods require labeled data in training. They do not work very well for UMA due to the absence of supervision from the labeled source data, by either collapsing in training or helping little in model adaptation performance. We design innovative historical contrastive learning to make up for the absence of the labeled source data, more details to be presented in the ensuing subsections.
+
+Contrastive learning [82, 87, 22, 54, 101, 24, 56, 75, 36, 11] learns discriminative representations from multiple views of the same instance. It works with certain dictionary look-up mechanism [22], where a given image $x$ is augmented into two views, query and key, and the query token $q$ should match its designated key $k _ { + }$ over a set of negative keys $k _ { - }$ from other images. Existing work can be broadly classified into three categories based on dictionary creation strategies. The first creates a memory bank [82] to store all the keys in the previous epoch. The second builds an end-to-end dictionary [87, 75, 36, 11] that generates keys using samples from the current mini-batch. The third employs a momentum encoder [22] that generates keys on-the-fly by a momentum-updated encoder.
+
+Other related source-free adaptation works. [7] considers supervised continual learning from previously learned tasks to a new task, which learns representations using the contrastive learning objective and preserves learned representations using a self-supervised distillation step, where the contrastively learned representations are more robust against the catastrophic forgetting for supervised continual learning. [38] addresses a source-free universal domain adaptation problem that does not guarantee that the classes in the target domain are the same as in the source domain. [39] propose a simple yet effective solution to realize inheritable models suitable for open-set source-free DA problem.
+
+![](images/fb878d7d22a0b009deca2f1081b39ae80e7f8442f02f9246c50db87303021e17.jpg)  
+Figure 2: The proposed historical contrastive learning consists of two key designs including historical contrastive instance discrimination (HCID) and historical contrastive category discrimination (HCCD). HCID learns from target samples by contrasting their embeddings generated by the current model (as queries) and historical models (as keys), which learns instance-discriminative target representations. HCCD pseudo-labels target samples to learn category-discriminative target representations, where the pseudo labels are re-weighted adaptively according to the prediction consistency across the current and historical models.
+
+# 3 Historical Contrastive Learning
+
+This section presents the proposed historical contrastive learning that memorizes source hypothesis to make up for the absence of source data as illustrated in Fig. 2. The proposed HCL consists of two key designs. The first is historical contrastive instance discrimination which encourages to learn instance-discriminative target representations that generalize well to unseen data [98]. The second is historical contrastive category discrimination that encourages to learn category-discriminative target representations which is well aligned with the objective of visual recognition tasks. More details to be described in the ensuring subsections.
+
+# 3.1 Historical Contrastive Instance Discrimination
+
+The proposed HCID learns from unlabeled target samples via contrastive learning over their embeddings generated from current and historical models: the positive pairs are pulled close while negative pairs are pushed apart. It is a new type of contrastive learning for UMA, which preserves sourcedomain hypothesis by generating positive keys from historical models. HCID works at instance level and encourages to learn instance-discriminative target representations that generalize well to unseen data [98].
+
+HCID loss. Given a query sample $x _ { q }$ and a set of key samples $X _ { k } = \{ x _ { k _ { 0 } } , x _ { k _ { 1 } } , x _ { k _ { 2 } } , . . . , x _ { k _ { N } } \}$ , HCID employs current model $E ^ { t }$ to encode the query $q ^ { t } = E ^ { t } ( x _ { q } )$ , and historical encoders $E ^ { t - m }$ to encode the keys $k _ { n } ^ { t - m } = E ^ { t - m } ( x _ { k _ { n } } ) , n = 0 , \cdots , \bar { N }$ . With the encoded embeddings, HCID is achieved via a historical contrastive loss $\mathcal { L } _ { \mathrm { H i s N C E } }$ , minimization of which pulls $q$ close to its positive key $k _ { + } ^ { t - m }$ while pushing it apart from all other (negative) keys:
+
+$$
+\mathcal { L } _ { \mathrm { H i s N C E } } = \sum _ { x _ { q } \in X _ { t g t } } - \log \frac { \exp ( q ^ { t } \cdot k _ { + } ^ { t - m } / \tau ) r _ { + } ^ { t - m } } { \sum _ { i = 0 } ^ { N } \exp ( q ^ { t } \cdot k _ { i } ^ { t - m } / \tau ) r _ { i } ^ { t - m } }
+$$
+
+where $\tau$ is a temperature parameter [82], $r$ indicates the reliability of each key $k _ { n } ^ { t - m }$ , with which we reweight the similarity loss of each key to encourage to memorize well-learnt instead of poorly-learnt historical embeddings. In this work, we use the classification entropy to estimate the reliability of each key. The positive key sample is the augmentation of the query sample[22, 36], and all the rest are negative keys.
+
+Remark 1 Note $\mathcal { L } _ { H i s N C E }$ in Eq.1 has a similar form as the InfoNCE loss[56, 22]. InfoNCE can actually be viewed as a special case of HisNCE, where all the query and keys are encoded by the current model ${ ' m = 0 }$ ) and the reliability is fixed ${ \bf \zeta } r _ { i } ^ { t - m } = 1 , \forall i )$ . For HisNCE, we assign each key a reliability score to encourage to memorize the well-learnt historical embeddings only. It is also worth noting that Eq. 1 only shows historical contrast with one historical model for simplicity. In practice, we could employ multiple historical models to comprehensively distill (memorize) the well-learnt embeddings from them. It could be achieved by computing Eq.1 multiple times.
+
+# 3.2 Historical contrastive category discrimination
+
+We design HCCD that generates pseudo labels and learns them conditioned on a historical consistency, i.e., the prediction consistency across the current and historical models. HCCD can be viewed as a new type of self-training, where pseudo labels are re-weighted by the historical consistency. It works at category level and encourages to learn category-discriminative target representations that are aligned with the objective of visual recognition tasks in UMA.
+
+Historical contrastive pseudo label generation. Given an unlabeled sample $x$ , the current and historical models predict $p ^ { t } = G ^ { t } ( x )$ (as the query) and $p ^ { t - m } = G ^ { t - m } ( x )$ (as the keys). The pseudo label and the historical consistency of the sample are computed by:
+
+$$
+\begin{array} { l } { { \hat { y } = \mathbf { { \Gamma } } ( p ^ { t } ) , } } \\ { { h _ { c o n } = 1 - \mathrm { S i g m o i d } ( | | p ^ { t } - p ^ { t - m } | | _ { 1 } ) , } } \end{array}
+$$
+
+where $p$ is a $K$ -class probability vector, $\mathbf { \delta T }$ is the pseudo label generation function [103, 102] and $\hat { y } = ( \hat { y } ^ { ( 1 ) } , \hat { y } ^ { ( 2 ) } , . . . , \hat { y } ^ { ( \bar { C } ) } )$ is the predicted category label.
+
+HCCD loss. Given the unlabeled data $x$ and its historical contrastive pseudo label $\left( \hat { y } , h _ { c o n } \right)$ , HCCD performs self-training on target data $x$ via a weighted cross-entropy loss:
+
+$$
+\mathcal { L } _ { \mathrm { H i s S T } } = - \sum _ { x \in X _ { t g t } } h _ { c o n } \times \hat { y } \log p _ { x } ,
+$$
+
+where $h _ { c o n }$ is the per-sample historical consistency and we use it to re-weight the self-training loss. If the predictions of a sample across the current and historical models are consistent, we consider it as a well-learnt sample and increase its influence in self-training. Otherwise, we consider it as pooly-learnt sample and decrease its influence in self-training.
+
+# 3.3 Theoretical Insights
+
+The two designs in Historical Contrastive Learning (HCL) are inherently connected with some probabilistic models and convergent under certain conditions:
+
+Proposition 1 The historical contrastive instance discrimination (HCID) can be modelled as a maximum likelihood problem optimized via Expectation Maximization.
+
+Proposition 2 The HCID is convergent under certain conditions.
+
+Proposition 3 The historical contrastive category discrimination (HCCD) can be modelled as a classification maximum likelihood problem optimized via Classification Expectation Maximization.
+
+Proposition 4 The HCCD is convergent under certain conditions.
+
+The proofs of Proposition 1, Proposition 2, Proposition 3 and Proposition 4 are provided in the appendix, respectively.
+
+# 4 Experiments
+
+This section presents experiments including datasets, implementation details, evaluations of the proposed HCL in semantic segmentation, object detection and image classification tasks as well as the discussion of its desirable features.
+
+Table 2: Experiments on semantic segmentation task GTA5 Cityscapes (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Method</td><td>SF</td><td>Road</td><td>SW</td><td>Build</td><td>Wall</td><td>Fence</td><td>Pole</td><td>TL</td><td>TS</td><td>Veg.</td><td>Terrain</td><td>Sky</td><td>PR</td><td>Rider</td><td>Car</td><td>Truck</td><td>Bus</td><td>Train</td><td>Motor</td><td>Bike</td><td>mIoU</td></tr><tr><td>AdaptSeg [76]</td><td>X</td><td>86.5</td><td>36.0</td><td>79.9</td><td>23.4</td><td>23.3</td><td>23.9</td><td>35.2</td><td>14.8</td><td>83.4</td><td>33.3</td><td>75.6</td><td>58.5</td><td>27.6</td><td>73.7</td><td>32.5</td><td>35.4</td><td>3.9</td><td>30.1</td><td>28.1</td><td>42.4</td></tr><tr><td>AdvEnt [79]</td><td>X</td><td>89.4</td><td>33.1</td><td>81.0</td><td>26.6</td><td>26.8</td><td>27.2</td><td>33.5</td><td>24.7</td><td>83.9</td><td>36.7</td><td>78.8</td><td>58.7</td><td>30.5</td><td>84.8</td><td>38.5</td><td>44.5</td><td>1.7</td><td>31.6</td><td>32.4</td><td>45.5</td></tr><tr><td>IDA [57]</td><td>X</td><td>90.6</td><td>37.1</td><td>82.6</td><td>30.1</td><td>19.1</td><td>29.5</td><td>32.4</td><td>20.6</td><td>85.7</td><td>40.5</td><td>79.7</td><td>58.7</td><td>31.1</td><td>86.3</td><td>31.5</td><td>48.3</td><td>0.0</td><td>30.2</td><td>35.8</td><td>46.3</td></tr><tr><td>CRST[102]</td><td>X</td><td>91.0</td><td>55.4</td><td>80.0</td><td>33.7</td><td>21.4</td><td>37.3</td><td>32.9</td><td>24.5</td><td>85.0</td><td>34.1</td><td>80.8</td><td>57.7</td><td>24.6</td><td>84.1</td><td>27.8</td><td>30.1</td><td>26.9</td><td>26.0</td><td>42.3</td><td>47.1</td></tr><tr><td>CrCDA [35]</td><td>X</td><td>92.4</td><td>55.3</td><td>82.3</td><td>31.2</td><td>29.1</td><td>32.5</td><td>33.2</td><td>35.6</td><td>83.5</td><td>34.8</td><td>84.2</td><td>58.9</td><td>32.2</td><td>84.7</td><td>40.6</td><td>46.1</td><td>2.1</td><td>31.1</td><td>32.7</td><td>48.6</td></tr><tr><td>UR [72]</td><td>√</td><td>92.3</td><td>55.2</td><td>81.6</td><td>30.8</td><td>18.8</td><td>37.1</td><td>17.7</td><td>12.1</td><td>84.2</td><td>35.9</td><td>83.8</td><td>57.7</td><td>24.1</td><td>81.7</td><td>27.5</td><td>44.3</td><td>6.9</td><td>24.1</td><td>40.4</td><td>45.1</td></tr><tr><td>+HCL</td><td>√</td><td>92.2</td><td>54.1</td><td>81.7</td><td>34.2</td><td>25.4</td><td>37.9</td><td>35.8</td><td>29.8</td><td>84.1</td><td>38.0</td><td>83.9</td><td>59.1</td><td>27.1</td><td>84.6</td><td>33.9</td><td>41.9</td><td>16.2</td><td>27.7</td><td>44.7</td><td>49.1</td></tr><tr><td>SFDA [48]</td><td>√</td><td>91.7</td><td>52.7</td><td>82.2</td><td>28.7</td><td>20.3</td><td></td><td>36.530.6</td><td></td><td>23.681.7</td><td>35.6</td><td>84.859.5</td><td></td><td>22.6</td><td>83.4</td><td>29.6</td><td>32.4</td><td>11.8</td><td>23.8</td><td>39.6</td><td>45.8</td></tr><tr><td>+HCL</td><td>√</td><td>92.3</td><td>54.5</td><td>82.6</td><td>33.1</td><td>26.2</td><td>38.9</td><td>37.9</td><td>31.7</td><td>83.5</td><td>38.1</td><td>84.4</td><td>60.9</td><td>30.0</td><td>84.5</td><td>32.6</td><td>41.2</td><td>14.2</td><td>26.4</td><td>43.2</td><td>49.3</td></tr><tr><td>HCID</td><td>√</td><td>89.5</td><td>53</td><td>80.3</td><td>33.9</td><td>22.9</td><td>36.2</td><td>32.7</td><td></td><td>23.882.3</td><td>36.5</td><td>73.7</td><td>60.0</td><td>22.4</td><td>83.8</td><td>28.9</td><td>34.7</td><td>13.5</td><td>21.2</td><td>38.0</td><td>45.6</td></tr><tr><td>HCCD</td><td>√</td><td>91.0</td><td>53.6</td><td>81.5</td><td>32.4</td><td>23.1</td><td>36.9</td><td>32.3</td><td>26.3</td><td>82.8</td><td>37.2</td><td>80.4</td><td>58.5</td><td>25.0</td><td>82.5</td><td>29.9</td><td>34.2</td><td>15.5</td><td>23.2</td><td>40.5</td><td>46.7</td></tr><tr><td>HCL</td><td></td><td>92.0</td><td>55.0</td><td>80.4</td><td>33.5</td><td>24.6</td><td>37.1</td><td>35.1</td><td></td><td>28.883.0</td><td>37.6</td><td>82.359.4</td><td></td><td>27.6</td><td>83.6</td><td>32.3</td><td>36.6</td><td>14.1</td><td>28.7</td><td>43.0</td><td>48.1</td></tr></table>
+
+Table 3: Experiments on semantic segmentation task SYNTHIA Cityscapes (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Method</td><td>SF</td><td>Road</td><td>SW</td><td>Build</td><td>Wall</td><td>Fence</td><td>Pole</td><td>TL</td><td>TS</td><td>Veg.</td><td>Sky</td><td>PR</td><td>Rider</td><td>Car</td><td>Bus</td><td>Motor</td><td>Bike</td><td>mIoU</td><td>mIoU</td></tr><tr><td>AdaptSeg[76]</td><td>X</td><td>84.3</td><td>42.7</td><td>77.5</td><td>，</td><td>1</td><td>-</td><td>4.7</td><td>7.0</td><td>77.9</td><td>82.5</td><td>54.3</td><td>21.0</td><td>72.3</td><td>32.2</td><td>18.9</td><td>32.3</td><td>1</td><td>46.7</td></tr><tr><td>AdvEnt [79]</td><td>X</td><td>85.6</td><td>42.2</td><td>79.7</td><td>8.7</td><td>0.4</td><td>25.9</td><td>5.4</td><td>8.1</td><td>80.4</td><td>84.1</td><td>57.9</td><td>23.8</td><td>73.3</td><td>36.4</td><td>14.2</td><td>33.0</td><td>41.2</td><td>48.0</td></tr><tr><td>IDA [57]</td><td>X</td><td>84.3</td><td>37.7</td><td>79.5</td><td>5.3</td><td>0.4</td><td>24.9</td><td>9.2</td><td>8.4</td><td>80.0</td><td>84.1</td><td>57.2</td><td>23.0</td><td>78.0</td><td>38.1</td><td>20.3</td><td>36.5</td><td>41.7</td><td>48.9</td></tr><tr><td>CRST[102]</td><td>×</td><td>67.7</td><td>32.2</td><td>73.9</td><td>10.7</td><td>1.6</td><td>37.4</td><td>22.2</td><td>31.2</td><td>80.8</td><td>80.5</td><td>60.8</td><td>29.1</td><td>82.8</td><td>25.0</td><td>19.4</td><td>45.3</td><td>43.8</td><td>50.1</td></tr><tr><td>CrCDA[35]</td><td>X</td><td>86.2</td><td>44.9</td><td>79.5</td><td>8.3</td><td>0.7</td><td>27.8</td><td>9.4</td><td>11.8</td><td>78.6</td><td>86.5</td><td>57.2</td><td>26.1</td><td>76.8</td><td>39.9</td><td>21.5</td><td>32.1</td><td>42.9</td><td>50.0</td></tr><tr><td>UR[72]</td><td>√</td><td>59.3</td><td>24.6</td><td>77.0</td><td>14.0</td><td>1.8</td><td>31.5</td><td>18.3</td><td>32.0</td><td>83.1</td><td>80.4</td><td>46.3</td><td>17.8</td><td>76.7</td><td>17.0</td><td>18.5</td><td>34.6</td><td>39.6</td><td>45.0</td></tr><tr><td>+HCL</td><td>√</td><td>76.7</td><td>33.7</td><td>78.7</td><td>7.2</td><td>0.1</td><td>34.4</td><td>23.2</td><td>31.6</td><td>80.5</td><td>84.3</td><td>54.4</td><td>26.6</td><td>79.5</td><td>35.9</td><td>24.8</td><td>34.4</td><td>44.1</td><td>51.1</td></tr><tr><td>SFDA [48]</td><td>√</td><td>67.8</td><td>31.9</td><td>77.1</td><td>8.3</td><td>1.1</td><td>35.9</td><td>21.2</td><td>26.7</td><td>79.8</td><td>79.4</td><td>58.8</td><td>27.3</td><td>80.4</td><td>25.3</td><td>19.5</td><td>37.4</td><td>42.4</td><td>48.7</td></tr><tr><td>+HCL</td><td>√</td><td>78.2</td><td>35.3</td><td>79.6</td><td>7.3</td><td>0.2</td><td>37.7</td><td>21</td><td>30.9</td><td>80.4</td><td>83.3</td><td>59.8</td><td>29.4</td><td>79.2</td><td>34.2</td><td>24.5</td><td>38.9</td><td>45.0</td><td>51.9</td></tr><tr><td>HCL</td><td>√</td><td>80.9</td><td>34.9</td><td>76.7</td><td>6.6</td><td>0.2</td><td>36.1</td><td>20.1</td><td>28.2</td><td>79.1</td><td>83.1</td><td>55.6</td><td>25.6</td><td>78.8</td><td>32.7</td><td>24.1</td><td>32.7</td><td>43.5</td><td>50.2</td></tr></table>
+
+# 4.1 Datasets
+
+UMA for semantic segmentation is evaluated on two challenging tasks GTA5 $[ 6 1 ]  $ Cityscapes [15] and SYNTHIA $[ 6 2 ]  \cdot$ Cityscapes. GTA5 has 24, 966 synthetic images and shares 19 categories with Cityscapes. SYNTHIA contains 9, 400 synthetic images and shares 16 categories with Cityscapes. Cityscapes has 2975 and 500 real-world images for training and validation, respectively.
+
+UMA for object detection is evaluated on tasks Cityscapes Foggy Cityscapes [68] and Cityscapes $ \mathrm { B D D 1 0 0 k }$ [89]. Foggy Cityscapes is derived by applying simulated fog to the 2, 975 Cityscapes images. BDD100k has $7 0 k$ training images and $1 0 k$ validation images, and shares 7 categories with Cityscapes. We evaluate a subset of BDD100k (i.e., daytime) as in [84, 65, 13] for fair comparisons.
+
+UMA for image classification is evaluated on benchmarks VisDA17 [58] and Office-31 [63]. VisDA17 has 152, 409 synthetic images as the source domain and 55, 400 real images of 12 shared categories as the target domain. Office-31 has 4110 images of three sources including 2817 from Amazon, 795 from Webcam and 498 from DSLR (with 31 shared categories). Following [102, 63, 70], the evaluation is on six adaptation tasks $\mathbf { A } {  } \mathbf { W } .$ , $\mathrm { D } \to \mathsf { W } .$ , $\mathrm { W } {  } \mathrm { D }$ , $\mathbf { A } {  } \mathbf { D }$ , $\mathrm { D } { \to } \mathsf { A }$ , and $\mathrm { W } { \to } \mathrm { A }$ .
+
+# 4.2 Implementation Details
+
+Semantic segmentation: Following [79, 103], we employ DeepLab-V2 [8] as the segmentation model. We adopt SGD [3] with momentum 0.9, weight decay $1 e - 4$ and learning rate $2 . 5 e \mathrm { ~ - ~ } 4$ where the learning rate is decayed by a polynomial annealing policy [8].
+
+Object detection: Following [84, 65, 13], we adopt Faster R-CNN [60] as the detection model. We use SGD [3] with momentum 0.9 and weight decay $5 e - 4$ . The learning rate is $1 e - 3$ for first $5 0 k$ iterations and then decreased to $1 e - 4$ for another $2 0 k$ iterations.
+
+Image classification: Following [102, 63, 70], we adopt ResNet-101 and ResNet-50 [23] as the classification models for VisDA17 and Office-31, respectively. We use SGD [3] with momentum 0.9, weight decay $5 e - 4$ , learning rate $1 e - 3$ and batch size 32.
+
+# 4.3 Unsupervised Domain Adaption for Semantic Segmentation
+
+We evaluated the proposed HCL in UMA-based semantic segmentation tasks $\mathrm { G T A } 5 $ Cityscapes and SYNTHIA Cityscapes. Tables 2 and 3 show experimental results in mean Intersectionover-Union (mIoU). We can see that HCL outperforms state-of-the-art UMA methods by large margins. In addition, HCL is complementary to existing UMA methods and incorporating it as denoted by $\mathrm { ^ { 6 6 } { + } H C L ^ { 9 3 } }$ improves the existing UMA methods clearly and consistently. Furthermore, HCL even achieves competitive performance as compared with state-of-the-art UDA methods (labeled by ✗in the column SF) which require to access the labeled source data in training. Further, We conduct ablation studies of the proposed HCL over the UMA-based semantic segmentation task $\mathrm { G T A } 5  $ Cityscapes. As the bottom of Table 2 shows, either HCID or HCCD achieves comparable performance. In addition, HCID and HCCD offer orthogonal self-supervision signals where HCID focuses on instance-level discrimination between queries and historical keys and HCCD focuses on category-level discrimination among samples with different pseudo category labels. The two designs are thus complementary and the combination of them in HCL produces the best segmentation.
+
+Table 4: Experiments on object detection task Cityscapes Foggy Cityscapes (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Method</td><td>SF</td><td>person</td><td>rider</td><td>car</td><td>truck</td><td>bus</td><td>train</td><td>mcycle</td><td>bicycle</td><td>mAP</td></tr><tr><td>DA [13]</td><td></td><td>25.0</td><td>31.0</td><td>40.5</td><td>22.1</td><td>35.3</td><td>20.2</td><td>20.0</td><td>27.1</td><td>27.6</td></tr><tr><td>MLDA [83]</td><td></td><td>33.2</td><td>44.2</td><td>44.8</td><td>28.2</td><td>41.8</td><td>28.7</td><td>30.5</td><td>36.5</td><td>36.0</td></tr><tr><td>DMA [37]</td><td></td><td>30.8</td><td>40.5</td><td>44.3</td><td>27.2</td><td>38.4</td><td>34.5</td><td>28.4</td><td>32.2</td><td>34.6</td></tr><tr><td>CAFA [27]</td><td></td><td>41.9</td><td>38.7</td><td>56.7</td><td>22.6</td><td>41.5</td><td>26.8</td><td>24.6</td><td>35.5</td><td>36.0</td></tr><tr><td>SWDA [65]</td><td></td><td>36.2</td><td>35.3</td><td>43.5</td><td>30.0</td><td>29.9</td><td>42.3</td><td>32.6</td><td>24.5</td><td>34.3</td></tr><tr><td>CRDA [84]</td><td>xxxxxx</td><td>32.9</td><td>43.8</td><td>49.2</td><td>27.2</td><td>45.1</td><td>36.4</td><td>30.3</td><td>34.6</td><td>37.4</td></tr><tr><td>SFOD [43]</td><td>1</td><td>25.5</td><td>44.5</td><td>40.7</td><td>33.2</td><td>22.2</td><td>28.4</td><td>34.1</td><td>39.0</td><td>33.5</td></tr><tr><td>+HCL</td><td></td><td>39.3</td><td>46.7</td><td>48.6</td><td>32.9</td><td>46.2</td><td>38.2</td><td>33.9</td><td>36.9</td><td>40.3</td></tr><tr><td>HCL</td><td>√</td><td>38.7</td><td>46.0</td><td>47.9</td><td>33.0</td><td>45.7</td><td>38.9</td><td>32.8</td><td>34.9</td><td>39.7</td></tr></table>
+
+Table 5: Experiments on object detection task Cityscapes $ \mathrm { B D D 1 0 0 k }$ (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Method</td><td>SF</td><td>person</td><td>rider</td><td>car</td><td>truck</td><td>bus</td><td>mcycle</td><td>bicycle</td><td>mAP</td></tr><tr><td>DA[13]</td><td>X</td><td>29.4</td><td>26.5</td><td>44.6</td><td>14.3</td><td>16.8</td><td>15.8</td><td>20.6</td><td>24.0</td></tr><tr><td rowspan="3">SWDA [65] CRDA [84]</td><td></td><td>30.2</td><td>29.5</td><td>45.7</td><td>15.2</td><td>18.4</td><td>17.1</td><td>21.2</td><td>25.3</td></tr><tr><td>X</td><td>31.4</td><td>31.3</td><td>46.3</td><td>19.5</td><td>18.9</td><td>17.3</td><td>23.8</td><td>26.9</td></tr><tr><td>·</td><td>32.4</td><td>32.6</td><td>50.4</td><td>20.6</td><td>23.4</td><td>18.9</td><td>25.0</td><td>29.0</td></tr><tr><td>SFOD [43] +HCL</td><td></td><td>33.9</td><td>34.4</td><td>52.8</td><td>22.1</td><td>25.3</td><td>22.6</td><td>26.7</td><td>31.1</td></tr><tr><td>HCL</td><td>√</td><td>32.7</td><td>33.2</td><td>52.0</td><td>21.3</td><td>25.6</td><td>21.5</td><td>26.0</td><td>30.3</td></tr></table>
+
+# 4.4 Unsupervised Domain Adaptation for Object Detection
+
+We evaluated the proposed HCL over the UMA-based object detection tasks Cityscapes $\mathrm { :  F o g g y }$ Cityscapes and Cityscapes $ \mathrm { B D D 1 0 0 k }$ . Tables 4 and 5 show experimental results. We can observe that HCL outperforms state-of-the-art UMA method SFOD clearly. Similar to the semantic segmentation experiments, HCL achieves competitive performance as compared with state-of-the-art UDA methods (labeled by ✗in column SF) which require to access labeled source data in training.
+
+# 4.5 Unsupervised Domain Adaptation for Image Classification
+
+We evaluate the proposed HCL over the UMA-based image classificat tasks VisDA17 and Office-31. Tables 6 and 7 show experimental results. We can observe that HCL outperforms state-of-the-art UMA methods clearly. Similar to the semantic segmentation and object detection experiments, HCL achieves competitive performance as compared with state-of-the-art UDA methods (labeled by $\pmb { \chi }$ ) which require to access labeled source data in training.
+
+# 4.6 Discussion
+
+Generalization across computer vision tasks: We study how HCL generalizes across computer vision tasks by evaluating it over three representative tasks on semantic segmentation, object detection and image classification. Experiments in Tables 2- 7 show that HCL achieves competitive performance consistently across all three visual tasks, demonstrating the generalization ability of HCL across computer vision tasks.
+
+Complementarity studies: We study the complementarity of our proposed HCL by combining it with existing UMA methods. Experiments in Table 2 (the row highlighted by $\ " + \mathrm { H C L } \ '$ ) shows that incorporating HCL boosts the existing UMA methods consistently.
+
+Table 6: Experiments on image classification benchmark VisDA17 (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Method</td><td>SF</td><td>Aero</td><td>Bike</td><td>Bus</td><td>Car</td><td>Horse</td><td>Knife</td><td>Motor</td><td>Person</td><td>Plant</td><td>Skateboard</td><td>Train</td><td>Truck</td><td>Mean</td></tr><tr><td>DANN [17]</td><td></td><td>81.9</td><td>77.7</td><td>82.8</td><td>44.3</td><td>81.2</td><td>29.5</td><td>65.1</td><td>28.6</td><td>51.9</td><td>54.6</td><td>82.8</td><td>7.8</td><td>57.4</td></tr><tr><td>ENT[18]</td><td></td><td>80.3</td><td>75.5</td><td>75.8</td><td>48.3</td><td>77.9</td><td>27.3</td><td>69.7</td><td>40.2</td><td>46.5</td><td>46.6</td><td>79.3</td><td>16.0</td><td>57.0</td></tr><tr><td>MCD [66]</td><td></td><td>87.0</td><td>60.9</td><td>83.7</td><td>64.0</td><td>88.9</td><td>79.6</td><td>84.7</td><td>76.9</td><td>88.6</td><td>40.3</td><td>83.0</td><td>25.8</td><td>71.9</td></tr><tr><td>CBST [103]</td><td></td><td>87.2</td><td>78.8</td><td>56.5</td><td>55.4</td><td>85.1</td><td>79.2</td><td>83.8</td><td>77.7</td><td>82.8</td><td>88.8</td><td>69.0</td><td>72.0</td><td>76.4</td></tr><tr><td>CRST[102]</td><td>xxxxx</td><td>88.0</td><td>79.2</td><td>61.0</td><td>60.0</td><td>87.5</td><td>81.4</td><td>86.3</td><td>78.8</td><td>85.6</td><td>86.6</td><td>73.9</td><td>68.8</td><td>78.1</td></tr><tr><td>3C-GAN [42]</td><td>√</td><td>94.8</td><td>73.4</td><td>68.8</td><td>74.8</td><td>93.1</td><td>95.4</td><td>88.6</td><td>84.7</td><td>89.1</td><td>84.7</td><td>83.5</td><td>48.1</td><td>81.6</td></tr><tr><td>+HCL</td><td>√</td><td>93.8</td><td>86.6</td><td>84.1</td><td>74.3</td><td>93.2</td><td>95.0</td><td>88.4</td><td>85.0</td><td>90.4</td><td>85.2</td><td>84.5</td><td>49.8</td><td>84.2</td></tr><tr><td>SHOT[45]</td><td>√</td><td>93.7</td><td>86.4</td><td>78.7</td><td>50.7</td><td>91.0</td><td>93.5</td><td>79.0</td><td>78.3</td><td>89.2</td><td>85.4</td><td>87.9</td><td>51.1</td><td>80.4</td></tr><tr><td>+HCL</td><td>√</td><td>94.3</td><td>87.0</td><td>82.6</td><td>70.6</td><td>92.0</td><td>93.2</td><td>87.0</td><td>80.6</td><td>89.6</td><td>86.8</td><td>84.6</td><td>58.7</td><td>83.9</td></tr><tr><td>HCL</td><td>√</td><td>93.3</td><td>85.4</td><td>80.7</td><td>68.5</td><td>91.0</td><td>88.1</td><td>86.0</td><td>78.6</td><td>86.6</td><td>88.8</td><td>80.0</td><td>74.7</td><td>83.5</td></tr></table>
+
+Table 7: Experiments on image classification benchmark Office-31 (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Method</td><td>SF</td><td>A→W</td><td>D→W</td><td>W→D</td><td>A→D</td><td>D→A</td><td>W→A</td><td>Mean</td></tr><tr><td>DAN [49]</td><td></td><td>80.5</td><td>97.1</td><td>99.6</td><td>78.6</td><td>63.6</td><td>62.8</td><td>80.4</td></tr><tr><td>DANN [17]</td><td></td><td>82.0</td><td>96.9</td><td>99.1</td><td>79.7</td><td>68.2</td><td>67.4</td><td>82.2</td></tr><tr><td>ADDA [78]</td><td></td><td>86.2</td><td>96.2</td><td>98.4</td><td>77.8</td><td>69.5</td><td>68.9</td><td>82.9</td></tr><tr><td>JAN [50]</td><td></td><td>85.4</td><td>97.4</td><td>99.8</td><td>84.7</td><td>68.6</td><td>70.0</td><td>84.3</td></tr><tr><td>CBST[103]</td><td></td><td>87.8</td><td>98.5</td><td>100</td><td>86.5</td><td>71.2</td><td>70.9</td><td>85.8</td></tr><tr><td>CRST[102]</td><td>xxxxxx</td><td>89.4</td><td>98.9</td><td>100</td><td>88.7</td><td>72.6</td><td>70.9</td><td>86.8</td></tr><tr><td>3C-GAN [42]</td><td></td><td>93.7</td><td>98.5</td><td>99.8</td><td>92.7</td><td>75.3</td><td>77.8</td><td>89.6</td></tr><tr><td>+HCL</td><td>V</td><td>93.4</td><td>99.3</td><td>100.0</td><td>94.6</td><td>77.1</td><td>79.0</td><td>90.6</td></tr><tr><td>SHOT[45]</td><td></td><td>91.2</td><td>98.3</td><td>99.9</td><td>90.6</td><td>72.5</td><td>71.4</td><td>87.3</td></tr><tr><td>+HCL</td><td>1</td><td>92.8</td><td>99.0</td><td>100.0</td><td>94.4</td><td>76.1</td><td>78.3</td><td>90.1</td></tr><tr><td>HCL</td><td>√</td><td>92.5</td><td>98.2</td><td>100.0</td><td>94.7</td><td>75.9</td><td>77.7</td><td>89.8</td></tr></table>
+
+Feature visualization: This paragraph presents the t-SNE [53] visualization of feature representation on GTA Cityscapes model adaptation task. We compare HCL with two state-of-the-art UMA methods, i.e., “UR" [72] and “SFDA" [48], and Fig.3 shows the visualization. We can observe that HCL can learn desirable instance-discriminative yet category-discriminative representations because it incorporates two key designs that work in a complementary manner: 1) HCID works at instance level, which encourages to learn instance-discriminative target representations that generalize well to unseen data [98]; 2) HCCD works at category level which encourages to learn category-discriminative target representations that are well aligned with the objective of down-stream visual tasks. In addition, qualitative illustrations are provided in Fig.4. It can be observed that our proposed HCL clearly outperforms UR and SFDA.
+
+Generalization across learning setups: We study how HCL generalizes across learning setups by adapting it into two adaptation setups, i.e., partial-set adaptation and open-set adaptation. Experiments in Table 8 show that HCL achieves competitive performance consistently across both setups.
+
+![](images/59c0d9e83b71f70ca19cbbba092b5f2d24b5854dd3a4148224d33b368f5d7aa6.jpg)  
+Figure 3: The t-SNE [53] visualization of feature representation on $\mathrm { { G T A }  }$ Cityscapes unsupervsied model adaptation task: Each color in the graphs stands for a category of samples (image pixels) with a digit representing the center of a category of samples. It can be observed that the proposed HCL outperforms “UR" and “SFDA" qualitatively, by generating instance-discriminative and categorydiscriminative representations for unlabeled target data.
+
+Table 8: Experiments on image classification benchmark Office-Home under the setup of partial-set DA (domain adaptation) and open-set DA (“SF” denotes source-data free, i.e., adaptation without source data).   
+
+<table><tr><td>Partial-set DA</td><td>SF</td><td>A→C</td><td>A→P</td><td>A→R</td><td>C→A</td><td>C→P</td><td>C→R</td><td>P→A</td><td>P→C</td><td>P→R</td><td>R→A</td><td>R→C</td><td>R→P</td><td>Mean</td></tr><tr><td>SAN[5]</td><td>X</td><td>44.4</td><td>68.7</td><td>74.6</td><td>67.5</td><td>65.0</td><td>77.8</td><td>59.8</td><td>44.7</td><td>80.1</td><td>72.2</td><td>50.2</td><td>78.7</td><td>65.3</td></tr><tr><td>ETN [6]</td><td>×</td><td>59.2</td><td>77.0</td><td>79.5</td><td>62.9</td><td>65.7</td><td>75.0</td><td>68.3</td><td>55.4</td><td>84.4</td><td>75.7</td><td>57.7</td><td>84.5</td><td>70.5</td></tr><tr><td>SAFN [85]</td><td>X</td><td>58.9</td><td>76.3</td><td>81.4</td><td>70.4</td><td>73.0</td><td>77.8</td><td>72.4</td><td>55.3</td><td>80.4</td><td>75.8</td><td>60.4</td><td>79.9</td><td>71.8</td></tr><tr><td>SHOT [45]</td><td></td><td>57.9</td><td>83.6</td><td>88.8</td><td>72.4</td><td>74.0</td><td>79.0</td><td>76.1</td><td>60.6</td><td>90.1</td><td>81.9</td><td>68.3</td><td>88.5</td><td>76.8</td></tr><tr><td>+HCL</td><td>·</td><td>66.9</td><td>85.5</td><td>92.5</td><td>78.3</td><td>77.2</td><td>87.1</td><td>78.3</td><td>65.1</td><td>90.7</td><td>82.4</td><td>68.7</td><td>88.4</td><td>80.1</td></tr><tr><td>HCL</td><td>√</td><td>65.6</td><td>85.2</td><td>92.7</td><td>77.3</td><td>76.2</td><td>87.2</td><td>78.2</td><td>66.0</td><td>89.1</td><td>81.5</td><td>68.4</td><td>87.3</td><td>79.6</td></tr><tr><td>Open-set DA</td><td>SF</td><td>A→C</td><td>A→P</td><td>A→R</td><td>C→A</td><td>C→P</td><td>C→R</td><td>P→A</td><td>P→C</td><td>P→R</td><td>R→A</td><td>R→C</td><td>R→P</td><td>Mean</td></tr><tr><td>OSBP[67]</td><td>X</td><td>56.7</td><td>51.5</td><td>49.2</td><td>67.5</td><td>65.5</td><td>74.0</td><td>62.5</td><td>64.8</td><td>69.3</td><td>80.6</td><td>74.7</td><td>71.5</td><td>65.7</td></tr><tr><td>OpenMax [2]</td><td></td><td>56.5</td><td>52.9</td><td>53.7</td><td>69.1</td><td>64.8</td><td>74.5</td><td>64.1</td><td>64.0</td><td>71.2</td><td>80.3</td><td>73.0</td><td>76.9</td><td>66.7</td></tr><tr><td>STA [47]</td><td>X</td><td>58.1</td><td>53.1</td><td>54.4</td><td>71.6</td><td>69.3</td><td>81.9</td><td>63.4</td><td>65.2</td><td>74.9</td><td>85.0</td><td>75.8</td><td>80.8</td><td>69.5</td></tr><tr><td>SHOT[45]</td><td></td><td>62.5</td><td>77.8</td><td>83.9</td><td>60.9</td><td>73.4</td><td>79.4</td><td>64.7</td><td>58.7</td><td>83.1</td><td>69.1</td><td>62.0</td><td>82.1</td><td>71.5</td></tr><tr><td>+HCL</td><td>1</td><td>64.2</td><td>78.3</td><td>83.0</td><td>61.1</td><td>72.2</td><td>79.6</td><td>65.5</td><td>59.3</td><td>80.6</td><td>80.1</td><td>72.0</td><td>82.8</td><td>73.2</td></tr><tr><td>HCL</td><td>√</td><td>64.0</td><td>78.6</td><td>82.4</td><td>64.5</td><td>73.1</td><td>80.1</td><td>64.8</td><td>59.8</td><td>75.3</td><td>78.1</td><td>69.3</td><td>81.5</td><td>72.6</td></tr></table>
+
+![](images/209e3b18de22c7d192c35d0e2839dfa1cd7f4269f1ba67662048a326e08c5ec7.jpg)  
+Figure 4: Qualitative illustrations and comparison over domain adaptive semantic segmentation task $\mathrm { G T A } 5 $ Cityscapes. Our historical contrastive learning (HCL) exploits historical source hypothesis to make up for the absence of source data in UMA, which produces better qualitative results (i.e., semantic segmentation) by preserving the source hypothesis. It can be observed that HCL generates better segmentation results, for example, the sidewalk in the first row, the road in the second row and the sky and sidewalk in the third row.
+
+# 5 Conclusion
+
+In this work, we studied historical contrastive learning, an innovative UMA technique that exploits historical source hypothesis to make up for the absence of source data in UMA. We achieve historical contrastive learning by novel designs of historical contrastive instance discrimination and historical contrastive category discrimination which learn discriminative representations for target data while preserving source hypothesis simultaneously. Extensive experiments over a variety of visual tasks and learning setups show that HCL outperforms state-of-the-art techniques consistently. Moving forward, we will explore memory-based learning in other transfer learning tasks.
+
+# Acknowledgement
+
+This research was conducted at Singtel Cognitive and Artificial Intelligence Lab for Enterprises (SCALE $@$ NTU), which is a collaboration between Singapore Telecommunications Limited (Singtel) and Nanyang Technological University (NTU) that is supported by $\mathbf { A } { ^ { * } \mathbf { S } } \mathbf { T } \mathbf { A } \mathbf { R }$ under its Industry Alignment Fund (LOA Award number: I1701E0013).
+
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parse/train/0zXJRJecC_/0zXJRJecC__content_list.json

@@ -0,0 +1,1064 @@
+[
+    {
+        "type": "text",
+        "text": "Model Adaptation: Historical Contrastive Learning for Unsupervised Domain Adaptation without Source Data ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiaxing Huang, Dayan Guan, Aoran Xiao, Shijian Lu∗ School of Computer Science Engineering, Nanyang Technological University {Jiaxing.Huang, Dayan.Guan, Aoran.Xiao, Shijian.Lu}@ntu.edu.sg ",
+        "bbox": [
+            245,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            324,
+            535,
+            340
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised domain adaptation aims to align a labeled source domain and an unlabeled target domain, but it requires to access the source data which often raises concerns in data privacy, data portability and data transmission efficiency. We study unsupervised model adaptation (UMA), or called Unsupervised Domain Adaptation without Source Data, an alternative setting that aims to adapt source-trained models towards target distributions without accessing source data. To this end, we design an innovative historical contrastive learning (HCL) technique that exploits historical source hypothesis to make up for the absence of source data in UMA. HCL addresses the UMA challenge from two perspectives. First, it introduces historical contrastive instance discrimination (HCID) that learns from target samples by contrasting their embeddings which are generated by the currently adapted model and the historical models. With the historical models, HCID encourages UMA to learn instance-discriminative target representations while preserving the source hypothesis. Second, it introduces historical contrastive category discrimination (HCCD) that pseudo-labels target samples to learn category-discriminative target representations. Specifically, HCCD re-weights pseudo labels according to their prediction consistency across the current and historical models. Extensive experiments show that HCL outperforms and state-of-the-art methods consistently across a variety of visual tasks and setups. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            650,
+            310,
+            667
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks (DNNs) [28, 73, 23] have achieved great success in various computer vision tasks [8, 55, 60, 59, 28, 73, 23] but often generalize poorly to new domains due to the inter-domain discrepancy [1]. Unsupervised domain adaptation (UDA) [78, 51, 76, 64, 66, 79, 77, 103, 102, 25, 71, 44, 26, 86] addresses the inter-domain discrepancy by aligning the source and target data distributions, but it requires to access the source-domain data which often raises concerns in data privacy, data portability, and data transmission efficiency. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study unsupervised model adaptation (UMA), an alternative setting that aims to adapt source-trained models to fit target data distribution without accessing the source-domain data. Under the UMA setting, the only information carried forward is a portable source-trained model which is usually much smaller than the source-domain data and can be transmitted more efficiently [45, 42, 43, 72, 48] as illustrated in Table 1. Beyond that, the UMA setting also alleviates the concern of data privacy and intellectual property effectively. On the other hand, the absence of the labeled source-domain data makes domain adaptation much more challenging and susceptible to collapse. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/1ce920a0a2b185e5c3db3ef1d6e40136100febf543a2af0e89856c1c233f4304.jpg",
+        "table_caption": [
+            "Table 1: Source data have much larger sizes than source-trained models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Storage size (MB)</td><td colspan=\"2\">Semantic segmentation</td><td>Object detection</td><td>Image classification</td></tr><tr><td>GTA5</td><td>SYNTHIA</td><td>Cityscapes</td><td>VisDA17</td></tr><tr><td>Source dataset</td><td>62,873.6</td><td>22,323.2</td><td>12,697.6</td><td>7,884.8</td></tr><tr><td>Source-trained model</td><td>179.1</td><td>179.1</td><td>553.4</td><td>172.6</td></tr></table>",
+        "bbox": [
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+            111,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To this end, we develop historical contrastive learning (HCL) that aims to make up for the absence of source data by adapting the source-trained model to fit target data distribution without forgetting source hypothesis, as illustrated in Fig. 1. HCL addresses the UMA challenge from two perspectives. First, it introduces historical contrastive instance discrimination (HCID) that learns target samples by comparing their embeddings generated by the current model (as queries) and those generated by historical models (as keys): a query is pulled close to its positive keys while pushed apart from its negative keys. HCID can thus be viewed as a new type of instance contrastive learning for the task of UMA with historical models, which learns instance-discriminative target representations without forgetting source-domain hypothesis. Second, it introduces historical contrastive category discrimination (HCCD) that pseudo-labels target samples for learning category-discriminative target representations. Specifically, HCCD re-weights the pseudo labels according to their consistency across the current and historical models. ",
+        "bbox": [
+            173,
+            205,
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+            369
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed HCL tackles UMA with three desirable features: 1) It introduces historical contrast and achieves UMA without forgetting source hypothesis; 2) The HCID works at instance level, which encourages to learn instance-discriminative target representations that generalize well to unseen data [98]; 3) The HCCD works at category level (i.e., output space) which encourages to learn category-discriminative target representation that is well aligned with the objective of down-stream tasks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The contributions of this work can be summarized in three aspects. First, we investigate memorybased learning for unsupervised model adaptation that learns discriminative representations for unlabeled target data without forgetting source hypothesis. To the best of our knowledge, this is the first work that explores memory-based learning for the task of UMA. Second, we design historical contrastive learning which introduces historical contrastive instance discrimination and category discrimination, the latter is naturally aligned with the objective of UMA. Third, extensive experiments show that the proposed historical contrastive learning outperforms state-of-the-art methods consistently across a variety of visual tasks and setups. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            601,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work is closely related to several branches of research in unsupervised model adaptation, domain adaptation, memory-based learning and contrastive learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised model adaptation aims to adapt a source-trained model to fit target data distributions without accessing source-domain data. This problem has attracted increasing attention recently with a few pioneer studies each of which focuses on a specific visual task. For example, [45, 46] freezes the classifier of source-trained model and performs information maximization on target data for classification model adaptation. [42] tackles classification model adaptation with a conditional GANs that generates training images with target-alike styles and source-alike semantics. [43] presents a self-entropy descent algorithm to improve model adaptation for object detection. [72] reduces the uncertainty of target predictions (by source-trained model) for segmentation model adaptation. [48] introduces data-free knowledge distillation to transfer source-domain knowledge for segmentation model adaptation. Despite the different designs for different tasks, the common motivation of these studies is to make up for the absence of source data in domain adaptation. [40] and [88] tackle source-free domain adaptation from a generative manner by generating samples from the source classes and generating reference distributions, respectively. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We tackle the absence of source data by a memory mechanism that encourages to memorize source hypothesis during model adaptation. Specifically, we design historical contrastive learning that learns target representations by contrasting historical and currently evolved models. To the best of our knowledge, this is the first work that explores memory mechanism for UMA. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "${ x _ { s r c } } / { x _ { t g t } }$ : Source/target data $G ^ { t } / G ^ { t - m }$ : Current/historical model ????????/???????? : Source/target feature $f ^ { t } / f ^ { t - m }$ Current/historical feature: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a7b73cae93218e46b16f87ef85b82645c03ffed18ea4ededb7c1b6796d7b9609.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of unsupervised domain adaptation, unsupervised model adaptation and the proposed historical contrastive learning which exploits historical source hypothesis (or memorized knowledge) to make up for the absence of source supervision in the process of UMA. Here the historical source hypothesis could be the original source hypothesis $G ^ { 0 }$ (i.e. $\\scriptstyle { \\mathrm { t } } = { \\mathrm { m } }$ , trained using the labeled source data only), the adapted source hypothesis $\\bar { G } ^ { t - m }$ (i.e. $m < t$ , trained in the last $m$ epoch), or other types of previous models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            472,
+            256
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f47b4088424c1bf835a66b73a5cf9927777552921bdaa0df571f46f0f833254.jpg",
+        "image_caption": [
+            ""
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Domain adaptation is related to UMA but it requires to access labeled source data in training. Most existing work handles UDA from three typical approaches. The first exploits adversarial training to align source and target distributions in the feature, output or latent space [78, 51, 76, 14, 96, 64, 66, 79, 34, 77, 35, 21, 94, 29, 10, 9, 80, 20]. The second employs self-training to generate pseudo labels to learn from unlabeled target data iteratively [103, 69, 100, 102, 31, 19, 90, 91]. The third leverages image translation to modify image styles to reduce domain gaps [25, 71, 44, 95, 26, 86, 32, 33, 93, 92]. In addition, [30] proposes a categorical contrastive learning for domain adaptation. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Memory-based learning has been studied extensively. Memory networks [81] as one of early efforts explores to use external modules to store memory for supervised learning. Temporal ensemble [41], as well as a few following works [74, 12] extend the memory mechanism to semi-supervised learning. It employs historical hypothesis/models to regularize the current model and produces stable and competitive predictions. Mean Teacher [74] leverages moving-average models as the memory model to regularize the training, and similar idea was extended for UDA [16, 99, 4, 52]. Mutual learning [97] has also been proposed for learning among multiple peer student models. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Most aforementioned methods require labeled data in training. They do not work very well for UMA due to the absence of supervision from the labeled source data, by either collapsing in training or helping little in model adaptation performance. We design innovative historical contrastive learning to make up for the absence of the labeled source data, more details to be presented in the ensuing subsections. ",
+        "bbox": [
+            174,
+            607,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Contrastive learning [82, 87, 22, 54, 101, 24, 56, 75, 36, 11] learns discriminative representations from multiple views of the same instance. It works with certain dictionary look-up mechanism [22], where a given image $x$ is augmented into two views, query and key, and the query token $q$ should match its designated key $k _ { + }$ over a set of negative keys $k _ { - }$ from other images. Existing work can be broadly classified into three categories based on dictionary creation strategies. The first creates a memory bank [82] to store all the keys in the previous epoch. The second builds an end-to-end dictionary [87, 75, 36, 11] that generates keys using samples from the current mini-batch. The third employs a momentum encoder [22] that generates keys on-the-fly by a momentum-updated encoder. ",
+        "bbox": [
+            174,
+            683,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Other related source-free adaptation works. [7] considers supervised continual learning from previously learned tasks to a new task, which learns representations using the contrastive learning objective and preserves learned representations using a self-supervised distillation step, where the contrastively learned representations are more robust against the catastrophic forgetting for supervised continual learning. [38] addresses a source-free universal domain adaptation problem that does not guarantee that the classes in the target domain are the same as in the source domain. [39] propose a simple yet effective solution to realize inheritable models suitable for open-set source-free DA problem. ",
+        "bbox": [
+            174,
+            800,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/fb878d7d22a0b009deca2f1081b39ae80e7f8442f02f9246c50db87303021e17.jpg",
+        "image_caption": [
+            "Figure 2: The proposed historical contrastive learning consists of two key designs including historical contrastive instance discrimination (HCID) and historical contrastive category discrimination (HCCD). HCID learns from target samples by contrasting their embeddings generated by the current model (as queries) and historical models (as keys), which learns instance-discriminative target representations. HCCD pseudo-labels target samples to learn category-discriminative target representations, where the pseudo labels are re-weighted adaptively according to the prediction consistency across the current and historical models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Historical Contrastive Learning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            395,
+            473,
+            412
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This section presents the proposed historical contrastive learning that memorizes source hypothesis to make up for the absence of source data as illustrated in Fig. 2. The proposed HCL consists of two key designs. The first is historical contrastive instance discrimination which encourages to learn instance-discriminative target representations that generalize well to unseen data [98]. The second is historical contrastive category discrimination that encourages to learn category-discriminative target representations which is well aligned with the objective of visual recognition tasks. More details to be described in the ensuring subsections. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Historical Contrastive Instance Discrimination ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            541,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proposed HCID learns from unlabeled target samples via contrastive learning over their embeddings generated from current and historical models: the positive pairs are pulled close while negative pairs are pushed apart. It is a new type of contrastive learning for UMA, which preserves sourcedomain hypothesis by generating positive keys from historical models. HCID works at instance level and encourages to learn instance-discriminative target representations that generalize well to unseen data [98]. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "HCID loss. Given a query sample $x _ { q }$ and a set of key samples $X _ { k } = \\{ x _ { k _ { 0 } } , x _ { k _ { 1 } } , x _ { k _ { 2 } } , . . . , x _ { k _ { N } } \\}$ , HCID employs current model $E ^ { t }$ to encode the query $q ^ { t } = E ^ { t } ( x _ { q } )$ , and historical encoders $E ^ { t - m }$ to encode the keys $k _ { n } ^ { t - m } = E ^ { t - m } ( x _ { k _ { n } } ) , n = 0 , \\cdots , \\bar { N }$ . With the encoded embeddings, HCID is achieved via a historical contrastive loss $\\mathcal { L } _ { \\mathrm { H i s N C E } }$ , minimization of which pulls $q$ close to its positive key $k _ { + } ^ { t - m }$ while pushing it apart from all other (negative) keys: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edb785e04771ab3b8fe5fbf09589dfe8598cbd17888d7a903753aa2443d58248.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { H i s N C E } } = \\sum _ { x _ { q } \\in X _ { t g t } } - \\log \\frac { \\exp ( q ^ { t } \\cdot k _ { + } ^ { t - m } / \\tau ) r _ { + } ^ { t - m } } { \\sum _ { i = 0 } ^ { N } \\exp ( q ^ { t } \\cdot k _ { i } ^ { t - m } / \\tau ) r _ { i } ^ { t - m } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            734,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is a temperature parameter [82], $r$ indicates the reliability of each key $k _ { n } ^ { t - m }$ , with which we reweight the similarity loss of each key to encourage to memorize well-learnt instead of poorly-learnt historical embeddings. In this work, we use the classification entropy to estimate the reliability of each key. The positive key sample is the augmentation of the query sample[22, 36], and all the rest are negative keys. ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 1 Note $\\mathcal { L } _ { H i s N C E }$ in Eq.1 has a similar form as the InfoNCE loss[56, 22]. InfoNCE can actually be viewed as a special case of HisNCE, where all the query and keys are encoded by the current model ${ ' m = 0 }$ ) and the reliability is fixed ${ \\bf \\zeta } r _ { i } ^ { t - m } = 1 , \\forall i )$ . For HisNCE, we assign each key a reliability score to encourage to memorize the well-learnt historical embeddings only. It is also worth noting that Eq. 1 only shows historical contrast with one historical model for simplicity. In practice, we could employ multiple historical models to comprehensively distill (memorize) the well-learnt embeddings from them. It could be achieved by computing Eq.1 multiple times. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Historical contrastive category discrimination ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            166,
+            532,
+            183
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We design HCCD that generates pseudo labels and learns them conditioned on a historical consistency, i.e., the prediction consistency across the current and historical models. HCCD can be viewed as a new type of self-training, where pseudo labels are re-weighted by the historical consistency. It works at category level and encourages to learn category-discriminative target representations that are aligned with the objective of visual recognition tasks in UMA. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Historical contrastive pseudo label generation. Given an unlabeled sample $x$ , the current and historical models predict $p ^ { t } = G ^ { t } ( x )$ (as the query) and $p ^ { t - m } = G ^ { t - m } ( x )$ (as the keys). The pseudo label and the historical consistency of the sample are computed by: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a05fef448944a895e04f1aaa84ff584917ef98e106a465c0a363e6eac368fc6f.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\hat { y } = \\mathbf { { \\Gamma } } ( p ^ { t } ) , } } \\\\ { { h _ { c o n } = 1 - \\mathrm { S i g m o i d } ( | | p ^ { t } - p ^ { t - m } | | _ { 1 } ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            321,
+            624,
+            362
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $p$ is a $K$ -class probability vector, $\\mathbf { \\delta T }$ is the pseudo label generation function [103, 102] and $\\hat { y } = ( \\hat { y } ^ { ( 1 ) } , \\hat { y } ^ { ( 2 ) } , . . . , \\hat { y } ^ { ( \\bar { C } ) } )$ is the predicted category label. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "HCCD loss. Given the unlabeled data $x$ and its historical contrastive pseudo label $\\left( \\hat { y } , h _ { c o n } \\right)$ , HCCD performs self-training on target data $x$ via a weighted cross-entropy loss: ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/992658b5a4609e186647af311e949498cec6ac02f5776f0f3b261c2392f0ac80.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { H i s S T } } = - \\sum _ { x \\in X _ { t g t } } h _ { c o n } \\times \\hat { y } \\log p _ { x } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $h _ { c o n }$ is the per-sample historical consistency and we use it to re-weight the self-training loss. If the predictions of a sample across the current and historical models are consistent, we consider it as a well-learnt sample and increase its influence in self-training. Otherwise, we consider it as pooly-learnt sample and decrease its influence in self-training. ",
+        "bbox": [
+            174,
+            491,
+            825,
+            547
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Theoretical Insights ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            566,
+            352,
+            582
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The two designs in Historical Contrastive Learning (HCL) are inherently connected with some probabilistic models and convergent under certain conditions: ",
+        "bbox": [
+            171,
+            593,
+            825,
+            622
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 The historical contrastive instance discrimination (HCID) can be modelled as a maximum likelihood problem optimized via Expectation Maximization. ",
+        "bbox": [
+            174,
+            637,
+            823,
+            666
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 The HCID is convergent under certain conditions. ",
+        "bbox": [
+            173,
+            680,
+            602,
+            696
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3 The historical contrastive category discrimination (HCCD) can be modelled as a classification maximum likelihood problem optimized via Classification Expectation Maximization. ",
+        "bbox": [
+            176,
+            710,
+            825,
+            739
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4 The HCCD is convergent under certain conditions. ",
+        "bbox": [
+            174,
+            753,
+            607,
+            770
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proofs of Proposition 1, Proposition 2, Proposition 3 and Proposition 4 are provided in the appendix, respectively. ",
+        "bbox": [
+            176,
+            784,
+            823,
+            813
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            835,
+            312,
+            853
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This section presents experiments including datasets, implementation details, evaluations of the proposed HCL in semantic segmentation, object detection and image classification tasks as well as the discussion of its desirable features. ",
+        "bbox": [
+            174,
+            869,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/2f7c4c8e18c3f385427be875880cd9ccdf4962e5a7999257b75ad32b42776b26.jpg",
+        "table_caption": [
+            "Table 2: Experiments on semantic segmentation task GTA5 Cityscapes (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>Road</td><td>SW</td><td>Build</td><td>Wall</td><td>Fence</td><td>Pole</td><td>TL</td><td>TS</td><td>Veg.</td><td>Terrain</td><td>Sky</td><td>PR</td><td>Rider</td><td>Car</td><td>Truck</td><td>Bus</td><td>Train</td><td>Motor</td><td>Bike</td><td>mIoU</td></tr><tr><td>AdaptSeg [76]</td><td>X</td><td>86.5</td><td>36.0</td><td>79.9</td><td>23.4</td><td>23.3</td><td>23.9</td><td>35.2</td><td>14.8</td><td>83.4</td><td>33.3</td><td>75.6</td><td>58.5</td><td>27.6</td><td>73.7</td><td>32.5</td><td>35.4</td><td>3.9</td><td>30.1</td><td>28.1</td><td>42.4</td></tr><tr><td>AdvEnt [79]</td><td>X</td><td>89.4</td><td>33.1</td><td>81.0</td><td>26.6</td><td>26.8</td><td>27.2</td><td>33.5</td><td>24.7</td><td>83.9</td><td>36.7</td><td>78.8</td><td>58.7</td><td>30.5</td><td>84.8</td><td>38.5</td><td>44.5</td><td>1.7</td><td>31.6</td><td>32.4</td><td>45.5</td></tr><tr><td>IDA [57]</td><td>X</td><td>90.6</td><td>37.1</td><td>82.6</td><td>30.1</td><td>19.1</td><td>29.5</td><td>32.4</td><td>20.6</td><td>85.7</td><td>40.5</td><td>79.7</td><td>58.7</td><td>31.1</td><td>86.3</td><td>31.5</td><td>48.3</td><td>0.0</td><td>30.2</td><td>35.8</td><td>46.3</td></tr><tr><td>CRST[102]</td><td>X</td><td>91.0</td><td>55.4</td><td>80.0</td><td>33.7</td><td>21.4</td><td>37.3</td><td>32.9</td><td>24.5</td><td>85.0</td><td>34.1</td><td>80.8</td><td>57.7</td><td>24.6</td><td>84.1</td><td>27.8</td><td>30.1</td><td>26.9</td><td>26.0</td><td>42.3</td><td>47.1</td></tr><tr><td>CrCDA [35]</td><td>X</td><td>92.4</td><td>55.3</td><td>82.3</td><td>31.2</td><td>29.1</td><td>32.5</td><td>33.2</td><td>35.6</td><td>83.5</td><td>34.8</td><td>84.2</td><td>58.9</td><td>32.2</td><td>84.7</td><td>40.6</td><td>46.1</td><td>2.1</td><td>31.1</td><td>32.7</td><td>48.6</td></tr><tr><td>UR [72]</td><td>√</td><td>92.3</td><td>55.2</td><td>81.6</td><td>30.8</td><td>18.8</td><td>37.1</td><td>17.7</td><td>12.1</td><td>84.2</td><td>35.9</td><td>83.8</td><td>57.7</td><td>24.1</td><td>81.7</td><td>27.5</td><td>44.3</td><td>6.9</td><td>24.1</td><td>40.4</td><td>45.1</td></tr><tr><td>+HCL</td><td>√</td><td>92.2</td><td>54.1</td><td>81.7</td><td>34.2</td><td>25.4</td><td>37.9</td><td>35.8</td><td>29.8</td><td>84.1</td><td>38.0</td><td>83.9</td><td>59.1</td><td>27.1</td><td>84.6</td><td>33.9</td><td>41.9</td><td>16.2</td><td>27.7</td><td>44.7</td><td>49.1</td></tr><tr><td>SFDA [48]</td><td>√</td><td>91.7</td><td>52.7</td><td>82.2</td><td>28.7</td><td>20.3</td><td></td><td>36.530.6</td><td></td><td>23.681.7</td><td>35.6</td><td>84.859.5</td><td></td><td>22.6</td><td>83.4</td><td>29.6</td><td>32.4</td><td>11.8</td><td>23.8</td><td>39.6</td><td>45.8</td></tr><tr><td>+HCL</td><td>√</td><td>92.3</td><td>54.5</td><td>82.6</td><td>33.1</td><td>26.2</td><td>38.9</td><td>37.9</td><td>31.7</td><td>83.5</td><td>38.1</td><td>84.4</td><td>60.9</td><td>30.0</td><td>84.5</td><td>32.6</td><td>41.2</td><td>14.2</td><td>26.4</td><td>43.2</td><td>49.3</td></tr><tr><td>HCID</td><td>√</td><td>89.5</td><td>53</td><td>80.3</td><td>33.9</td><td>22.9</td><td>36.2</td><td>32.7</td><td></td><td>23.882.3</td><td>36.5</td><td>73.7</td><td>60.0</td><td>22.4</td><td>83.8</td><td>28.9</td><td>34.7</td><td>13.5</td><td>21.2</td><td>38.0</td><td>45.6</td></tr><tr><td>HCCD</td><td>√</td><td>91.0</td><td>53.6</td><td>81.5</td><td>32.4</td><td>23.1</td><td>36.9</td><td>32.3</td><td>26.3</td><td>82.8</td><td>37.2</td><td>80.4</td><td>58.5</td><td>25.0</td><td>82.5</td><td>29.9</td><td>34.2</td><td>15.5</td><td>23.2</td><td>40.5</td><td>46.7</td></tr><tr><td>HCL</td><td></td><td>92.0</td><td>55.0</td><td>80.4</td><td>33.5</td><td>24.6</td><td>37.1</td><td>35.1</td><td></td><td>28.883.0</td><td>37.6</td><td>82.359.4</td><td></td><td>27.6</td><td>83.6</td><td>32.3</td><td>36.6</td><td>14.1</td><td>28.7</td><td>43.0</td><td>48.1</td></tr></table>",
+        "bbox": [
+            174,
+            123,
+            823,
+            233
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/ad2e3a4e9f1655f462f3a117065e1d4335bb1470dc0dec8b60ae6148f59a840e.jpg",
+        "table_caption": [
+            "Table 3: Experiments on semantic segmentation task SYNTHIA Cityscapes (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>Road</td><td>SW</td><td>Build</td><td>Wall</td><td>Fence</td><td>Pole</td><td>TL</td><td>TS</td><td>Veg.</td><td>Sky</td><td>PR</td><td>Rider</td><td>Car</td><td>Bus</td><td>Motor</td><td>Bike</td><td>mIoU</td><td>mIoU</td></tr><tr><td>AdaptSeg[76]</td><td>X</td><td>84.3</td><td>42.7</td><td>77.5</td><td>，</td><td>1</td><td>-</td><td>4.7</td><td>7.0</td><td>77.9</td><td>82.5</td><td>54.3</td><td>21.0</td><td>72.3</td><td>32.2</td><td>18.9</td><td>32.3</td><td>1</td><td>46.7</td></tr><tr><td>AdvEnt [79]</td><td>X</td><td>85.6</td><td>42.2</td><td>79.7</td><td>8.7</td><td>0.4</td><td>25.9</td><td>5.4</td><td>8.1</td><td>80.4</td><td>84.1</td><td>57.9</td><td>23.8</td><td>73.3</td><td>36.4</td><td>14.2</td><td>33.0</td><td>41.2</td><td>48.0</td></tr><tr><td>IDA [57]</td><td>X</td><td>84.3</td><td>37.7</td><td>79.5</td><td>5.3</td><td>0.4</td><td>24.9</td><td>9.2</td><td>8.4</td><td>80.0</td><td>84.1</td><td>57.2</td><td>23.0</td><td>78.0</td><td>38.1</td><td>20.3</td><td>36.5</td><td>41.7</td><td>48.9</td></tr><tr><td>CRST[102]</td><td>×</td><td>67.7</td><td>32.2</td><td>73.9</td><td>10.7</td><td>1.6</td><td>37.4</td><td>22.2</td><td>31.2</td><td>80.8</td><td>80.5</td><td>60.8</td><td>29.1</td><td>82.8</td><td>25.0</td><td>19.4</td><td>45.3</td><td>43.8</td><td>50.1</td></tr><tr><td>CrCDA[35]</td><td>X</td><td>86.2</td><td>44.9</td><td>79.5</td><td>8.3</td><td>0.7</td><td>27.8</td><td>9.4</td><td>11.8</td><td>78.6</td><td>86.5</td><td>57.2</td><td>26.1</td><td>76.8</td><td>39.9</td><td>21.5</td><td>32.1</td><td>42.9</td><td>50.0</td></tr><tr><td>UR[72]</td><td>√</td><td>59.3</td><td>24.6</td><td>77.0</td><td>14.0</td><td>1.8</td><td>31.5</td><td>18.3</td><td>32.0</td><td>83.1</td><td>80.4</td><td>46.3</td><td>17.8</td><td>76.7</td><td>17.0</td><td>18.5</td><td>34.6</td><td>39.6</td><td>45.0</td></tr><tr><td>+HCL</td><td>√</td><td>76.7</td><td>33.7</td><td>78.7</td><td>7.2</td><td>0.1</td><td>34.4</td><td>23.2</td><td>31.6</td><td>80.5</td><td>84.3</td><td>54.4</td><td>26.6</td><td>79.5</td><td>35.9</td><td>24.8</td><td>34.4</td><td>44.1</td><td>51.1</td></tr><tr><td>SFDA [48]</td><td>√</td><td>67.8</td><td>31.9</td><td>77.1</td><td>8.3</td><td>1.1</td><td>35.9</td><td>21.2</td><td>26.7</td><td>79.8</td><td>79.4</td><td>58.8</td><td>27.3</td><td>80.4</td><td>25.3</td><td>19.5</td><td>37.4</td><td>42.4</td><td>48.7</td></tr><tr><td>+HCL</td><td>√</td><td>78.2</td><td>35.3</td><td>79.6</td><td>7.3</td><td>0.2</td><td>37.7</td><td>21</td><td>30.9</td><td>80.4</td><td>83.3</td><td>59.8</td><td>29.4</td><td>79.2</td><td>34.2</td><td>24.5</td><td>38.9</td><td>45.0</td><td>51.9</td></tr><tr><td>HCL</td><td>√</td><td>80.9</td><td>34.9</td><td>76.7</td><td>6.6</td><td>0.2</td><td>36.1</td><td>20.1</td><td>28.2</td><td>79.1</td><td>83.1</td><td>55.6</td><td>25.6</td><td>78.8</td><td>32.7</td><td>24.1</td><td>32.7</td><td>43.5</td><td>50.2</td></tr></table>",
+        "bbox": [
+            176,
+            282,
+            823,
+            382
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 Datasets ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            411,
+            272,
+            425
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "UMA for semantic segmentation is evaluated on two challenging tasks GTA5 $[ 6 1 ]  $ Cityscapes [15] and SYNTHIA $[ 6 2 ]  \\cdot$ Cityscapes. GTA5 has 24, 966 synthetic images and shares 19 categories with Cityscapes. SYNTHIA contains 9, 400 synthetic images and shares 16 categories with Cityscapes. Cityscapes has 2975 and 500 real-world images for training and validation, respectively. ",
+        "bbox": [
+            174,
+            436,
+            825,
+            492
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "UMA for object detection is evaluated on tasks Cityscapes Foggy Cityscapes [68] and Cityscapes $ \\mathrm { B D D 1 0 0 k }$ [89]. Foggy Cityscapes is derived by applying simulated fog to the 2, 975 Cityscapes images. BDD100k has $7 0 k$ training images and $1 0 k$ validation images, and shares 7 categories with Cityscapes. We evaluate a subset of BDD100k (i.e., daytime) as in [84, 65, 13] for fair comparisons. ",
+        "bbox": [
+            173,
+            498,
+            825,
+            554
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "UMA for image classification is evaluated on benchmarks VisDA17 [58] and Office-31 [63]. VisDA17 has 152, 409 synthetic images as the source domain and 55, 400 real images of 12 shared categories as the target domain. Office-31 has 4110 images of three sources including 2817 from Amazon, 795 from Webcam and 498 from DSLR (with 31 shared categories). Following [102, 63, 70], the evaluation is on six adaptation tasks $\\mathbf { A } {  } \\mathbf { W } .$ , $\\mathrm { D } \\to \\mathsf { W } .$ , $\\mathrm { W } {  } \\mathrm { D }$ , $\\mathbf { A } {  } \\mathbf { D }$ , $\\mathrm { D } { \\to } \\mathsf { A }$ , and $\\mathrm { W } { \\to } \\mathrm { A }$ . ",
+        "bbox": [
+            174,
+            560,
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+            630
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            647,
+            377,
+            662
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Semantic segmentation: Following [79, 103], we employ DeepLab-V2 [8] as the segmentation model. We adopt SGD [3] with momentum 0.9, weight decay $1 e - 4$ and learning rate $2 . 5 e \\mathrm { ~ - ~ } 4$ where the learning rate is decayed by a polynomial annealing policy [8]. ",
+        "bbox": [
+            174,
+            672,
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+            715
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Object detection: Following [84, 65, 13], we adopt Faster R-CNN [60] as the detection model. We use SGD [3] with momentum 0.9 and weight decay $5 e - 4$ . The learning rate is $1 e - 3$ for first $5 0 k$ iterations and then decreased to $1 e - 4$ for another $2 0 k$ iterations. ",
+        "bbox": [
+            174,
+            722,
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+            763
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Image classification: Following [102, 63, 70], we adopt ResNet-101 and ResNet-50 [23] as the classification models for VisDA17 and Office-31, respectively. We use SGD [3] with momentum 0.9, weight decay $5 e - 4$ , learning rate $1 e - 3$ and batch size 32. ",
+        "bbox": [
+            174,
+            770,
+            825,
+            813
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Unsupervised Domain Adaption for Semantic Segmentation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            829,
+            629,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the proposed HCL in UMA-based semantic segmentation tasks $\\mathrm { G T A } 5 $ Cityscapes and SYNTHIA Cityscapes. Tables 2 and 3 show experimental results in mean Intersectionover-Union (mIoU). We can see that HCL outperforms state-of-the-art UMA methods by large margins. In addition, HCL is complementary to existing UMA methods and incorporating it as denoted by $\\mathrm { ^ { 6 6 } { + } H C L ^ { 9 3 } }$ improves the existing UMA methods clearly and consistently. Furthermore, HCL even achieves competitive performance as compared with state-of-the-art UDA methods (labeled by ✗in the column SF) which require to access the labeled source data in training. Further, We conduct ablation studies of the proposed HCL over the UMA-based semantic segmentation task $\\mathrm { G T A } 5  $ Cityscapes. As the bottom of Table 2 shows, either HCID or HCCD achieves comparable performance. In addition, HCID and HCCD offer orthogonal self-supervision signals where HCID focuses on instance-level discrimination between queries and historical keys and HCCD focuses on category-level discrimination among samples with different pseudo category labels. The two designs are thus complementary and the combination of them in HCL produces the best segmentation. ",
+        "bbox": [
+            174,
+            856,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/2af0398575810e0d235c9033b8ce137d16b876a7ea5d4a9a19fa3da13f724c80.jpg",
+        "table_caption": [
+            "Table 4: Experiments on object detection task Cityscapes Foggy Cityscapes (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>person</td><td>rider</td><td>car</td><td>truck</td><td>bus</td><td>train</td><td>mcycle</td><td>bicycle</td><td>mAP</td></tr><tr><td>DA [13]</td><td></td><td>25.0</td><td>31.0</td><td>40.5</td><td>22.1</td><td>35.3</td><td>20.2</td><td>20.0</td><td>27.1</td><td>27.6</td></tr><tr><td>MLDA [83]</td><td></td><td>33.2</td><td>44.2</td><td>44.8</td><td>28.2</td><td>41.8</td><td>28.7</td><td>30.5</td><td>36.5</td><td>36.0</td></tr><tr><td>DMA [37]</td><td></td><td>30.8</td><td>40.5</td><td>44.3</td><td>27.2</td><td>38.4</td><td>34.5</td><td>28.4</td><td>32.2</td><td>34.6</td></tr><tr><td>CAFA [27]</td><td></td><td>41.9</td><td>38.7</td><td>56.7</td><td>22.6</td><td>41.5</td><td>26.8</td><td>24.6</td><td>35.5</td><td>36.0</td></tr><tr><td>SWDA [65]</td><td></td><td>36.2</td><td>35.3</td><td>43.5</td><td>30.0</td><td>29.9</td><td>42.3</td><td>32.6</td><td>24.5</td><td>34.3</td></tr><tr><td>CRDA [84]</td><td>xxxxxx</td><td>32.9</td><td>43.8</td><td>49.2</td><td>27.2</td><td>45.1</td><td>36.4</td><td>30.3</td><td>34.6</td><td>37.4</td></tr><tr><td>SFOD [43]</td><td>1</td><td>25.5</td><td>44.5</td><td>40.7</td><td>33.2</td><td>22.2</td><td>28.4</td><td>34.1</td><td>39.0</td><td>33.5</td></tr><tr><td>+HCL</td><td></td><td>39.3</td><td>46.7</td><td>48.6</td><td>32.9</td><td>46.2</td><td>38.2</td><td>33.9</td><td>36.9</td><td>40.3</td></tr><tr><td>HCL</td><td>√</td><td>38.7</td><td>46.0</td><td>47.9</td><td>33.0</td><td>45.7</td><td>38.9</td><td>32.8</td><td>34.9</td><td>39.7</td></tr></table>",
+        "bbox": [
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+            123,
+            717,
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/6dedae6b361bf5b490df5f9c456df7822471fcab805409cb3ca00df2aac1d12c.jpg",
+        "table_caption": [
+            "Table 5: Experiments on object detection task Cityscapes $ \\mathrm { B D D 1 0 0 k }$ (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>person</td><td>rider</td><td>car</td><td>truck</td><td>bus</td><td>mcycle</td><td>bicycle</td><td>mAP</td></tr><tr><td>DA[13]</td><td>X</td><td>29.4</td><td>26.5</td><td>44.6</td><td>14.3</td><td>16.8</td><td>15.8</td><td>20.6</td><td>24.0</td></tr><tr><td rowspan=\"3\">SWDA [65] CRDA [84]</td><td></td><td>30.2</td><td>29.5</td><td>45.7</td><td>15.2</td><td>18.4</td><td>17.1</td><td>21.2</td><td>25.3</td></tr><tr><td>X</td><td>31.4</td><td>31.3</td><td>46.3</td><td>19.5</td><td>18.9</td><td>17.3</td><td>23.8</td><td>26.9</td></tr><tr><td>·</td><td>32.4</td><td>32.6</td><td>50.4</td><td>20.6</td><td>23.4</td><td>18.9</td><td>25.0</td><td>29.0</td></tr><tr><td>SFOD [43] +HCL</td><td></td><td>33.9</td><td>34.4</td><td>52.8</td><td>22.1</td><td>25.3</td><td>22.6</td><td>26.7</td><td>31.1</td></tr><tr><td>HCL</td><td>√</td><td>32.7</td><td>33.2</td><td>52.0</td><td>21.3</td><td>25.6</td><td>21.5</td><td>26.0</td><td>30.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.4 Unsupervised Domain Adaptation for Object Detection ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            536,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the proposed HCL over the UMA-based object detection tasks Cityscapes $\\mathrm { :  F o g g y }$ Cityscapes and Cityscapes $ \\mathrm { B D D 1 0 0 k }$ . Tables 4 and 5 show experimental results. We can observe that HCL outperforms state-of-the-art UMA method SFOD clearly. Similar to the semantic segmentation experiments, HCL achieves competitive performance as compared with state-of-the-art UDA methods (labeled by ✗in column SF) which require to access labeled source data in training. ",
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 Unsupervised Domain Adaptation for Image Classification ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            651,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed HCL over the UMA-based image classificat tasks VisDA17 and Office-31. Tables 6 and 7 show experimental results. We can observe that HCL outperforms state-of-the-art UMA methods clearly. Similar to the semantic segmentation and object detection experiments, HCL achieves competitive performance as compared with state-of-the-art UDA methods (labeled by $\\pmb { \\chi }$ ) which require to access labeled source data in training. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.6 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
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+            781
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Generalization across computer vision tasks: We study how HCL generalizes across computer vision tasks by evaluating it over three representative tasks on semantic segmentation, object detection and image classification. Experiments in Tables 2- 7 show that HCL achieves competitive performance consistently across all three visual tasks, demonstrating the generalization ability of HCL across computer vision tasks. ",
+        "bbox": [
+            174,
+            794,
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+            863
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Complementarity studies: We study the complementarity of our proposed HCL by combining it with existing UMA methods. Experiments in Table 2 (the row highlighted by $\\ \" + \\mathrm { H C L } \\ '$ ) shows that incorporating HCL boosts the existing UMA methods consistently. ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/81d57b4003733910479fea22a5569ce2e82411b213111b329ee4ff690e932778.jpg",
+        "table_caption": [
+            "Table 6: Experiments on image classification benchmark VisDA17 (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>Aero</td><td>Bike</td><td>Bus</td><td>Car</td><td>Horse</td><td>Knife</td><td>Motor</td><td>Person</td><td>Plant</td><td>Skateboard</td><td>Train</td><td>Truck</td><td>Mean</td></tr><tr><td>DANN [17]</td><td></td><td>81.9</td><td>77.7</td><td>82.8</td><td>44.3</td><td>81.2</td><td>29.5</td><td>65.1</td><td>28.6</td><td>51.9</td><td>54.6</td><td>82.8</td><td>7.8</td><td>57.4</td></tr><tr><td>ENT[18]</td><td></td><td>80.3</td><td>75.5</td><td>75.8</td><td>48.3</td><td>77.9</td><td>27.3</td><td>69.7</td><td>40.2</td><td>46.5</td><td>46.6</td><td>79.3</td><td>16.0</td><td>57.0</td></tr><tr><td>MCD [66]</td><td></td><td>87.0</td><td>60.9</td><td>83.7</td><td>64.0</td><td>88.9</td><td>79.6</td><td>84.7</td><td>76.9</td><td>88.6</td><td>40.3</td><td>83.0</td><td>25.8</td><td>71.9</td></tr><tr><td>CBST [103]</td><td></td><td>87.2</td><td>78.8</td><td>56.5</td><td>55.4</td><td>85.1</td><td>79.2</td><td>83.8</td><td>77.7</td><td>82.8</td><td>88.8</td><td>69.0</td><td>72.0</td><td>76.4</td></tr><tr><td>CRST[102]</td><td>xxxxx</td><td>88.0</td><td>79.2</td><td>61.0</td><td>60.0</td><td>87.5</td><td>81.4</td><td>86.3</td><td>78.8</td><td>85.6</td><td>86.6</td><td>73.9</td><td>68.8</td><td>78.1</td></tr><tr><td>3C-GAN [42]</td><td>√</td><td>94.8</td><td>73.4</td><td>68.8</td><td>74.8</td><td>93.1</td><td>95.4</td><td>88.6</td><td>84.7</td><td>89.1</td><td>84.7</td><td>83.5</td><td>48.1</td><td>81.6</td></tr><tr><td>+HCL</td><td>√</td><td>93.8</td><td>86.6</td><td>84.1</td><td>74.3</td><td>93.2</td><td>95.0</td><td>88.4</td><td>85.0</td><td>90.4</td><td>85.2</td><td>84.5</td><td>49.8</td><td>84.2</td></tr><tr><td>SHOT[45]</td><td>√</td><td>93.7</td><td>86.4</td><td>78.7</td><td>50.7</td><td>91.0</td><td>93.5</td><td>79.0</td><td>78.3</td><td>89.2</td><td>85.4</td><td>87.9</td><td>51.1</td><td>80.4</td></tr><tr><td>+HCL</td><td>√</td><td>94.3</td><td>87.0</td><td>82.6</td><td>70.6</td><td>92.0</td><td>93.2</td><td>87.0</td><td>80.6</td><td>89.6</td><td>86.8</td><td>84.6</td><td>58.7</td><td>83.9</td></tr><tr><td>HCL</td><td>√</td><td>93.3</td><td>85.4</td><td>80.7</td><td>68.5</td><td>91.0</td><td>88.1</td><td>86.0</td><td>78.6</td><td>86.6</td><td>88.8</td><td>80.0</td><td>74.7</td><td>83.5</td></tr></table>",
+        "bbox": [
+            199,
+            123,
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+            237
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/af3f58cc1f01c8a97aafbe0526b89e4bb43b36f67b1bc3279de45df4eb3d6de8.jpg",
+        "table_caption": [
+            "Table 7: Experiments on image classification benchmark Office-31 (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SF</td><td>A→W</td><td>D→W</td><td>W→D</td><td>A→D</td><td>D→A</td><td>W→A</td><td>Mean</td></tr><tr><td>DAN [49]</td><td></td><td>80.5</td><td>97.1</td><td>99.6</td><td>78.6</td><td>63.6</td><td>62.8</td><td>80.4</td></tr><tr><td>DANN [17]</td><td></td><td>82.0</td><td>96.9</td><td>99.1</td><td>79.7</td><td>68.2</td><td>67.4</td><td>82.2</td></tr><tr><td>ADDA [78]</td><td></td><td>86.2</td><td>96.2</td><td>98.4</td><td>77.8</td><td>69.5</td><td>68.9</td><td>82.9</td></tr><tr><td>JAN [50]</td><td></td><td>85.4</td><td>97.4</td><td>99.8</td><td>84.7</td><td>68.6</td><td>70.0</td><td>84.3</td></tr><tr><td>CBST[103]</td><td></td><td>87.8</td><td>98.5</td><td>100</td><td>86.5</td><td>71.2</td><td>70.9</td><td>85.8</td></tr><tr><td>CRST[102]</td><td>xxxxxx</td><td>89.4</td><td>98.9</td><td>100</td><td>88.7</td><td>72.6</td><td>70.9</td><td>86.8</td></tr><tr><td>3C-GAN [42]</td><td></td><td>93.7</td><td>98.5</td><td>99.8</td><td>92.7</td><td>75.3</td><td>77.8</td><td>89.6</td></tr><tr><td>+HCL</td><td>V</td><td>93.4</td><td>99.3</td><td>100.0</td><td>94.6</td><td>77.1</td><td>79.0</td><td>90.6</td></tr><tr><td>SHOT[45]</td><td></td><td>91.2</td><td>98.3</td><td>99.9</td><td>90.6</td><td>72.5</td><td>71.4</td><td>87.3</td></tr><tr><td>+HCL</td><td>1</td><td>92.8</td><td>99.0</td><td>100.0</td><td>94.4</td><td>76.1</td><td>78.3</td><td>90.1</td></tr><tr><td>HCL</td><td>√</td><td>92.5</td><td>98.2</td><td>100.0</td><td>94.7</td><td>75.9</td><td>77.7</td><td>89.8</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Feature visualization: This paragraph presents the t-SNE [53] visualization of feature representation on GTA Cityscapes model adaptation task. We compare HCL with two state-of-the-art UMA methods, i.e., “UR\" [72] and “SFDA\" [48], and Fig.3 shows the visualization. We can observe that HCL can learn desirable instance-discriminative yet category-discriminative representations because it incorporates two key designs that work in a complementary manner: 1) HCID works at instance level, which encourages to learn instance-discriminative target representations that generalize well to unseen data [98]; 2) HCCD works at category level which encourages to learn category-discriminative target representations that are well aligned with the objective of down-stream visual tasks. In addition, qualitative illustrations are provided in Fig.4. It can be observed that our proposed HCL clearly outperforms UR and SFDA. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Generalization across learning setups: We study how HCL generalizes across learning setups by adapting it into two adaptation setups, i.e., partial-set adaptation and open-set adaptation. Experiments in Table 8 show that HCL achieves competitive performance consistently across both setups. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/59c0d9e83b71f70ca19cbbba092b5f2d24b5854dd3a4148224d33b368f5d7aa6.jpg",
+        "image_caption": [
+            "Figure 3: The t-SNE [53] visualization of feature representation on $\\mathrm { { G T A }  }$ Cityscapes unsupervsied model adaptation task: Each color in the graphs stands for a category of samples (image pixels) with a digit representing the center of a category of samples. It can be observed that the proposed HCL outperforms “UR\" and “SFDA\" qualitatively, by generating instance-discriminative and categorydiscriminative representations for unlabeled target data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/617a24c2b7a268d606f636752ace876e752c5f29111b46cf0e85dec07d475770.jpg",
+        "table_caption": [
+            "Table 8: Experiments on image classification benchmark Office-Home under the setup of partial-set DA (domain adaptation) and open-set DA (“SF” denotes source-data free, i.e., adaptation without source data). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Partial-set DA</td><td>SF</td><td>A→C</td><td>A→P</td><td>A→R</td><td>C→A</td><td>C→P</td><td>C→R</td><td>P→A</td><td>P→C</td><td>P→R</td><td>R→A</td><td>R→C</td><td>R→P</td><td>Mean</td></tr><tr><td>SAN[5]</td><td>X</td><td>44.4</td><td>68.7</td><td>74.6</td><td>67.5</td><td>65.0</td><td>77.8</td><td>59.8</td><td>44.7</td><td>80.1</td><td>72.2</td><td>50.2</td><td>78.7</td><td>65.3</td></tr><tr><td>ETN [6]</td><td>×</td><td>59.2</td><td>77.0</td><td>79.5</td><td>62.9</td><td>65.7</td><td>75.0</td><td>68.3</td><td>55.4</td><td>84.4</td><td>75.7</td><td>57.7</td><td>84.5</td><td>70.5</td></tr><tr><td>SAFN [85]</td><td>X</td><td>58.9</td><td>76.3</td><td>81.4</td><td>70.4</td><td>73.0</td><td>77.8</td><td>72.4</td><td>55.3</td><td>80.4</td><td>75.8</td><td>60.4</td><td>79.9</td><td>71.8</td></tr><tr><td>SHOT [45]</td><td></td><td>57.9</td><td>83.6</td><td>88.8</td><td>72.4</td><td>74.0</td><td>79.0</td><td>76.1</td><td>60.6</td><td>90.1</td><td>81.9</td><td>68.3</td><td>88.5</td><td>76.8</td></tr><tr><td>+HCL</td><td>·</td><td>66.9</td><td>85.5</td><td>92.5</td><td>78.3</td><td>77.2</td><td>87.1</td><td>78.3</td><td>65.1</td><td>90.7</td><td>82.4</td><td>68.7</td><td>88.4</td><td>80.1</td></tr><tr><td>HCL</td><td>√</td><td>65.6</td><td>85.2</td><td>92.7</td><td>77.3</td><td>76.2</td><td>87.2</td><td>78.2</td><td>66.0</td><td>89.1</td><td>81.5</td><td>68.4</td><td>87.3</td><td>79.6</td></tr><tr><td>Open-set DA</td><td>SF</td><td>A→C</td><td>A→P</td><td>A→R</td><td>C→A</td><td>C→P</td><td>C→R</td><td>P→A</td><td>P→C</td><td>P→R</td><td>R→A</td><td>R→C</td><td>R→P</td><td>Mean</td></tr><tr><td>OSBP[67]</td><td>X</td><td>56.7</td><td>51.5</td><td>49.2</td><td>67.5</td><td>65.5</td><td>74.0</td><td>62.5</td><td>64.8</td><td>69.3</td><td>80.6</td><td>74.7</td><td>71.5</td><td>65.7</td></tr><tr><td>OpenMax [2]</td><td></td><td>56.5</td><td>52.9</td><td>53.7</td><td>69.1</td><td>64.8</td><td>74.5</td><td>64.1</td><td>64.0</td><td>71.2</td><td>80.3</td><td>73.0</td><td>76.9</td><td>66.7</td></tr><tr><td>STA [47]</td><td>X</td><td>58.1</td><td>53.1</td><td>54.4</td><td>71.6</td><td>69.3</td><td>81.9</td><td>63.4</td><td>65.2</td><td>74.9</td><td>85.0</td><td>75.8</td><td>80.8</td><td>69.5</td></tr><tr><td>SHOT[45]</td><td></td><td>62.5</td><td>77.8</td><td>83.9</td><td>60.9</td><td>73.4</td><td>79.4</td><td>64.7</td><td>58.7</td><td>83.1</td><td>69.1</td><td>62.0</td><td>82.1</td><td>71.5</td></tr><tr><td>+HCL</td><td>1</td><td>64.2</td><td>78.3</td><td>83.0</td><td>61.1</td><td>72.2</td><td>79.6</td><td>65.5</td><td>59.3</td><td>80.6</td><td>80.1</td><td>72.0</td><td>82.8</td><td>73.2</td></tr><tr><td>HCL</td><td>√</td><td>64.0</td><td>78.6</td><td>82.4</td><td>64.5</td><td>73.1</td><td>80.1</td><td>64.8</td><td>59.8</td><td>75.3</td><td>78.1</td><td>69.3</td><td>81.5</td><td>72.6</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/209e3b18de22c7d192c35d0e2839dfa1cd7f4269f1ba67662048a326e08c5ec7.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative illustrations and comparison over domain adaptive semantic segmentation task $\\mathrm { G T A } 5 $ Cityscapes. Our historical contrastive learning (HCL) exploits historical source hypothesis to make up for the absence of source data in UMA, which produces better qualitative results (i.e., semantic segmentation) by preserving the source hypothesis. It can be observed that HCL generates better segmentation results, for example, the sidewalk in the first row, the road in the second row and the sky and sidewalk in the third row. "
+        ],
+        "image_footnote": [],
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+        "text": "5 Conclusion ",
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+        "text": "In this work, we studied historical contrastive learning, an innovative UMA technique that exploits historical source hypothesis to make up for the absence of source data in UMA. We achieve historical contrastive learning by novel designs of historical contrastive instance discrimination and historical contrastive category discrimination which learn discriminative representations for target data while preserving source hypothesis simultaneously. Extensive experiments over a variety of visual tasks and learning setups show that HCL outperforms state-of-the-art techniques consistently. Moving forward, we will explore memory-based learning in other transfer learning tasks. ",
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+        "text": "Acknowledgement ",
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+        "text": "This research was conducted at Singtel Cognitive and Artificial Intelligence Lab for Enterprises (SCALE $@$ NTU), which is a collaboration between Singapore Telecommunications Limited (Singtel) and Nanyang Technological University (NTU) that is supported by $\\mathbf { A } { ^ { * } \\mathbf { S } } \\mathbf { T } \\mathbf { A } \\mathbf { R }$ under its Industry Alignment Fund (LOA Award number: I1701E0013). ",
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parse/train/2j_cut38wv/2j_cut38wv.md

@@ -0,0 +1,288 @@
+# Backdoor Attack with Imperceptible Input and Latent Modification
+
+Khoa Doan, Yingjie Lao, Ping Li Cognitive Computing Lab Baidu Research 10900 NE 8th St. Bellevue, WA 98004, USA {khoadoan106, laoyingjie, pingli98}@gmail.com
+
+# Abstract
+
+Recent studies have shown that deep neural networks (DNN) are vulnerable to various adversarial attacks. In particular, an adversary can inject a stealthy backdoor into a model such that the compromised model will behave normally without the presence of the trigger. Techniques for generating backdoor images that are visually imperceptible from clean images have also been developed recently, which further enhance the stealthiness of the backdoor attacks from the input space. Along with the development of attacks, defense against backdoor attacks is also evolving. Many existing countermeasures found that backdoor tends to leave tangible footprints in the latent or feature space, which can be utilized to mitigate backdoor attacks.
+
+In this paper, we extend the concept of imperceptible backdoor from the input space to the latent representation, which significantly improves the effectiveness against the existing defense mechanisms, especially those relying on the distinguishability between clean inputs and backdoor inputs in latent space. In the proposed framework, the trigger function will learn to manipulate the input by injecting imperceptible input noise while matching the latent representations of the clean and manipulated inputs via a Wasserstein-based regularization of the corresponding empirical distributions. We formulate such an objective as a non-convex and constrained optimization problem and solve the problem with an efficient stochastic alternating optimization procedure. We name the proposed backdoor attack as Wasserstein Backdoor (WB), which achieves a high attack success rate while being stealthy from both the input and latent spaces, as tested in several benchmark datasets, including MNIST, CIFAR10, GTSRB, and TinyImagenet.
+
+# 1 Introduction
+
+In the past years, deep neural network (DNN) has successfully transformed many technological fields, such as object classification [26, 20], face recognition [31, 1], autonomous driving [53], security applications [19, 3], etc. Meanwhile, due to the underlying black-box nature, its security and privacy implications have also raised serious concerns recently. Efforts in the research community have exposed the vulnerability of DNN classifiers to various attacks [50, 41, 33]. For instance, adversarial examples leverage the difference between the classifier and human to misclassify specific inputs by adding imperceptible perturbations without altering the model [17]. Such attacks during the inference phase are categorized as evasion attacks [27, 5]. On the other hand, poisoning attacks attempt to inject malicious data points or manipulate the training process to either degrade the model accuracy [37, 45, 60] or cause misclassification for specific inputs (a.k.a. backdoor attacks) [8, 36, 34, 18].
+
+In general, backdoor attacks aim at injecting a malicious behavior into a DNN model so that the model would perform normally on clean inputs but yield misclassification in the presence of the backdoor trigger (e.g., a specific pattern such as a small square [18]). Later on, many works adopt the concepts and techniques in adversarial examples to improve the stealthiness of the trigger against human observers [34, 2, 35]. Recent works have demonstrated more powerful backdoor attacks that are capable of mounting attacks with visual indistinguishable backdoor images [29, 55, 59, 39, 13]. For instance, WaNet [39] generates backdoor images with warping transformation to minimize input difference while LIRA [13] generates backdoor images with imperceptible conditional noise addition, resulting in much stealthier triggers.
+
+To alleviate the threats originated from the ever-growing powerful backdoor attacks, several categories of countermeasures have been developed. One promising direction for backdoor detection entails identifying backdoor images by characterizing the distinguishable dissimilarity in the feature or latent representation between backdoor images and clean images [6, 54, 42, 47, 52]. These methods rely on the assumption that the injected backdoor would leave a noticeable fingerprint in the latent space. For example, activation clustering [6] and spectral signature [54] detect malicious samples by inspecting the clusters of the latent space and the spectrum of the covariance of latent representations, respectively. Thus, a stronger adaptive backdoor attack should also ensure its stealthiness from the latent space.
+
+In this paper, we present a novel methodology for a backdoor attack that is imperceptible from both the input and latent spaces. We extend the concept of generating imperceptible backdoor triggers to the latent space by minimizing the Wasserstein distance between the latent representations of the clean and backdoor data, which significantly improves the effectiveness against the existing defense mechanisms, especially those aforementioned that rely on the distinguishability in latent space. We name the proposed method Wasserstein Backdoor, or WB. Our technical contributions are summarized below:
+
+• We propose a non-convex, constrained optimization problem, which learns to poison the classifier with a backdoor whose trigger is visually imperceptible in the input space and whose poisoned samples have indistinguishable latent distribution to the latent distribution of the clean samples. The latent constraint is formulated via a variant of Wasserstein distance, called sliced-Wasserstein distance [24], between the two sets of clean and backdoor data. • We then develop an efficient estimation of the sliced-Wasserstein distance by exploiting the discriminant directions of the trained classifier, instead of randomly sampling from the unit sphere. The proposed distance is a valid distance metric and requires significantly less computation, while yielding a better estimate than the existing calculations of the sliced-Wasserstein distance. • Finally, we demonstrate the superior attack performance of the proposed method and its robustness against several representative defense mechanisms. Specifically, we show that the proposed method outperforms the state-of-the-art attacks in terms of latent indistinguishability, while maintaining similar attack success rates and input indistinguishability.
+
+The rest of the paper is organized as follows. We review the background and related work in Section 2. In Section 3, we define the threat model. Section 4 presents the details of the proposed methodology. We evaluate the performance and compare to prior works in Section 5. Finally, Section 6 presents remarks and concludes this paper. We present more details about experimental settings and results as well as supporting proofs in the supplementary material.
+
+# 2 Background and Related Work
+
+# 2.1 Backdoor Attack
+
+The increasing popularity of training outsourcing and machine learning as a service (MLaaS) has created potential security risks in the supply chain [10, 58]. One important security threat is backdoor attacks against DNNs, which have recently attracted a lot of attention. Backdoor attacks inject a malicious behavior by leveraging the redundancies inside the model such that the model responds to inputs with triggers maliciously (e.g., classify as a target class that would normally be considered as a wrong class by manual annotation), while preserving the benign behavior for clean inputs without the triggers. Hence, a typical backdoor embedding process is to train the model by minimizing the loss of the clean inputs and the corresponding labels as well as backdoor inputs (with triggers) and the target class(es). A trigger is typically applied on a clean image by superimposing at a certain location (i.e., patch-based) [18, 34] or adding perturbations [44]. Various forms of the triggers have been investigated in the literature, including blended [8], sinusoidal strips (SIG) [2], reflection (ReFool) [35], and warping-based (WaNet) [39]. As we mentioned above, several techniques have been developed recently that can significantly reduce the visibility of the trigger in the input space to enhance the stealthiness of the backdoor attack [34, 2, 35]. In particular, WaNet uses a smooth warping field to generate backdoor images with unnoticeable modifications [39], while LIRA [13] alternates between the processes of trigger generation and backdoor injection to learn visually stealthy triggers. One prior work, Adversarial Embedding [51], also attempted to improve the latent indistinguishability of the backdoor attack by using adversarial regularization to minimize the distance between the latent distributions of the backdoor inputs and clean inputs.
+
+# 2.2 Backdoor Defense
+
+By exploring specific characteristics of the injected backdoor, various countermeasures have been proposed [6, 54, 16, 47, 9, 7, 42], although they are often circumvented by subsequent adaptive attacks. For instance, based on the property that a backdoor attack usually targets redundant weights or neurons based on the clean images, model pruning can be used to eliminate the injected backdoor [32]. In contrast, Neural Cleanse assumes a known subset of clean inputs to reverse-engineer possible trigger patches [56]. It is also possible to filter the images to nullify the presence of triggers at the test phase to defend against backdoor attacks [36, 30].
+
+In this paper, we focus on optimizing the characteristics of backdoor attacks in the latent space. As we discussed above, the rationale behind this is that prior works have demonstrated backdoor images cause distinctive activations in the latent space from those of clean inputs. Hence, this distinguishable dissimilarity between clean images and backdoor images can be utilized for defense in both training [6, 54] and test phases [49, 23, 22]. Most of these approaches compute an outlier score to detect abnormal inputs that will be filtered afterward. For example, spectral signature [54] computes the outlier score based on the singular value decomposition of the covariance matrix of the latent representations, while CleaNN [22] leverages a concentration inequality to detect anomalous reconstruction errors that are then suppressed before the input entering the victim DNN.
+
+This work proposes a method to minimize the difference between clean images and backdoor images in the latent space to improve the attack stealthiness. While doing this, we also optimize the visual imperceptibility in the input space, so that our proposed method can bypass visual inspection.
+
+# 3 Threat Model
+
+We consider the same threat model as in prior studies [44, 51, 39], which assumes the backdoor injection is performed at training and the adversary can access to the victim model including both structures and parameters. A successful backdoor attack over an image classification task should produce malicious behavior on images with the trigger, while otherwise working normally on clean images. However, in typical backdoor attacks, the poisoned images are visually inconsistent with natural images, which can be identified easily by human observers. Besides, these attacks usually leave a tangible trace in the latent space of the poisoned classifier; thus, some defense methods can easily detect and discard the poisoned models. To this end, we propose a stronger backdoor attack where the poisoned images are crafted with imperceptible perturbation in the input space to clean images as well as unnoticeable trace in the latent space. We advance the state-of-the-art by significantly enhancing the imperceptibility and robustness of the backdoor attack.
+
+# 4 Proposed Methodology: Wasserstein Backdoor (WB)
+
+# 4.1 Preliminaries
+
+Consider the standard supervised classification task where one seeks to learn a mapping function $f _ { \theta } : \mathcal { X } \longrightarrow \mathcal { C }$ where $\mathcal { X }$ is the input domain and $\mathcal { C }$ is the set of target classes. The task is to learn the parameters $\theta$ by using the training dataset $\mathcal { S } = \{ ( x _ { i } , y _ { i } ) : x _ { i } \in \bar { \mathcal { X } } , y _ { i } \in \mathcal { C } , i = 1 , . . , N \}$ .
+
+Following the standard training scheme of backdoor attacks, the classifier is trained with the combination of the clean and poisoned subsets of $S$ . To create a poisoned sample, a clean training sample $( x , y )$ is transformed into a backdoor sample $( T ( x ) , \eta ( y ) )$ , where $T$ is a backdoor injection function (also called the trigger function) and $\eta$ is the target label function. When training $f$ with the clean and poison samples, we alter the behavior of $f$ so that:
+
+$$
+f ( x ) = y , \quad f ( T ( x ) ) = \eta ( y ) ,
+$$
+
+for any pair of clean data $x \in \mathcal { X }$ and its corresponding label $y \in { \mathcal { C } }$ . There are two commonly studied backdoor attack settings [18, 39, 51]: all-to-one and all-to-all. In the all-to-one attack, the label is changed to a constant target, i.e. $\eta ( y ) = c$ ; while for the all-to-all attack, the true label is one-shifted, i.e. $\bar { \eta ( y ) } = ( y + 1 )$ mod $| { \mathcal { C } } |$ . In the existing works, the trigger function $T$ is usually selected before training $f$ and fixed during the training process of $f$ .
+
+# 4.2 Learning to Backdoor
+
+Given the training dataset $s$ and a loss function $\mathcal { L }$ , e.g., cross entropy loss, empirical risk minimization can be used to learn the parameters $\theta$ , as follows:
+
+$$
+\theta ^ { * } = \arg \operatorname* { m i n } _ { \theta } \sum _ { i = 1 } ^ { N } \mathcal { L } ( f _ { \theta } ( x _ { i } ) , y _ { i } ) .
+$$
+
+The goal of this work is to learn a trigger function $T _ { \xi } : \mathcal { X } \longrightarrow \mathcal { X }$ and a classification model $f _ { \theta }$ in such a way that the clean image $x$ and its corresponding backdoor image $T ( x )$ are visually consistent in the input space while the backdoor attack does not leave a detectable trace in the latent space of the poisoned classifier. When $f$ is a neural network, $\phi ( x )$ can be the output of an intermediate, hidden layer of $f$ , which captures some high-level abstractions of the input. Note that we require the classifier to perform normally on the clean sample, $x$ , compared to the classifier’s vanilla version, but change its prediction on the poisoned image, $T ( x )$ , to the target class $\eta ( y )$ .
+
+To generate a trigger and poison the image, we follow the prior work [13] and formulate the trigger function as a conditional noise generator $g$ , as follows:
+
+$$
+T _ { \xi } ( x ) = x + g _ { \xi } ( x ) , \quad | | g _ { \xi } ( x ) | | _ { \infty } \leq \epsilon \forall x
+$$
+
+The generator function $g _ { \xi }$ takes an input $x$ and generates an artificially imperceptible noise on the same input space, which guarantees the stealthiness of the backdoor attack. We can design such generator function as an autoencoder or the more complex U-Net architecture [43].
+
+With the above objectives and notations, similar to [13, 4], we can formulate the task into the following constrained optimization problem:
+
+$$
+\begin{array} { r l } { \displaystyle } & { { } \displaystyle \operatorname* { m i n } _ { \theta } \sum _ { i = 1 } ^ { N } \alpha \mathcal { L } \big ( f _ { \theta } ( x _ { i } ) , y _ { i } \big ) + \beta \mathcal { L } \big ( f _ { \theta } ( T _ { \xi ^ { * } ( \theta ) } ( x _ { i } ) \big ) , \eta ( y _ { i } ) \big ) } \\ { \displaystyle s . t . \quad } & { { } \xi ^ { * } = \arg \operatorname* { m i n } _ { \xi } \sum _ { i = 1 } ^ { N } \mathcal { L } \big ( f _ { \theta } ( T _ { \xi } ( x _ { i } ) ) , \eta ( y _ { i } ) \big ) + \mathcal { R } _ { \phi } ( \mathcal { F } _ { c } , \mathcal { F } _ { b } ) } \end{array}
+$$
+
+where $\mathcal { R } _ { \phi }$ is the regularization constraint of the clean and poisoned representations, denoted as $\mathcal { F } _ { c } = \{ \phi ( x _ { i } ) : i = 1 , . . , N \}$ and $\mathcal { F } _ { b } \{ \phi ( T ( x _ { i } ) ) : i = 1 , . . , N \}$ , respectively.
+
+In this problem, a learned classification model with a specific parameter configuration $\theta$ is associated with an optimal yet stealthy backdoor trigger function, which is trained to poison the model. The classifier is trained to minimize a linear combination of clean and targeted backdoor objectives. The parameters $\alpha$ and $\beta$ control the mixing strengths of the clean and backdoor loss signals. The trigger function is trained to perturb an image within its $\ell _ { \infty }$ ball in the input space, so that the loss towards the attack target class is minimized while regularizing the latent representations of the backdoor images.
+
+# 4.3 Stealthy Latent Representation via Wasserstein Regularization
+
+In practical applications, latent-space defense methods investigate the abnormal trace of incoming data points with respect to the previous stream of data. These traces exist primarily because of the fact that the clean and backdoor latent representations are separated or distributed differently (e.g., the separated clusters of the clean and poison representations that can be seen in Figures 2 and 3). Thus, we aim to minimize such distributional difference through the regularization constraint $\mathcal { R } _ { \phi }$ . Since we cannot assume that the two latent distributions have common support or their density functions are known, commonly-used divergences, such as $f$ -divergences [40, 15] (which include KL and JSD), are difficult to minimize. Instead, we consider the Wasserstein-2 distance and formulate the regularization constraint as follows:
+
+$$
+\mathcal { R } \phi ( \mu , \nu ) = \left( \operatorname* { i n f } _ { \gamma \in \Pi ( \mu , \nu ) } \int _ { ( x , z ) \sim \gamma } p ( x , z ) | | x - z | | _ { 2 } d x d z \right) ^ { 1 / 2 }
+$$
+
+where $\mu$ and $\nu$ are marginal probability measures defined by empirical samples $\mathcal { F } _ { c }$ and $\mathcal { F } _ { b }$ of the latent representations of the clean and poisoned data, respectively.
+
+Estimating the Wasserstein distance also has some challenges. From the primal domain, computing the infimum in Equation (4) is particularly difficult since the data distributions are not fixed or known. On the other hand, employing the Kantorovich-Rubinstein duality requires a separate, parameterized Lipschitz function and a minimax solver, which increases the complexity of the proposed problem. Fortunately, for one-dimensional continuous measures, the Wasserstein distance has an elegant yet closed-form solution. Let $q _ { \mu }$ and $q _ { \nu }$ be the corresponding density functions of $\mu$ and $\nu$ , respectively. The Wasserstein-2 distance between one-dimensional measures $\mu$ and $\nu$ can be given by:
+
+$$
+\mathcal { W } ( \mu , \nu ) = \left( \int _ { 0 } ^ { 1 } | | ( F _ { \mu } ^ { - 1 } ( z ) - F _ { \nu } ^ { - 1 } ( z ) | | _ { 2 } d z \right) ^ { 1 / 2 }
+$$
+
+where $\begin{array} { r } { F _ { \mu } ( z ) = \int _ { \infty } ^ { z } q _ { \mu } ( \rho ) d \rho } \end{array}$ and $\begin{array} { r } { F _ { \nu } ( z ) = \int _ { \infty } ^ { z } q _ { \nu } ( \rho ) d \rho } \end{array}$ are the cumulative distribution functions. 1 1Inspired by the efficiency of this solution and its successful applications in a variety of tasks [12, 24, 14], we propose to first find a family of one-dimensional representations, e.g., through the linear projections, and approximate the Wasserstein distance as a function of these one-dimensional marginals, as follows:
+
+$$
+\mathcal { R } _ { \phi } ( \mathcal { F } _ { c } , \mathcal { F } _ { b } ) \approx \left( \frac { 1 } { L } \sum _ { l = 1 } ^ { L } [ \mathcal { W } ( \mathcal { F } _ { c } ^ { \theta _ { l } } , \mathcal { F } _ { b } ^ { \theta _ { l } } ) ] ^ { 2 } \right) ^ { 1 / 2 }
+$$
+
+where $\mathcal { F } _ { c } ^ { \theta _ { l } } = \{ \theta _ { l } ^ { T } \phi ( x _ { i } ) : i = 1 , . . , N \}$ and $\mathcal { F } _ { b } ^ { \theta _ { l } } = \{ \theta _ { l } ^ { T } \phi ( T ( x _ { i } ) ) : i = 1 , . . , N \}$ contains the projections of the clean and poisoned datasets into a one-dimensional direction defined by $\theta _ { l }$ (a slice). Typically, $\theta _ { l }$ is drawn from a uniform distribution on the unit sphere. This formulation is also known as the sliced-Wasserstein distance (SWD) [12, 24]. One particular problem with this approach is that the random nature of the slices could lead to several non-informative directions; i.e., the sliced distances are close to 0 in directions that do not lie on the manifolds of the data. Consequently, a large number $L$ of random directions are needed to approximate the sliced-Wasserstein distance, which increases the computational complexity of the estimation.
+
+To remedy this issue, we avoid the uniform sampling of the unit sphere and select directions that contain discriminant information of the two data sources, by exploiting the following fact in the classification task. For backdoor samples of a target class $c _ { 1 } \in { \mathcal { C } }$ , created from clean samples of some other class $c _ { 2 } \in { \mathcal { C } }$ , the projections into an output dimension represent meaningful discriminant information that distinguishes the backdoor samples (from class $c _ { 2 }$ ) and the clean samples (from class $c _ { 1 }$ ). Thus, we propose to replace the uniform linear projections of SWD with the projections into the output layer. When the latent space is the penultimate layer of the classifier, such projections are equivalent to the following approximation:
+
+$$
+\mathcal { R } _ { \phi } ( \mathcal { F } _ { c } , \mathcal { F } _ { b } ) \approx \left( \frac { 1 } { | \mathcal { C } | } \sum _ { c = 1 } ^ { | \mathcal { C } | } \left[ \mathcal { W } ( \mathcal { F } _ { c } ^ { W _ { c , : } } , \mathcal { F } _ { c } ^ { W _ { c , : } } ) \right] ^ { 2 } \right) ^ { 1 / 2 } .
+$$
+
+where $W _ { c , : }$ is a row of the matrix $W \in \mathbb { R } ^ { | \mathcal { C } | \times d }$ ( $d$ is the dimension of the latent space), which is the normalized parameter matrix between the penultimate and the output layers.
+
+![](images/c7af9caf5c809e25a62d365bbd85a4c2242c21166b3b94b61d1406b07a5c6433.jpg)  
+Figure 1: Distance estimates (normalized) in the latent space for SWD with different number of sampled directions (between 10 to 10,000) and DSWD.
+
+Empirically, Figure 1 shows the estimated SWD with different numbers of random directions and the proposed calculation, so called DSWD, when the latent space is defined at the penultimate layer of the classifier. The dimension of the latent space is 512 for both MNIST and CIFAR10 datasets. Each distance is computed on a random sample of 1000 clean and 1000 backdoor images, and each calculation is repeated 100 times. It can be seen that with only a fraction of slices, DSWD achieves a significantly smaller variance than that of the SWD estimates. Furthermore, in MNIST, the selected directions of DSWD leads to higher distance estimates than SWD, which means that DSWD selects more discriminant directions than SWD while SWD underestimates the distance between the two empirical samples. In addition, we show that DSWD is a valid distance metric of the latent distributions. The detailed proof is presented in the supplementary material.
+
+Theorem 1. When the latent space is the penultimate layer of a neural network, the proposed DSWD distance is a valid distance function of probability measures in this space.
+
+Remark 1. Since existing defense methods choose the penultimate layer of a neural network. as the space to perform the defense analysis, in most cases, we can employ the proposed DSWD calculation.
+
+Remark 2. To preserve the clean classification performance, the classifier seeks optimal parameters that lead to similar predictions of clean samples from the same class. The goal of the trigger function is to make the poisoned samples classified toward a different class. This leads to an adversarial game between the classifier and the trigger functions.
+
+DSWD also has a significantly better computational efficiency than SWD. In most problems, SWD requires a large number of random directions, typically between 1000 to 10,000, in order to provide a reliable estimate of the distance [38, 14]. In DSWD, the number of random directions is fixed to the number of possible output labels, which is typically small for many classification problems.
+
+# 4.4 Optimization
+
+The non-convex, constrained optimization in Equation (3) is challenging because of its non-linear constraint. In general, we can alternately update one of $f$ and $T$ while keeping the other fixed, similar to training GANs. However, it is difficult and slow for the classifier to reach an acceptable performance on the clean data, i.e., similar accuracy to that of the vanilla classifier.
+
+Under the alternating update scheme, we observe that on MNIST, the poisoned classifier can reach the acceptable clean-data performance after several epochs; while on other more complex datasets (i.e. CIFAR10, GTSRB, and TinyImagenet), this procedure results in sub-optimal clean-data performance. One possible explanation is that training the vanilla classifier with complex architecture and dataset to reach a decent accuracy is already a difficult and time-consuming task (e.g., 2 to 3 epochs to reach the optimal performance on MNIST but several hundreds of epochs on the other datasets).
+
+Fortunately, we observe that after training the classifier and the trigger functions in an alternating update scheme for a certain number of epochs (denoted as Stage I), we can fix the trigger function and only train the classifier for the remaining epochs (denoted as Stage II). This two-stage training scheme is adopted in our experiments.
+
+# 5 Experimental Results
+
+# 5.1 Experimental Setup
+
+We demonstrate the effectiveness of the proposed method through a range of experiments on four widely-used datasets for backdoor attack study: MNIST, CIFAR10, GTSRB and TinyImagenet. For these experiments, we follow the previous works [51, 54, 6, 39] and select the penultimate layers of the classifiers as the latent space for the defense experiments. The implementation of WB was based on the PaddlePaddle deep learning platform.
+
+Architectures: For the classifier $f$ , we consider several popular models: Pre-activation Resnet18 [20], VGG [46], DenseNet [21] for CIFAR10 and GTSRB datasets, and Resnet-18 for TinyImagenet. For the MNIST dataset, we employ a CNN model.
+
+Hyperparameters: For the baselines, we train the classifiers using the SGD optimizer with an initial learning rate of 0.01 and a learning rate decay of 0.1 after every 100 epochs. For other hyperparameters, we follow the proposed setup in [39] for all datasets. We use the same configurations for WB. We train the classifier and trigger functions alternately (Stage I) for 10 and 50 epochs for MNIST and the other datasets, respectively, and fine-tune the classifier (Stage II) for another 40 epochs and 450 epochs for MNIST and the other datasets, respectively. To achieve a high-degree stealthiness of WB, we pick $\epsilon$ as small as 0.01 for all datasets. In general, the larger the value of $\epsilon$ the easier the trigger functions can be learned and the more successful the attacks are.
+
+# 5.2 Attack Performance
+
+We present the attack success rates of the proposed WB method, along with a comparison to two state-of-the-art methods, i.e., WaNet [39] and LIRA [13]. Both LIRA and Wanet’s attack performances are significantly better than other approaches, including BadNets [18], and are two of the strongest existing methods that generate very stealthy triggers on the images. We first poison the classifier using the backdoor attack methods in both all-to-one and all-to-all settings and record the performance of the classifier on both clean and backdoor test samples. For all-to-one, we randomly pick the target label $\hat { c }$ (i.e., $\eta ( y ) = \hat { c } \forall y )$ , while for all-to-all, the target label function is defined as $\eta ( y ) = { \bar { ( y + 1 ) } }$ mod $| { \mathcal { C } } | \ \forall y$ , which is widely used to evaluate the backdoor-related works [39, 18, 6, 13]. Note that this all-to-all attack setting is more challenging than the all-to-one setting, especially on datasets with a large number of classes such as TinyImagenet.
+
+The classification accuracy on the clean test samples and the attack success rate for each method is represented in Table 1 and Table 2 for the all-to-one and all-to-all settings, respectively. As we can observe from these tables, all the methods can achieve high clean-data accuracies and attack success rates. While WB’s attack performance slightly drops compared to LIRA’s performance, WB is significantly more stealthy in the latent space, as being discussed next.
+
+Table 1: Attack Performance: All-to-one Attack   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">WaNet</td><td colspan="2">LIRA</td><td colspan="2">WB</td></tr><tr><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td></tr><tr><td>MNIST</td><td>0.99</td><td>0.99</td><td>0.99</td><td>1.00</td><td>0.99</td><td>0.99</td></tr><tr><td>CIFAR10</td><td>0.94</td><td>0.99</td><td>0.94</td><td>1.00</td><td>0.94</td><td>0.99</td></tr><tr><td>GTSRB</td><td>0.99</td><td>0.98</td><td>0.99</td><td>1.00</td><td>0.99</td><td>0.99</td></tr><tr><td>TinyImagenet</td><td>0.57</td><td>0.99</td><td>0.58</td><td>1.00</td><td>0.57</td><td>0.99</td></tr></table>
+
+Table 2: Attack Performance: All-to-all Attack   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">WaNet</td><td colspan="2">LIRA</td><td colspan="2">WB</td></tr><tr><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td></tr><tr><td>MNIST</td><td>0.99</td><td>0.95</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.96</td></tr><tr><td>CIFAR10</td><td>0.94</td><td>0.93</td><td>0.94</td><td>0.94</td><td>0.94</td><td>0.94</td></tr><tr><td>GTSRB</td><td>0.99</td><td>0.98</td><td>0.99</td><td>1.00</td><td>0.99</td><td>0.98</td></tr><tr><td>TinyImagenet</td><td>0.58</td><td>0.58</td><td>0.58</td><td>0.59</td><td>0.58</td><td>0.58</td></tr></table>
+
+# 5.3 Latent-Space Defense
+
+Recent works on backdoor defense have found that backdoor attacks tend to leave a tangible trace in the latent space of the poisoned classifier. Activation Clustering [6] and Spectral Signature [54] are two representative defenses used for analyzing the latent space in prior work [51]. In this section, we also examine the latent space of the backdoor-injected classifiers through the lens of these defense methods.
+
+# 5.3.1 Learned Latent Representation and Activation Clustering
+
+It has been shown in [6] that in a poisoned classifier, the latent representations of the clean and backdoor samples form separate clusters, which can be easily detected using clustering methods such as K-means. The authors also recommend a process called exclusionary reclassification to determine which cluster is poisoned and re-train the poisoned classifier.
+
+In Figure 2 and Figure 3, we can observe highly separated clusters (for samples with the sample predictions of $y = 0$ ) in the latent space when we omit the latent regularization term $\mathcal { R } _ { \phi }$ in WB (Baseline), which is similar to LIRA [13]. However, when $\mathcal { R } _ { \phi }$ is included, the latent representations of the clean and backdoor samples are distributed similarly. Without well-separated clusters of the clean and poisoned samples, the exclusionary reclassification process in the activation clustering is not effective against the attacks.
+
+![](images/06e6d3444d42a0f1ae4a52a74f76903477b0bdaf9bc0b3cb802283e00926243d.jpg)  
+Figure 2: MNIST: t-SNE embedding in the latent space. Baseline is WB without $\mathcal { R } _ { \phi }$ .
+
+![](images/b60bd196e6d6df38879720961f333230fa538a26c3db4859eca00fcaeef41932.jpg)  
+Figure 3: CIFAR10: t-SNE embedding in the latent space. Baseline is WB without $\mathcal { R } _ { \phi }$ .
+
+Quantitatively, we present the quality scores (i.e., the adjusted Rand Index) of the clustering step in Table 3. The adjusted Rand Index is 1 when the samples form two distinct clusters and is close to 0 for a random separation. We compare WB with BadNets [18] and Adversarial Embedding [51], which is the state-of-the-art backdoor attack method with stealthy latent space. As we can observe in this table, the defense is most successful on BadNets since there exists a perfect clustering of the clean and poisoned samples (Rand Index $\geq 0 . 9 5$ ). While Adversarial Embedding is more resistant against the defense, WB is significantly more stealthy against the defense since the values of Rand Index are all very close to 0. Note that, similar to BadNets, WaNet also does not pass this defense (please see the supplementary material).
+
+Table 3: Adjusted Rand Index in All-to-one Attack   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Dataset</td><td rowspan="2">Rand Index (BadNets)</td><td colspan="2">Adversarial Embedding</td><td colspan="2">WB</td></tr><tr><td>Rand Index</td><td>Attack</td><td>Rand Index</td><td>Attack</td></tr><tr><td>DenseNet</td><td>CIFAR10</td><td>0.979</td><td>0.1820</td><td>0.764</td><td>0.0382</td><td>0.998</td></tr><tr><td>DenseNet</td><td>GTSRB</td><td>0.997</td><td>0.2710</td><td>0.914</td><td>0.0135</td><td>0.997</td></tr><tr><td>VGG</td><td>CIFAR10</td><td>0.998</td><td>0.0006</td><td>0.962</td><td>0.0002</td><td>0.999</td></tr><tr><td>VGG</td><td>GTSRB</td><td>0.997</td><td>0.6420</td><td>0.743</td><td>0.1010</td><td>0.999</td></tr></table>
+
+# 5.3.2 Spectral Signature Defense
+
+The work in [54] proposes a defense method that identifies and removes backdoor samples using the Spectral Signature. For data from each predicted class, Spectral Signature first finds the top singular value of the covariance matrix of the latent vectors of the data. Then it computes the correlation score to this singular value for each sample and those samples with the outlier scores are flagged as backdoor samples. While Spectral Signature is a sample filtering-based defense method, the inspection of the correlation scores can also be useful to verify whether there is a tangible trace in the latent space of the classifier.
+
+Following the same experiments in [54], we first pick 5,000 clean samples and 500 backdoor samples for each dataset. Then, we plot the histograms of the correlation scores for both sets of samples. As we can observe in Figure 4, there is no clear separation between the scores of the backdoor samples and those of the clean samples.
+
+![](images/92dceef2b089708025cd466c7e81445da9fe91338f47a7c368502aa6b41ae451.jpg)  
+Figure 4: Defense experiments of the all-to-one attack against Spectral Signature. The correlations of the clean and backdoor samples with the top singular vector of the covariance matrix in the latent space are not separable.
+
+# 5.4 Model Mitigation Defense
+
+In this section, we evaluate the robustness of WB against another popular defense, Neural Cleanse [56], which is model mitigation defense based on a pattern optimization approach. Specifically, Neural Cleanse searches for the optimal patch pattern for each possible target label that induces a misclassification to that label. It then quantifies whether any of the optimal backdoor trigger pattern is an outlier via a metric called Anomaly Index. The model has a backdoor if the Anomaly Index is greater than 2 for any class. The anomaly indices are presented in Figure 5.
+
+It can be seen that both WaNet and WB can pass the detection of Neural Cleanse, similar to that of the vanilla classifier (Clean). In MNIST and CIFAR10, WB even achieves smaller Anomaly Indices than those of the vanilla models. Note that popular backdoor attacks, such as BadNets, can be defended by Neural Cleanse in most of these datasets [56].
+
+![](images/8cf52aada9604635bbac1b3e8db94e8deb55a98abd1bd801f6b427337a81fcbd.jpg)  
+Figure 5: Backdoor attacks against Neural Cleanse defense.
+
+# 5.5 Input Perturbation Defense
+
+In this section, we study the stealthiness of WB against STRIP [16], a representative detection based backdoor defense mechanism. Given the classifier and an input image, STRIP first perturbs the image and determines the presence of a backdoor in the model according to the entropy of the predictions of these perturbed images (i.e., if the predictions are consistent or not).
+
+In Figure 6, we plot the entropy of clean and backdoor images, which are computed by STRIP. We can observe that the distribution of entropy of the backdoor samples is similar to that of the clean samples. In other words, STRIP fails to detect backdoor samples generated by WB, which further validates the advantage of the proposed method.
+
+![](images/0e55b555ba734983cb5aead631bac73c591a889d0922fbd0ca99fa3092065999.jpg)  
+Figure 6: Performance against STRIP defense.
+
+Additional experiments for demonstrating the stealthiness of WB against several other defense approaches can be found in the supplementary material.
+
+# 6 Conclusion
+
+This paper presented a novel methodology for a backdoor attack that is imperceptible from both the input and latent spaces, i.e., Wasserstein Backdoor (WB). WB learns a trigger function that adds visually imperceptible noise to an input image and minimizes the distributional difference via a novel sliced Wasserstein distance formulation between representations of the clean and backdoor images in the latent space of the trained classifier. We comprehensively evaluated the performance of the proposed method on various image classification benchmark models over a wide range of datasets. Our experimental results demonstrated that the proposed method could significantly improve the effectiveness against the existing defense mechanisms, especially those relying on the distinguishability in latent space.
+
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+        "text": "Backdoor Attack with Imperceptible Input and Latent Modification ",
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+        "text": "Khoa Doan, Yingjie Lao, Ping Li Cognitive Computing Lab Baidu Research 10900 NE 8th St. Bellevue, WA 98004, USA {khoadoan106, laoyingjie, pingli98}@gmail.com ",
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+        "text": "Abstract ",
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+        "text": "Recent studies have shown that deep neural networks (DNN) are vulnerable to various adversarial attacks. In particular, an adversary can inject a stealthy backdoor into a model such that the compromised model will behave normally without the presence of the trigger. Techniques for generating backdoor images that are visually imperceptible from clean images have also been developed recently, which further enhance the stealthiness of the backdoor attacks from the input space. Along with the development of attacks, defense against backdoor attacks is also evolving. Many existing countermeasures found that backdoor tends to leave tangible footprints in the latent or feature space, which can be utilized to mitigate backdoor attacks. ",
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+        "text": "In this paper, we extend the concept of imperceptible backdoor from the input space to the latent representation, which significantly improves the effectiveness against the existing defense mechanisms, especially those relying on the distinguishability between clean inputs and backdoor inputs in latent space. In the proposed framework, the trigger function will learn to manipulate the input by injecting imperceptible input noise while matching the latent representations of the clean and manipulated inputs via a Wasserstein-based regularization of the corresponding empirical distributions. We formulate such an objective as a non-convex and constrained optimization problem and solve the problem with an efficient stochastic alternating optimization procedure. We name the proposed backdoor attack as Wasserstein Backdoor (WB), which achieves a high attack success rate while being stealthy from both the input and latent spaces, as tested in several benchmark datasets, including MNIST, CIFAR10, GTSRB, and TinyImagenet. ",
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+        "text": "1 Introduction ",
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+        "text": "In the past years, deep neural network (DNN) has successfully transformed many technological fields, such as object classification [26, 20], face recognition [31, 1], autonomous driving [53], security applications [19, 3], etc. Meanwhile, due to the underlying black-box nature, its security and privacy implications have also raised serious concerns recently. Efforts in the research community have exposed the vulnerability of DNN classifiers to various attacks [50, 41, 33]. For instance, adversarial examples leverage the difference between the classifier and human to misclassify specific inputs by adding imperceptible perturbations without altering the model [17]. Such attacks during the inference phase are categorized as evasion attacks [27, 5]. On the other hand, poisoning attacks attempt to inject malicious data points or manipulate the training process to either degrade the model accuracy [37, 45, 60] or cause misclassification for specific inputs (a.k.a. backdoor attacks) [8, 36, 34, 18]. ",
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+        "text": "In general, backdoor attacks aim at injecting a malicious behavior into a DNN model so that the model would perform normally on clean inputs but yield misclassification in the presence of the backdoor trigger (e.g., a specific pattern such as a small square [18]). Later on, many works adopt the concepts and techniques in adversarial examples to improve the stealthiness of the trigger against human observers [34, 2, 35]. Recent works have demonstrated more powerful backdoor attacks that are capable of mounting attacks with visual indistinguishable backdoor images [29, 55, 59, 39, 13]. For instance, WaNet [39] generates backdoor images with warping transformation to minimize input difference while LIRA [13] generates backdoor images with imperceptible conditional noise addition, resulting in much stealthier triggers. ",
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+        "text": "To alleviate the threats originated from the ever-growing powerful backdoor attacks, several categories of countermeasures have been developed. One promising direction for backdoor detection entails identifying backdoor images by characterizing the distinguishable dissimilarity in the feature or latent representation between backdoor images and clean images [6, 54, 42, 47, 52]. These methods rely on the assumption that the injected backdoor would leave a noticeable fingerprint in the latent space. For example, activation clustering [6] and spectral signature [54] detect malicious samples by inspecting the clusters of the latent space and the spectrum of the covariance of latent representations, respectively. Thus, a stronger adaptive backdoor attack should also ensure its stealthiness from the latent space. ",
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+        "text": "In this paper, we present a novel methodology for a backdoor attack that is imperceptible from both the input and latent spaces. We extend the concept of generating imperceptible backdoor triggers to the latent space by minimizing the Wasserstein distance between the latent representations of the clean and backdoor data, which significantly improves the effectiveness against the existing defense mechanisms, especially those aforementioned that rely on the distinguishability in latent space. We name the proposed method Wasserstein Backdoor, or WB. Our technical contributions are summarized below: ",
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+        "text": "• We propose a non-convex, constrained optimization problem, which learns to poison the classifier with a backdoor whose trigger is visually imperceptible in the input space and whose poisoned samples have indistinguishable latent distribution to the latent distribution of the clean samples. The latent constraint is formulated via a variant of Wasserstein distance, called sliced-Wasserstein distance [24], between the two sets of clean and backdoor data. • We then develop an efficient estimation of the sliced-Wasserstein distance by exploiting the discriminant directions of the trained classifier, instead of randomly sampling from the unit sphere. The proposed distance is a valid distance metric and requires significantly less computation, while yielding a better estimate than the existing calculations of the sliced-Wasserstein distance. • Finally, we demonstrate the superior attack performance of the proposed method and its robustness against several representative defense mechanisms. Specifically, we show that the proposed method outperforms the state-of-the-art attacks in terms of latent indistinguishability, while maintaining similar attack success rates and input indistinguishability. ",
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+        "text": "The rest of the paper is organized as follows. We review the background and related work in Section 2. In Section 3, we define the threat model. Section 4 presents the details of the proposed methodology. We evaluate the performance and compare to prior works in Section 5. Finally, Section 6 presents remarks and concludes this paper. We present more details about experimental settings and results as well as supporting proofs in the supplementary material. ",
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+        "text": "2 Background and Related Work ",
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+        "text": "2.1 Backdoor Attack ",
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+        "type": "text",
+        "text": "The increasing popularity of training outsourcing and machine learning as a service (MLaaS) has created potential security risks in the supply chain [10, 58]. One important security threat is backdoor attacks against DNNs, which have recently attracted a lot of attention. Backdoor attacks inject a malicious behavior by leveraging the redundancies inside the model such that the model responds to inputs with triggers maliciously (e.g., classify as a target class that would normally be considered as a wrong class by manual annotation), while preserving the benign behavior for clean inputs without the triggers. Hence, a typical backdoor embedding process is to train the model by minimizing the loss of the clean inputs and the corresponding labels as well as backdoor inputs (with triggers) and the target class(es). A trigger is typically applied on a clean image by superimposing at a certain location (i.e., patch-based) [18, 34] or adding perturbations [44]. Various forms of the triggers have been investigated in the literature, including blended [8], sinusoidal strips (SIG) [2], reflection (ReFool) [35], and warping-based (WaNet) [39]. As we mentioned above, several techniques have been developed recently that can significantly reduce the visibility of the trigger in the input space to enhance the stealthiness of the backdoor attack [34, 2, 35]. In particular, WaNet uses a smooth warping field to generate backdoor images with unnoticeable modifications [39], while LIRA [13] alternates between the processes of trigger generation and backdoor injection to learn visually stealthy triggers. One prior work, Adversarial Embedding [51], also attempted to improve the latent indistinguishability of the backdoor attack by using adversarial regularization to minimize the distance between the latent distributions of the backdoor inputs and clean inputs. ",
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+        "page_idx": 1
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 Backdoor Defense ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "By exploring specific characteristics of the injected backdoor, various countermeasures have been proposed [6, 54, 16, 47, 9, 7, 42], although they are often circumvented by subsequent adaptive attacks. For instance, based on the property that a backdoor attack usually targets redundant weights or neurons based on the clean images, model pruning can be used to eliminate the injected backdoor [32]. In contrast, Neural Cleanse assumes a known subset of clean inputs to reverse-engineer possible trigger patches [56]. It is also possible to filter the images to nullify the presence of triggers at the test phase to defend against backdoor attacks [36, 30]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on optimizing the characteristics of backdoor attacks in the latent space. As we discussed above, the rationale behind this is that prior works have demonstrated backdoor images cause distinctive activations in the latent space from those of clean inputs. Hence, this distinguishable dissimilarity between clean images and backdoor images can be utilized for defense in both training [6, 54] and test phases [49, 23, 22]. Most of these approaches compute an outlier score to detect abnormal inputs that will be filtered afterward. For example, spectral signature [54] computes the outlier score based on the singular value decomposition of the covariance matrix of the latent representations, while CleaNN [22] leverages a concentration inequality to detect anomalous reconstruction errors that are then suppressed before the input entering the victim DNN. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This work proposes a method to minimize the difference between clean images and backdoor images in the latent space to improve the attack stealthiness. While doing this, we also optimize the visual imperceptibility in the input space, so that our proposed method can bypass visual inspection. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Threat Model ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider the same threat model as in prior studies [44, 51, 39], which assumes the backdoor injection is performed at training and the adversary can access to the victim model including both structures and parameters. A successful backdoor attack over an image classification task should produce malicious behavior on images with the trigger, while otherwise working normally on clean images. However, in typical backdoor attacks, the poisoned images are visually inconsistent with natural images, which can be identified easily by human observers. Besides, these attacks usually leave a tangible trace in the latent space of the poisoned classifier; thus, some defense methods can easily detect and discard the poisoned models. To this end, we propose a stronger backdoor attack where the poisoned images are crafted with imperceptible perturbation in the input space to clean images as well as unnoticeable trace in the latent space. We advance the state-of-the-art by significantly enhancing the imperceptibility and robustness of the backdoor attack. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Proposed Methodology: Wasserstein Backdoor (WB) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider the standard supervised classification task where one seeks to learn a mapping function $f _ { \\theta } : \\mathcal { X } \\longrightarrow \\mathcal { C }$ where $\\mathcal { X }$ is the input domain and $\\mathcal { C }$ is the set of target classes. The task is to learn the parameters $\\theta$ by using the training dataset $\\mathcal { S } = \\{ ( x _ { i } , y _ { i } ) : x _ { i } \\in \\bar { \\mathcal { X } } , y _ { i } \\in \\mathcal { C } , i = 1 , . . , N \\}$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following the standard training scheme of backdoor attacks, the classifier is trained with the combination of the clean and poisoned subsets of $S$ . To create a poisoned sample, a clean training sample $( x , y )$ is transformed into a backdoor sample $( T ( x ) , \\eta ( y ) )$ , where $T$ is a backdoor injection function (also called the trigger function) and $\\eta$ is the target label function. When training $f$ with the clean and poison samples, we alter the behavior of $f$ so that: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b2c35522ecd149c5d1de18b3f731b65976fd366c18d5aafaf0b647895719537.jpg",
+        "text": "$$\nf ( x ) = y , \\quad f ( T ( x ) ) = \\eta ( y ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for any pair of clean data $x \\in \\mathcal { X }$ and its corresponding label $y \\in { \\mathcal { C } }$ . There are two commonly studied backdoor attack settings [18, 39, 51]: all-to-one and all-to-all. In the all-to-one attack, the label is changed to a constant target, i.e. $\\eta ( y ) = c$ ; while for the all-to-all attack, the true label is one-shifted, i.e. $\\bar { \\eta ( y ) } = ( y + 1 )$ mod $| { \\mathcal { C } } |$ . In the existing works, the trigger function $T$ is usually selected before training $f$ and fixed during the training process of $f$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 Learning to Backdoor ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the training dataset $s$ and a loss function $\\mathcal { L }$ , e.g., cross entropy loss, empirical risk minimization can be used to learn the parameters $\\theta$ , as follows: ",
+        "bbox": [
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+            299,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b39e2afb8b2b1064709d23b9ad0db65e06a68bf99e370c75be1a95edbe28436c.jpg",
+        "text": "$$\n\\theta ^ { * } = \\arg \\operatorname* { m i n } _ { \\theta } \\sum _ { i = 1 } ^ { N } \\mathcal { L } ( f _ { \\theta } ( x _ { i } ) , y _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The goal of this work is to learn a trigger function $T _ { \\xi } : \\mathcal { X } \\longrightarrow \\mathcal { X }$ and a classification model $f _ { \\theta }$ in such a way that the clean image $x$ and its corresponding backdoor image $T ( x )$ are visually consistent in the input space while the backdoor attack does not leave a detectable trace in the latent space of the poisoned classifier. When $f$ is a neural network, $\\phi ( x )$ can be the output of an intermediate, hidden layer of $f$ , which captures some high-level abstractions of the input. Note that we require the classifier to perform normally on the clean sample, $x$ , compared to the classifier’s vanilla version, but change its prediction on the poisoned image, $T ( x )$ , to the target class $\\eta ( y )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To generate a trigger and poison the image, we follow the prior work [13] and formulate the trigger function as a conditional noise generator $g$ , as follows: ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b6e8cea74a7a51a1d7aa3bf947c76f002e4cca8de2f326160caaf7620d09c59.jpg",
+        "text": "$$\nT _ { \\xi } ( x ) = x + g _ { \\xi } ( x ) , \\quad | | g _ { \\xi } ( x ) | | _ { \\infty } \\leq \\epsilon \\forall x\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The generator function $g _ { \\xi }$ takes an input $x$ and generates an artificially imperceptible noise on the same input space, which guarantees the stealthiness of the backdoor attack. We can design such generator function as an autoencoder or the more complex U-Net architecture [43]. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the above objectives and notations, similar to [13, 4], we can formulate the task into the following constrained optimization problem: ",
+        "bbox": [
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+            597,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be740e4a7f5610727ec4dbdf248fbe90803a7f323690f1bfdad390491e396985.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\displaystyle } & { { } \\displaystyle \\operatorname* { m i n } _ { \\theta } \\sum _ { i = 1 } ^ { N } \\alpha \\mathcal { L } \\big ( f _ { \\theta } ( x _ { i } ) , y _ { i } \\big ) + \\beta \\mathcal { L } \\big ( f _ { \\theta } ( T _ { \\xi ^ { * } ( \\theta ) } ( x _ { i } ) \\big ) , \\eta ( y _ { i } ) \\big ) } \\\\ { \\displaystyle s . t . \\quad } & { { } \\xi ^ { * } = \\arg \\operatorname* { m i n } _ { \\xi } \\sum _ { i = 1 } ^ { N } \\mathcal { L } \\big ( f _ { \\theta } ( T _ { \\xi } ( x _ { i } ) ) , \\eta ( y _ { i } ) \\big ) + \\mathcal { R } _ { \\phi } ( \\mathcal { F } _ { c } , \\mathcal { F } _ { b } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            632,
+            692,
+            720
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { R } _ { \\phi }$ is the regularization constraint of the clean and poisoned representations, denoted as $\\mathcal { F } _ { c } = \\{ \\phi ( x _ { i } ) : i = 1 , . . , N \\}$ and $\\mathcal { F } _ { b } \\{ \\phi ( T ( x _ { i } ) ) : i = 1 , . . , N \\}$ , respectively. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this problem, a learned classification model with a specific parameter configuration $\\theta$ is associated with an optimal yet stealthy backdoor trigger function, which is trained to poison the model. The classifier is trained to minimize a linear combination of clean and targeted backdoor objectives. The parameters $\\alpha$ and $\\beta$ control the mixing strengths of the clean and backdoor loss signals. The trigger function is trained to perturb an image within its $\\ell _ { \\infty }$ ball in the input space, so that the loss towards the attack target class is minimized while regularizing the latent representations of the backdoor images. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 Stealthy Latent Representation via Wasserstein Regularization ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In practical applications, latent-space defense methods investigate the abnormal trace of incoming data points with respect to the previous stream of data. These traces exist primarily because of the fact that the clean and backdoor latent representations are separated or distributed differently (e.g., the separated clusters of the clean and poison representations that can be seen in Figures 2 and 3). Thus, we aim to minimize such distributional difference through the regularization constraint $\\mathcal { R } _ { \\phi }$ . Since we cannot assume that the two latent distributions have common support or their density functions are known, commonly-used divergences, such as $f$ -divergences [40, 15] (which include KL and JSD), are difficult to minimize. Instead, we consider the Wasserstein-2 distance and formulate the regularization constraint as follows: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f786c048fad701e2b4e438757cfcd5a82f31ec2a37143bf6fefb91067ca47a31.jpg",
+        "text": "$$\n\\mathcal { R } \\phi ( \\mu , \\nu ) = \\left( \\operatorname* { i n f } _ { \\gamma \\in \\Pi ( \\mu , \\nu ) } \\int _ { ( x , z ) \\sim \\gamma } p ( x , z ) | | x - z | | _ { 2 } d x d z \\right) ^ { 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            195,
+            696,
+            241
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mu$ and $\\nu$ are marginal probability measures defined by empirical samples $\\mathcal { F } _ { c }$ and $\\mathcal { F } _ { b }$ of the latent representations of the clean and poisoned data, respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Estimating the Wasserstein distance also has some challenges. From the primal domain, computing the infimum in Equation (4) is particularly difficult since the data distributions are not fixed or known. On the other hand, employing the Kantorovich-Rubinstein duality requires a separate, parameterized Lipschitz function and a minimax solver, which increases the complexity of the proposed problem. Fortunately, for one-dimensional continuous measures, the Wasserstein distance has an elegant yet closed-form solution. Let $q _ { \\mu }$ and $q _ { \\nu }$ be the corresponding density functions of $\\mu$ and $\\nu$ , respectively. The Wasserstein-2 distance between one-dimensional measures $\\mu$ and $\\nu$ can be given by: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44bf3de363d541a7f634e2ed30c73dfc27f2da7fb08e2952224f813cdce56bee.jpg",
+        "text": "$$\n\\mathcal { W } ( \\mu , \\nu ) = \\left( \\int _ { 0 } ^ { 1 } | | ( F _ { \\mu } ^ { - 1 } ( z ) - F _ { \\nu } ^ { - 1 } ( z ) | | _ { 2 } d z \\right) ^ { 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            385,
+            661,
+            424
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { F _ { \\mu } ( z ) = \\int _ { \\infty } ^ { z } q _ { \\mu } ( \\rho ) d \\rho } \\end{array}$ and $\\begin{array} { r } { F _ { \\nu } ( z ) = \\int _ { \\infty } ^ { z } q _ { \\nu } ( \\rho ) d \\rho } \\end{array}$ are the cumulative distribution functions. 1 1Inspired by the efficiency of this solution and its successful applications in a variety of tasks [12, 24, 14], we propose to first find a family of one-dimensional representations, e.g., through the linear projections, and approximate the Wasserstein distance as a function of these one-dimensional marginals, as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6c049c21118a2a0e26bea05eb3ccfb292f4c76bd5d9cefee7bee75c208518e8.jpg",
+        "text": "$$\n\\mathcal { R } _ { \\phi } ( \\mathcal { F } _ { c } , \\mathcal { F } _ { b } ) \\approx \\left( \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } [ \\mathcal { W } ( \\mathcal { F } _ { c } ^ { \\theta _ { l } } , \\mathcal { F } _ { b } ^ { \\theta _ { l } } ) ] ^ { 2 } \\right) ^ { 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } _ { c } ^ { \\theta _ { l } } = \\{ \\theta _ { l } ^ { T } \\phi ( x _ { i } ) : i = 1 , . . , N \\}$ and $\\mathcal { F } _ { b } ^ { \\theta _ { l } } = \\{ \\theta _ { l } ^ { T } \\phi ( T ( x _ { i } ) ) : i = 1 , . . , N \\}$ contains the projections of the clean and poisoned datasets into a one-dimensional direction defined by $\\theta _ { l }$ (a slice). Typically, $\\theta _ { l }$ is drawn from a uniform distribution on the unit sphere. This formulation is also known as the sliced-Wasserstein distance (SWD) [12, 24]. One particular problem with this approach is that the random nature of the slices could lead to several non-informative directions; i.e., the sliced distances are close to 0 in directions that do not lie on the manifolds of the data. Consequently, a large number $L$ of random directions are needed to approximate the sliced-Wasserstein distance, which increases the computational complexity of the estimation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To remedy this issue, we avoid the uniform sampling of the unit sphere and select directions that contain discriminant information of the two data sources, by exploiting the following fact in the classification task. For backdoor samples of a target class $c _ { 1 } \\in { \\mathcal { C } }$ , created from clean samples of some other class $c _ { 2 } \\in { \\mathcal { C } }$ , the projections into an output dimension represent meaningful discriminant information that distinguishes the backdoor samples (from class $c _ { 2 }$ ) and the clean samples (from class $c _ { 1 }$ ). Thus, we propose to replace the uniform linear projections of SWD with the projections into the output layer. When the latent space is the penultimate layer of the classifier, such projections are equivalent to the following approximation: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/181d093b6bca4cfc07552c2697be6486e8a191895ffaf624f89d8825e7fa530b.jpg",
+        "text": "$$\n\\mathcal { R } _ { \\phi } ( \\mathcal { F } _ { c } , \\mathcal { F } _ { b } ) \\approx \\left( \\frac { 1 } { | \\mathcal { C } | } \\sum _ { c = 1 } ^ { | \\mathcal { C } | } \\left[ \\mathcal { W } ( \\mathcal { F } _ { c } ^ { W _ { c , : } } , \\mathcal { F } _ { c } ^ { W _ { c , : } } ) \\right] ^ { 2 } \\right) ^ { 1 / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            799,
+            673,
+            853
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { c , : }$ is a row of the matrix $W \\in \\mathbb { R } ^ { | \\mathcal { C } | \\times d }$ ( $d$ is the dimension of the latent space), which is the normalized parameter matrix between the penultimate and the output layers. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7af9caf5c809e25a62d365bbd85a4c2242c21166b3b94b61d1406b07a5c6433.jpg",
+        "image_caption": [
+            "Figure 1: Distance estimates (normalized) in the latent space for SWD with different number of sampled directions (between 10 to 10,000) and DSWD. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            89,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Empirically, Figure 1 shows the estimated SWD with different numbers of random directions and the proposed calculation, so called DSWD, when the latent space is defined at the penultimate layer of the classifier. The dimension of the latent space is 512 for both MNIST and CIFAR10 datasets. Each distance is computed on a random sample of 1000 clean and 1000 backdoor images, and each calculation is repeated 100 times. It can be seen that with only a fraction of slices, DSWD achieves a significantly smaller variance than that of the SWD estimates. Furthermore, in MNIST, the selected directions of DSWD leads to higher distance estimates than SWD, which means that DSWD selects more discriminant directions than SWD while SWD underestimates the distance between the two empirical samples. In addition, we show that DSWD is a valid distance metric of the latent distributions. The detailed proof is presented in the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. When the latent space is the penultimate layer of a neural network, the proposed DSWD distance is a valid distance function of probability measures in this space. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. Since existing defense methods choose the penultimate layer of a neural network. as the space to perform the defense analysis, in most cases, we can employ the proposed DSWD calculation. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 2. To preserve the clean classification performance, the classifier seeks optimal parameters that lead to similar predictions of clean samples from the same class. The goal of the trigger function is to make the poisoned samples classified toward a different class. This leads to an adversarial game between the classifier and the trigger functions. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "DSWD also has a significantly better computational efficiency than SWD. In most problems, SWD requires a large number of random directions, typically between 1000 to 10,000, in order to provide a reliable estimate of the distance [38, 14]. In DSWD, the number of random directions is fixed to the number of possible output labels, which is typically small for many classification problems. ",
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+    {
+        "type": "text",
+        "text": "4.4 Optimization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The non-convex, constrained optimization in Equation (3) is challenging because of its non-linear constraint. In general, we can alternately update one of $f$ and $T$ while keeping the other fixed, similar to training GANs. However, it is difficult and slow for the classifier to reach an acceptable performance on the clean data, i.e., similar accuracy to that of the vanilla classifier. ",
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+        "type": "text",
+        "text": "Under the alternating update scheme, we observe that on MNIST, the poisoned classifier can reach the acceptable clean-data performance after several epochs; while on other more complex datasets (i.e. CIFAR10, GTSRB, and TinyImagenet), this procedure results in sub-optimal clean-data performance. One possible explanation is that training the vanilla classifier with complex architecture and dataset to reach a decent accuracy is already a difficult and time-consuming task (e.g., 2 to 3 epochs to reach the optimal performance on MNIST but several hundreds of epochs on the other datasets). ",
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+    {
+        "type": "text",
+        "text": "Fortunately, we observe that after training the classifier and the trigger functions in an alternating update scheme for a certain number of epochs (denoted as Stage I), we can fix the trigger function and only train the classifier for the remaining epochs (denoted as Stage II). This two-stage training scheme is adopted in our experiments. ",
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+        "type": "text",
+        "text": "5 Experimental Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "5.1 Experimental Setup ",
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+        "type": "text",
+        "text": "We demonstrate the effectiveness of the proposed method through a range of experiments on four widely-used datasets for backdoor attack study: MNIST, CIFAR10, GTSRB and TinyImagenet. For these experiments, we follow the previous works [51, 54, 6, 39] and select the penultimate layers of the classifiers as the latent space for the defense experiments. The implementation of WB was based on the PaddlePaddle deep learning platform. ",
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+        "type": "text",
+        "text": "Architectures: For the classifier $f$ , we consider several popular models: Pre-activation Resnet18 [20], VGG [46], DenseNet [21] for CIFAR10 and GTSRB datasets, and Resnet-18 for TinyImagenet. For the MNIST dataset, we employ a CNN model. ",
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+        "type": "text",
+        "text": "Hyperparameters: For the baselines, we train the classifiers using the SGD optimizer with an initial learning rate of 0.01 and a learning rate decay of 0.1 after every 100 epochs. For other hyperparameters, we follow the proposed setup in [39] for all datasets. We use the same configurations for WB. We train the classifier and trigger functions alternately (Stage I) for 10 and 50 epochs for MNIST and the other datasets, respectively, and fine-tune the classifier (Stage II) for another 40 epochs and 450 epochs for MNIST and the other datasets, respectively. To achieve a high-degree stealthiness of WB, we pick $\\epsilon$ as small as 0.01 for all datasets. In general, the larger the value of $\\epsilon$ the easier the trigger functions can be learned and the more successful the attacks are. ",
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+    {
+        "type": "text",
+        "text": "5.2 Attack Performance ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present the attack success rates of the proposed WB method, along with a comparison to two state-of-the-art methods, i.e., WaNet [39] and LIRA [13]. Both LIRA and Wanet’s attack performances are significantly better than other approaches, including BadNets [18], and are two of the strongest existing methods that generate very stealthy triggers on the images. We first poison the classifier using the backdoor attack methods in both all-to-one and all-to-all settings and record the performance of the classifier on both clean and backdoor test samples. For all-to-one, we randomly pick the target label $\\hat { c }$ (i.e., $\\eta ( y ) = \\hat { c } \\forall y )$ , while for all-to-all, the target label function is defined as $\\eta ( y ) = { \\bar { ( y + 1 ) } }$ mod $| { \\mathcal { C } } | \\ \\forall y$ , which is widely used to evaluate the backdoor-related works [39, 18, 6, 13]. Note that this all-to-all attack setting is more challenging than the all-to-one setting, especially on datasets with a large number of classes such as TinyImagenet. ",
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+    {
+        "type": "text",
+        "text": "The classification accuracy on the clean test samples and the attack success rate for each method is represented in Table 1 and Table 2 for the all-to-one and all-to-all settings, respectively. As we can observe from these tables, all the methods can achieve high clean-data accuracies and attack success rates. While WB’s attack performance slightly drops compared to LIRA’s performance, WB is significantly more stealthy in the latent space, as being discussed next. ",
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+    {
+        "type": "table",
+        "img_path": "images/4ba0b994c5f6d95f14d8c79dc1be6d42ee8f3da7a066f57e12a386f7b172a325.jpg",
+        "table_caption": [
+            "Table 1: Attack Performance: All-to-one Attack "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"2\">WaNet</td><td colspan=\"2\">LIRA</td><td colspan=\"2\">WB</td></tr><tr><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td></tr><tr><td>MNIST</td><td>0.99</td><td>0.99</td><td>0.99</td><td>1.00</td><td>0.99</td><td>0.99</td></tr><tr><td>CIFAR10</td><td>0.94</td><td>0.99</td><td>0.94</td><td>1.00</td><td>0.94</td><td>0.99</td></tr><tr><td>GTSRB</td><td>0.99</td><td>0.98</td><td>0.99</td><td>1.00</td><td>0.99</td><td>0.99</td></tr><tr><td>TinyImagenet</td><td>0.57</td><td>0.99</td><td>0.58</td><td>1.00</td><td>0.57</td><td>0.99</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d071ecb977eb62c2db625f066ac4dbb5a7844840fe39ef1ebdfa64859d4e1b12.jpg",
+        "table_caption": [
+            "Table 2: Attack Performance: All-to-all Attack "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"2\">WaNet</td><td colspan=\"2\">LIRA</td><td colspan=\"2\">WB</td></tr><tr><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td><td>Clean</td><td>Attack</td></tr><tr><td>MNIST</td><td>0.99</td><td>0.95</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.96</td></tr><tr><td>CIFAR10</td><td>0.94</td><td>0.93</td><td>0.94</td><td>0.94</td><td>0.94</td><td>0.94</td></tr><tr><td>GTSRB</td><td>0.99</td><td>0.98</td><td>0.99</td><td>1.00</td><td>0.99</td><td>0.98</td></tr><tr><td>TinyImagenet</td><td>0.58</td><td>0.58</td><td>0.58</td><td>0.59</td><td>0.58</td><td>0.58</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.3 Latent-Space Defense ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Recent works on backdoor defense have found that backdoor attacks tend to leave a tangible trace in the latent space of the poisoned classifier. Activation Clustering [6] and Spectral Signature [54] are two representative defenses used for analyzing the latent space in prior work [51]. In this section, we also examine the latent space of the backdoor-injected classifiers through the lens of these defense methods. ",
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+    {
+        "type": "text",
+        "text": "5.3.1 Learned Latent Representation and Activation Clustering ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "It has been shown in [6] that in a poisoned classifier, the latent representations of the clean and backdoor samples form separate clusters, which can be easily detected using clustering methods such as K-means. The authors also recommend a process called exclusionary reclassification to determine which cluster is poisoned and re-train the poisoned classifier. ",
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+    {
+        "type": "text",
+        "text": "In Figure 2 and Figure 3, we can observe highly separated clusters (for samples with the sample predictions of $y = 0$ ) in the latent space when we omit the latent regularization term $\\mathcal { R } _ { \\phi }$ in WB (Baseline), which is similar to LIRA [13]. However, when $\\mathcal { R } _ { \\phi }$ is included, the latent representations of the clean and backdoor samples are distributed similarly. Without well-separated clusters of the clean and poisoned samples, the exclusionary reclassification process in the activation clustering is not effective against the attacks. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/06e6d3444d42a0f1ae4a52a74f76903477b0bdaf9bc0b3cb802283e00926243d.jpg",
+        "image_caption": [
+            "Figure 2: MNIST: t-SNE embedding in the latent space. Baseline is WB without $\\mathcal { R } _ { \\phi }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b60bd196e6d6df38879720961f333230fa538a26c3db4859eca00fcaeef41932.jpg",
+        "image_caption": [
+            "Figure 3: CIFAR10: t-SNE embedding in the latent space. Baseline is WB without $\\mathcal { R } _ { \\phi }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Quantitatively, we present the quality scores (i.e., the adjusted Rand Index) of the clustering step in Table 3. The adjusted Rand Index is 1 when the samples form two distinct clusters and is close to 0 for a random separation. We compare WB with BadNets [18] and Adversarial Embedding [51], which is the state-of-the-art backdoor attack method with stealthy latent space. As we can observe in this table, the defense is most successful on BadNets since there exists a perfect clustering of the clean and poisoned samples (Rand Index $\\geq 0 . 9 5$ ). While Adversarial Embedding is more resistant against the defense, WB is significantly more stealthy against the defense since the values of Rand Index are all very close to 0. Note that, similar to BadNets, WaNet also does not pass this defense (please see the supplementary material). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/78450786e5a83397722dd5e283d8e50b8bba4ba34a91265598eda77c8aa5b2f9.jpg",
+        "table_caption": [
+            "Table 3: Adjusted Rand Index in All-to-one Attack "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Rand Index (BadNets)</td><td colspan=\"2\">Adversarial Embedding</td><td colspan=\"2\">WB</td></tr><tr><td>Rand Index</td><td>Attack</td><td>Rand Index</td><td>Attack</td></tr><tr><td>DenseNet</td><td>CIFAR10</td><td>0.979</td><td>0.1820</td><td>0.764</td><td>0.0382</td><td>0.998</td></tr><tr><td>DenseNet</td><td>GTSRB</td><td>0.997</td><td>0.2710</td><td>0.914</td><td>0.0135</td><td>0.997</td></tr><tr><td>VGG</td><td>CIFAR10</td><td>0.998</td><td>0.0006</td><td>0.962</td><td>0.0002</td><td>0.999</td></tr><tr><td>VGG</td><td>GTSRB</td><td>0.997</td><td>0.6420</td><td>0.743</td><td>0.1010</td><td>0.999</td></tr></table>",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.3.2 Spectral Signature Defense ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The work in [54] proposes a defense method that identifies and removes backdoor samples using the Spectral Signature. For data from each predicted class, Spectral Signature first finds the top singular value of the covariance matrix of the latent vectors of the data. Then it computes the correlation score to this singular value for each sample and those samples with the outlier scores are flagged as backdoor samples. While Spectral Signature is a sample filtering-based defense method, the inspection of the correlation scores can also be useful to verify whether there is a tangible trace in the latent space of the classifier. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Following the same experiments in [54], we first pick 5,000 clean samples and 500 backdoor samples for each dataset. Then, we plot the histograms of the correlation scores for both sets of samples. As we can observe in Figure 4, there is no clear separation between the scores of the backdoor samples and those of the clean samples. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/92dceef2b089708025cd466c7e81445da9fe91338f47a7c368502aa6b41ae451.jpg",
+        "image_caption": [
+            "Figure 4: Defense experiments of the all-to-one attack against Spectral Signature. The correlations of the clean and backdoor samples with the top singular vector of the covariance matrix in the latent space are not separable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4 Model Mitigation Defense ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate the robustness of WB against another popular defense, Neural Cleanse [56], which is model mitigation defense based on a pattern optimization approach. Specifically, Neural Cleanse searches for the optimal patch pattern for each possible target label that induces a misclassification to that label. It then quantifies whether any of the optimal backdoor trigger pattern is an outlier via a metric called Anomaly Index. The model has a backdoor if the Anomaly Index is greater than 2 for any class. The anomaly indices are presented in Figure 5. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It can be seen that both WaNet and WB can pass the detection of Neural Cleanse, similar to that of the vanilla classifier (Clean). In MNIST and CIFAR10, WB even achieves smaller Anomaly Indices than those of the vanilla models. Note that popular backdoor attacks, such as BadNets, can be defended by Neural Cleanse in most of these datasets [56]. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/8cf52aada9604635bbac1b3e8db94e8deb55a98abd1bd801f6b427337a81fcbd.jpg",
+        "image_caption": [
+            "Figure 5: Backdoor attacks against Neural Cleanse defense. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5.5 Input Perturbation Defense ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we study the stealthiness of WB against STRIP [16], a representative detection based backdoor defense mechanism. Given the classifier and an input image, STRIP first perturbs the image and determines the presence of a backdoor in the model according to the entropy of the predictions of these perturbed images (i.e., if the predictions are consistent or not). ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In Figure 6, we plot the entropy of clean and backdoor images, which are computed by STRIP. We can observe that the distribution of entropy of the backdoor samples is similar to that of the clean samples. In other words, STRIP fails to detect backdoor samples generated by WB, which further validates the advantage of the proposed method. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e55b555ba734983cb5aead631bac73c591a889d0922fbd0ca99fa3092065999.jpg",
+        "image_caption": [
+            "Figure 6: Performance against STRIP defense. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Additional experiments for demonstrating the stealthiness of WB against several other defense approaches can be found in the supplementary material. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper presented a novel methodology for a backdoor attack that is imperceptible from both the input and latent spaces, i.e., Wasserstein Backdoor (WB). WB learns a trigger function that adds visually imperceptible noise to an input image and minimizes the distributional difference via a novel sliced Wasserstein distance formulation between representations of the clean and backdoor images in the latent space of the trained classifier. We comprehensively evaluated the performance of the proposed method on various image classification benchmark models over a wide range of datasets. Our experimental results demonstrated that the proposed method could significantly improve the effectiveness against the existing defense mechanisms, especially those relying on the distinguishability in latent space. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Tadas Baltrusaitis, Amir Zadeh, Yao Chong Lim, and Louis-Philippe Morency. Openface 2.0: Facial behavior analysis toolkit. In Proceedings of the 13th IEEE International Conference on Automatic Face & Gesture Recognition (FG), pages 59–66, Xi’an, China, 2018.   \n[2] Mauro Barni, Kassem Kallas, and Benedetta Tondi. A new backdoor attack in CNNS by training set corruption without label poisoning. In Proceedings of the 2019 IEEE International Conference on Image Processing (ICIP), pages 101–105, Taipei, 2019.   \n[3] Daniel S Berman, Anna L Buczak, Jeffrey S Chavis, and Cherita L Corbett. A survey of deep learning methods for cyber security. Information, 10(4):122, 2019.   \n[4] Avishek Joey Bose, Gauthier Gidel, Hugo Berard, Andre Cianflone, Pascal Vincent, Simon Lacoste-Julien, and William L. Hamilton. Adversarial example games. In Advances in Neural Information Processing Systems (NeurIPS), virtual, 2020.   \n[5] Nicholas Carlini and David A. 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parse/train/H113pWZRb/H113pWZRb.md

@@ -0,0 +1,298 @@
+# TOPOLOGY ADAPTIVE GRAPH CONVOLUTIONAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Convolution acts as a local feature extractor in convolutional neural networks (CNNs). However, the convolution operation is not applicable when the input data is supported on an irregular graph such as with social networks, citation networks, or knowledge graphs. This paper proposes the topology adaptive graph convolutional network (TAGCN), a novel graph convolutional network that generalizes CNN architectures to graph-structured data and provides a systematic way to design a set of fixed-size learnable filters to perform convolutions on graphs. The topologies of these filters are adaptive to the topology of the graph when they scan the graph to perform convolution, replacing the square filter for the grid-structured data in traditional CNNs. The outputs are the weighted sum of these filters’ outputs, extraction of both vertex features and strength of correlation between vertices. It can be used with both directed and undirected graphs. The proposed TAGCN not only inherits the properties of convolutions in CNN for grid-structured data, but it is also consistent with convolution as defined in graph signal processing. Further, as no approximation to the convolution is needed, TAGCN exhibits better performance than existing graph-convolution-approximation methods on a number of data sets. As only the polynomials of degree two of the adjacency matrix are used, TAGCN is also computationally simpler than other recent methods.
+
+# 1 INTRODUCTION
+
+Convolutional neural network (CNN) architectures exhibit state-of-the-art performance on a variety of learning tasks dealing with 1D, 2D, and 3D grid-structured data such as acoustic signals, images, and videos, in which convolution serves as a feature extractor (LeCun et al., 2015). However, the (usual) convolution operation is not applicable when applying CNN to data that is supported on an arbitrary graph rather than on a regular grid structure, since the number of neighbors of each vertex on the graph varies, and it is difficult to design a fixed-size filter scanning over the graph-structured data for feature extraction.
+
+Recently, there has been an increasing interest in graph CNNs (Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2017; Monti et al., 2017; Levie et al., 2017), attempting to generalize deep learning methods to graph-structured data, specifically focusing on the design of graph CNN . In this paper, we propose the topology adaptive graph convolutional network (TAGCN), a unified convolutional neural network to learn nonlinear representations for the graph-structured data. It slides a set of fixed-size learnable filters on the graph simultaneously, and the output is the weighted sum of these filters’ outputs, which extract both vertex features and strength of correlation between vertices. Each filter is adaptive to the topology of the local region on the graph where it is applied. TAGCN unifies filtering in both the spectrum and vertex domains; and applies to both directed and undirected graphs.
+
+In general, the existing graph CNNs can be grouped into two types: spectral domain techniques and vertex domain techniques. In Bruna et al. (2014), CNNs have been generalized to graph-structured data, where convolution is achieved by a pointwise product in the spectrum domain according to the convolution theorem. Later, Defferrard et al. (2016) and Levie et al. (2017) proposed spectrum filtering based methods that utilize Chebyshev polynomials and Cayley polynomials, respectively. The assumption of symmetric adjacency matrix in these spectrum based methods restrict the application to undirected graphs. Kipf & Welling (2017) simplified this spectrum method and obtained a filter in the vertex domain, which achieves state-of-the-art performance. Other researchers (Atwood & Towsley, 2016; Monti et al., 2017) worked on designing feature propagation models in the vertex domain for graph CNNs. Yang et al. (2016); Dai et al. (2016); Grover & Leskovec (2016); Du et al. (2016) study transforming graph-structured data to embedding vectors for learning problems. Nevertheless, it still remains open how to extend CNNs from grid-structured data to arbitrary graph-structured data with local feature extraction capability.
+
+This paper proposes a modification to the graph convolution step in CNNs that is particularly relevant for graph structured data. Our proposed TAGCN is graph-based convolution and draws on techniques from graph signal processing. We define rigorously the graph convolution operation on the vertex domain as multiplication by polynomials of the graph adjacency matrix, which is consistent with the notion of convolution in graph signal processing. In graph signal processing, polynomials of the adjacency matrix are graph filters, extending to graph based data from the usual concept of filters in traditional time or image based signal processing. Thus, comparing ours with existing work on graph CNNs, our paper provides a solid theoretical foundation for our proposed convolution step instead of an ad-hoc approach to convolution in CNNs for graph structured data.
+
+Further, our method avoids computing the spectrum of the graph Laplacian as in Bruna et al. (2014), or approximating the spectrum using high degree Chebyshev polynomials of the graph Laplacian matrix (in Defferrard et al. (2016), it is suggested that one needs a $2 5 ^ { \mathrm { t h } }$ degree Chebyshev polynomial to provide a good approximation to the graph Laplacian spectrum) or using high degree Cayley polynomials of the graph Laplacian matrix (in Levie et al. (2017), $1 2 ^ { \mathrm { t h } }$ degree Cayley polynomials are needed). We also clarify that the GCN method in Kipf & Welling (2017) is a first order approximation of the Chebyshev polynomials approximation in Defferrard et al. (2016), which is very different from our method. Our method has a much lower computational complexity than the complexity of the methods proposed in Bruna et al. (2014); Defferrard et al. (2016); Levie et al. (2017), since our method only uses polynomials of the adjacency matrix with maximum degree 2 as shown in our experiments. Finally, the method that we propose exhibits better performance than existing methods. Our contributions are summarized follows:
+
+• We propose a general $K$ -localized filter for graph convolution in the vertex domain to extract local features on a set of size-1 up to size- $K$ receptive fields. The topologies of these filters are adaptive to the topology of the graph as they scan the graph to perform convolution. It replaces the fixed square filters in traditional CNNs for the input gridstructured data volumes in traditional CNNs. Thus, our convolution definition that we use in the convolution step for the vertex domain is consistent with convolution in traditional CNNs. TAGCN is based on the graph signal processing and it is consistent with the convolution in graph signal processing. It applies to both directed and undirected graphs. Moreover, it has a much lower computational complexity compared with recent methods since it only needs polynomials of the adjacency matrix with maximum degree 2 compared with the $2 5 ^ { \mathrm { { i } \tilde { h } } }$ and $1 \bar { 2 } ^ { \mathrm { t h } }$ degree Laplacian matrix polynomials in Defferrard et al. (2016) and Levie et al. (2017). As no approximation to the convolution is needed in TAGCN, it achieves better performance compared with existing methods. We contrast TAGCN with recently proposed graph CNN including both spectrum filtering methods (Bruna et al., 2014; Defferrard et al., 2016) and vertex domain propagation methods (Kipf & Welling, 2017; Monti et al., 2017; Atwood & Towsley, 2016), evaluating their performances on three commonly used data sets for graph vertices classification. Our experimental tests show that TAGCN outperforms consistently all other approaches for each of these data sets.
+
+# 2 GRAPH POLYNOMIAL BASED CONVOLUTION ON GRAPHS
+
+We use boldface uppercase and lowercase letters to represent matrices and vectors, respectively. The information and their relationship on a graph $\mathcal { G }$ can be represented by $\mathcal { G } = ( \nu , \mathcal { E } , \vec { \mathbf { A } } )$ , where $\nu$ is the set of vertices, $\mathcal { E }$ is the set of edges, and $\bar { \bf A }$ is the weighted adjacency matrix of the graph; the graph can be weighted or unweighted, directed or undirected. We assume there is no isolated vertex in $\mathcal { G }$ . If $\mathcal { G }$ is a directed weighted graph, the weight $\bar { \mathbf { A } } _ { n , m }$ is on the directed edge from vertex $m$ to $n$ . The entry $\bar { \mathbf { A } } _ { n , m }$ reveals the dependency between node $n$ and $m$ and can take arbitrary real or complex values. The graph convolution is general and can be adapted to graph CNNs for particular tasks. In this paper, we focus on the vertex semisupervised learning problem, where we have access to limited labeled vertices, and the task is to classify the remaining unlabeled vertices.
+
+# 2.1 GRAPH CONVOLUTION
+
+Without loss of generality, we demonstrate graph convolution on the $\ell \cdot$ -th hidden layer. The results apply to any other hidden layers. Suppose on the $\ell \cdot$ -th hidden layer, the input feature map for each vertex of the graph has $C _ { \ell }$ features. We collect the $\ell$ -th hidden layer input data on all vertices for the $c$ -th feature by the vector $\mathbf { x } _ { c } ^ { ( \ell ) } \in \mathbb { R } ^ { N _ { \ell } }$ , where $c = 1 , 2 , . . . C _ { \ell }$ and $N _ { \ell }$ is the number of vertices1. The components of $\mathbf { x } _ { c } ^ { ( \ell ) }$ are indexed by vertices of the data graph representation $\mathcal { G } = ( \gamma , \mathcal { E } , \bar { \mathbf { A } } ) ^ { 2 }$ . Let $\mathbf { G } _ { c , f } ^ { ( \ell ) } \in \mathbb { R } ^ { N _ { \ell } \times N _ { \ell } }$ denote the $f$ -th graph filter. The graph convolution is the matrix-vector product, i.e., $\mathbf { G } _ { c , f } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) }$ . Then the $f$ -th output feature map is
+
+$$
+\mathbf { y } _ { f } ^ { ( \ell ) } = \sum _ { c = 1 } ^ { C _ { \ell } } \mathbf { G } _ { c , f } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) } + b _ { f } \mathbf { 1 } _ { N _ { \ell } } ,
+$$
+
+where $b _ { f } ^ { ( \ell ) }$ is a learnable bias, and ${ \bf 1 } _ { N _ { \ell } }$ is the $N _ { \ell }$ dimension vector of all ones. We design $\mathbf { G } _ { c , f } ^ { ( \ell ) }$ such that $\mathbf { G } _ { c , f } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) }$ is a meaningful convolution on a graph with arbitrary topology.
+
+In the recent theory on graph signal processing (Sandryhaila & Moura, 2013), the graph shift is defined as a local operation that replaces a graph signal at a graph vertex by a linear weighted combination of the values of the graph signal at the neighboring vertices:
+
+$$
+\tilde { \mathbf { x } } _ { c } ^ { ( \ell ) } = \bar { \mathbf { A } } \mathbf { x } _ { c } ^ { ( \ell ) } .
+$$
+
+The graph shift $\bar { \bf A }$ extends the time shift in traditional signal processing to graph-structured data. Following Sandryhaila & Moura (2013), a graph filter $\mathbf { G } _ { c , f } ^ { \overline { { ( \ell ) } } }$ is shift-invariant, i.e., the shift $\bar { \bf A }$ and the filter $\mathbf { G } _ { c , f } ^ { ( \ell ) }$ commute, $\bar { \mathbf { A } } ( \mathbf { G } _ { c , f } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) } ) = \mathbf { G } _ { c , f } ^ { ( \ell ) } ( \bar { \mathbf { A } } \mathbf { x } _ { c } ^ { ( \ell ) } )$ , if under appropriate assumption $\mathbf { G } _ { c , f } ^ { ( \ell ) }$ is a polynomial in A,
+
+$$
+\mathbf { G } _ { c , f } ^ { ( \ell ) } = \sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \ell ) } \mathbf { A } ^ { k } .
+$$
+
+In (2), the g(\`)c,f,k are the graph filter polynomial coefficients; the quantity $\mathbf { A } = \mathbf { D } ^ { - \frac { 1 } { 2 } } \bar { \mathbf { A } } \mathbf { D } ^ { - \frac { 1 } { 2 } }$ is the normalized adjacency matrix of the graph, and $\mathbf { D } = \mathrm { d i a g } [ \mathbf { d } ]$ with the $i$ th component being ${ \bf d } ( i ) =$ $\textstyle \sum _ { j } \mathbf { A } _ { i , j }$ .3 We adopt the normalized adjacency matrix to guarantee that all the eigenvalues of $\mathbf { A }$ are inside the unit circle, and therefore $\mathbf { G } _ { c , f } ^ { ( \ell ) }$ is computationally stable. The next subsection shows we , and $K \times C _ { \ell }$   
+coincides with GoogLeNet (Szegedy et al., 2015), in which a set of filters with different sizes are used in each convolutional layer.
+
+Following the CNN architecture, an additional nonlinear operation, e.g, rectified linear unit (ReLU) is used after every graph convolution operation:
+
+$$
+\begin{array} { r } { \mathbf { x } _ { f } ^ { ( \ell + 1 ) } = \sigma \left( \mathbf { y } _ { f } ^ { ( \ell ) } \right) , } \end{array}
+$$
+
+where $\sigma ( \cdot )$ denotes the ReLU activation function applied to the vertex values.
+
+In the following subsection, we demonstrate that convolution operator shown in (2) exhibits the nice properties of a square filter in traditional CNNs and generalize the convolution theorem from classical signal processing to signal processing on graphs (Sandryhaila & Moura, 2013).
+
+# 2.2 FILTER DESIGN FOR TAGCN CONVOLUTIONAL LAYER
+
+In this section, we would like to understand the proposed convolution as a feature extraction operator in traditional CNN rather than as propagating labeled data on the graph. Taking this point of view helps us to profit from the design knowledge/experience from traditional CNN and apply it to grid structured data. Our definition of weight of a path and the following filter size for graph convolution in this section make it possible to design a graph CNN architecture similar to GoogLeNet (Szegedy et al., 2015), in which a set of filters with different sizes are used in each convolutional layer. In fact, we found that a combination of size 1 and size 2 filters gives the best performance in all three data sets studied, which is a polynomial with maximum order 2.
+
+In traditional CNN, a $K \times K \times C _ { \ell }$ filter scans over the input grid-structured data for feature extraction. For image classification problems, the value $K$ varies for different CNN architectures and tasks to achieve better performance. For example, in VGG-Verydeep-16 CNN model (Simonyan & Zisserman, 2015), only $3 \times 3 \times C _ { \ell }$ filters are used; in ImageNet CNN model (Krizhevsky et al., 2012), $1 1 \times 1 1 \times C _ { \ell }$ filters are adopted; and in GoogLeNet (Szegedy et al., 2015), rather than using the same size filter in each convolutional layer, different size filters, for example, $1 \times 1 \times C _ { \ell }$ , $3 \times 3 \times C _ { \ell }$ and $5 \times 5 \times C _ { \ell }$ filters, are concatenated in each convolution layer. Similarly, we propose a general $K$ -localized filter for graph CNN.
+
+For a graph-structured data, we cannot use a square filter window since the graph topology is no longer a grid. In the following, we demonstrate that the convolution operation $\mathbf { G } _ { c , f } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) }$ with $\mathbf { G } _ { c , f } ^ { ( \ell ) }$ a polyn ial filt $\begin{array} { r } { \mathbf { G } _ { c , f } ^ { ( \ell ) } = \sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \ell ) } \mathbf { A } ^ { k } } \end{array}$ is equivalent to using a set of filters with filter size from 1 $K$ $k$ -size filter, which is used for local feature extraction on the graph, is $k$ -localized in the vertex domain.
+
+Define a path of length $m$ on a graph $\mathcal { G }$ as a sequence $v = ( v _ { 0 } , v _ { 1 } , . . . , v _ { m } )$ of vertices $v _ { k } \in \mathcal V$ such that each step of the path $( v _ { k } , v _ { k + 1 } )$ corresponds to an (directed) edge of the graph, i.e., $( v _ { k } , v _ { k + 1 } ) \in$ $\mathcal { E }$ . Here one path may visit the same vertex or cross the same edge multiple times. The following adjacency matrix $\mathbf { A }$ is one such example:
+
+$$
+\mathbf { A } = { \left[ \begin{array} { l l l l l l l l l } { 0 } & { 1 } & { 0 } & { 2 } & { 3 } & { 0 } & { 0 } & { \cdots } \\ { 1 } & { 0 } & { 4 } & { 5 } & { 0 } & { 0 } & { 0 } & { \cdots } \\ { 0 } & { 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { \cdots } \\ { 1 } & { 1 } & { 0 } & { 0 } & { 6 } & { 0 } & { 0 } & { \cdots } \\ { 1 } & { 0 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { \cdots } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { 0 } & { 0 } & { \cdots } \\ { 0 } & { 0 } & { 0 } & { 1 } & { 0 } & { 0 } & { 0 } & { \cdots } \\ { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \cdots } \end{array} \right] } .
+$$
+
+Since A is asymmetric, it represents a directed graph, given in Fig. 2. In this example, there are 6 different length 3-paths on the graph from vertex 2 to vertex 1, namely, $( 2 , 1 , 4 , 1 )$ , $( 2 , 1 , 2 , 1 )$ , $( 2 , 1 , 5 , 1 ) , ( 2 , \bar { 3 } , 2 , 1 \bar { ) } , ( 2 , 4 , 2 , 1 )$ , and $( 2 , 4 , 5 , 1 )$ .
+
+We further define the weight of a path to be the product of the edge weights along the path, i.e., $\begin{array} { r } { \phi ( p _ { 0 , m } ) = \prod _ { k = 1 } ^ { m } \mathbf { A } _ { v _ { k - 1 } , v _ { k } } } \end{array}$ , where $p _ { 0 , m } = ( v _ { 0 } , v _ { 1 } , \dots v _ { m } )$ . For example, the weight of the path $( 2 , 1 , 4 , 1 )$ is $1 \times 1 \times 2 = 2$ . Then, the $( i , j )$ th entry of $\mathbf { A } ^ { k }$ in (2), denoted by $\omega ( p _ { j , i } ^ { k } )$ , can be interpreted as the sum of the weights of all the length- $k$ paths from $j$ to $i$ , which is
+
+$$
+\omega ( p _ { j , i } ^ { k } ) = \sum _ { j \in \{ \tilde { j } | \tilde { j } \mathrm { i s } k \mathrm { p a t h s t o } i \} } \phi ( p _ { j , i } ) .
+$$
+
+In the above example, it can be easily verified that $\mathbf { A } _ { 1 , 2 } ^ { 3 } = 1 8$ by summing up the weights of all the above six paths from vertex 2 to vertex 1 with length 3. Then, the $i$ th component of $\mathbf { A } ^ { k } \mathbf { x } _ { c } ^ { ( \ell ) }$ is the weighted sum of the input features of each vertex $\mathbf { x } _ { c } ^ { ( \ell ) }$ that are length- $k$ paths away to vertex $i$ . Here, $k$ is defined as the filter size. The output feature map is a vector with each component given by the size- $k$ filter sliding on the graph following a fixed order of the vertex indices.
+
+The output at the $i$ -th component can be written explicitly as $\begin{array} { r l } { \sum _ { c = 1 } ^ { C _ { \ell } } \sum _ { j } g _ { c , f , k } ^ { ( \ell ) } \omega ( p _ { j , i } ^ { k } ) \mathbf { x } _ { c } ^ { ( \ell ) } ( j ) } & { { } } \end{array}$ . This weighted sum is similar to the dot product for convolution for a grid-structured data in traditional
+
+![](images/eb78b28841ad6e4469230c3a3d293d4e5717dd4c69b348648686bd534fceac2f.jpg)  
+Figure 1: Graph convolution operation at each neuron in the graph convolution layer.
+
+CNNs. Finnaly, the output feature map is a weighted sum of convolution results from filters with different sizes, which is
+
+$$
+\mathbf { y } _ { f } ^ { ( \ell ) } ( i ) = \sum _ { k = 1 } ^ { K _ { \ell } } \sum _ { c = 1 } ^ { C _ { \ell } } \sum _ { \substack { j \in \{ \tilde { j } | \tilde { j } \mathrm { ~ i s ~ } k \mathrm { ~ p a t h s ~ t o ~ } i \} } } g _ { c , f , k } ^ { ( \ell ) } \omega ( p _ { j , i } ^ { k } ) \mathbf { x } _ { c } ^ { ( \ell ) } ( j ) + b _ { f } \mathbf { 1 } _ { N _ { \ell } } ,
+$$
+
+The above equation shows that each neuron in the graph convolutional layer is connected only to a local region (local vertices and edges) in the vertex domain of the input data volume, which is adaptive to the graph topology. The strength of correlation is explicitly utilized in $\omega ( p _ { j , i } ^ { k } )$ . Fig. 1 summarizes the graph convolution operation at each neuron. We refer to this method as topology adaptive graph convolutional network (TAGCN).
+
+In Fig. 2, we show TAGCN with an example of 2-size filter sliding from vertex 1 (figure on the lefthand-side) to vertex 2 (figure on the right-hand-side). The filter is first placed at vertex 1. Since paths $( 1 , 2 , 1 ) \ ( 5 , 4 , 1 )$ and so on (paths with red glow) are all 2-length paths to vertex 1, they are covered by this 2-size filter. Since paths $( 2 , 3 )$ and $( 7 , 3 )$ are not on any 2-length path to vertex 1, they are not covered by this filter. Further, when this 2-size filter moves to vertex 2, paths $( 1 , 5 )$ , $( 4 , 5 )$ and $( 6 , 5 )$ are no longer covered, but paths $( 2 , 3 )$ and $( 7 , 3 )$ are first time covered and contribute to the convolution with output at vertex 2.
+
+Further, ${ \bf y } _ { f } ^ { ( \ell ) } ( i )$ is the weighted sum of the input features of vertices in $\mathbf { x } _ { c } ^ { ( \ell ) }$ that are within $k$ -paths away to vertex $i$ , for $k = 0 , 1 , \ldots K$ , with weights given by the products of components of $\mathbf { A } ^ { k }$ and $g _ { c , f , k } ^ { ( \ell ) ^ { \top } }$ . Thus the output is the weighted sum of the feature map given by the filtered results from 1-size up to $K$ -size filters. It is evident that the vertex convolution on the graph using $K$ th order polynomials is $K$ -paths localized. Moreover, different vertices on the graph share $g _ { c , f , k } ^ { ( \ell ) }$ . The above local convolution and weight sharing properties of the convolution (3) on a graph are very similar to those in traditional CNN.
+
+Though the convolution operator defined in (2) is defined on the vertex domain, it can also be understood as a filter in the spectrum domain, and it is consistent with the definition of convolution in graph signal processing. We provide detailed discussion in the appendix.
+
+# 2.3 COMPUTATION COMPLEXITY OF THE CONVOLUTION OPERATION
+
+Equation (7) shows that TAGCN is equivalent to performing graph convolution either on the vertex domain or on the spectrum domain. We next show that it is preferred to implement TAGCN in the vertex domain.
+
+To obtain F, $\mathbf { F } ^ { - 1 }$ and J, the eigendecomposition for a diagonalizable matrix is in general $\mathcal { O } ( N ^ { 3 } )$ . If the graph is directed, A may only be block diagonalizable, needing the Jordan decomposition, which is sensitive to round-off errors. Additional matrix and vector multiplications may be needed to project the filtered out results from the spectrum domain back to the vertex domain. In contrast, filtering directly in the vertex domain that involves powers of the graph adjacency matrix with $N$ vertices and $M$ edges can be computed in $\mathcal { O } ( M N )$ by performing a breadth-first search starting from each vertex to determine the distances to all other vertices Hammack et al. (2011). For a sparse graph, which is often the case in practical problems, $M \ll N$ . Since the motivation for performing graph convolution is to detect local features for graph-structured data, the optimal $K$ usually is very small. The experiments section shows that a filter size $K = 2$ leads to the best performance is selected for the data sets and specific tasks tested. In summary, using polynomial filters in the vertex domain rather than in the spectrum domain saves significantly computational costs.
+
+![](images/25af37966a6ee9d9d3d18b6528a58b0bb43ad5dd6fa24bee254ee0eacd8239b5.jpg)  
+Figure 2: An example of a directed graph with weights along directed edges corresponding to A. The parts with glow on left(right)-hand-side represent filters at different locations. The figure on the left-hand-side denotes the filtering/convolution starting from vertex 1, then the filter slides to vertex 2 as shown on the right-hand-side with filter topology adaptive to the new local region.
+
+# 3 RELATION WITH OTHER EXISTING FORMULATIONS
+
+In this section, we show the connections and differences between the proposed TAGCN and existing methods. In general, there are two types of graph convolution operators for the CNN architecture. One defines the convolution in the spectrum domain, whose output feature map is the multiplication of the inverse Fourier transform matrix with the filtered results in the spectrum domain (Bruna et al., 2014; Defferrard et al., 2016; Levie et al., 2017). By doing further approximations based on this spectrum domain operator, a simplified convolution was obtained in Kipf & Welling (2017). The other defines convolution by a feature propagation model in the vertex domain such as MoNet in Monti et al. (2017) and the diffusion CNN (DCNN) in Atwood & Towsley (2016). We investigate in detail each alternative.
+
+In Bruna et al. (2014); Defferrard et al. (2016); Levie et al. (2017), the convolution operation was defined using the convolution theorem and filtering operation in the spectrum domain by computing the eigendecomposition of the normalized Laplacian matrix of the graph. The Laplacian matrix $\mathbf { L }$ is defined as $\mathbf { L } = \mathbf { D } - \mathbf { A }$ with the further assumption that $\mathbf { A }$ is symmetric to guarantee that $\mathbf { L }$ is positive semi-definite. The convolution defined by the multiplication in the spectrum domain is approximated by Defferrard et al. (2016) by
+
+$$
+\mathbf { U } g _ { \theta } \mathbf { U } ^ { T } \mathbf { x } \approx \sum _ { k = 0 } ^ { K } \theta _ { k } T _ { k } \left[ \frac { 2 } { \lambda _ { \operatorname* { m a x } } } \mathbf { L } - \mathbf { I } \right] \mathbf { x } _ { c } ^ { ( \ell ) } ,
+$$
+
+where $T _ { k } \left[ \cdot \right]$ is the $k$ th order matrix Chebyshev polynomial (Shuman et al., 2013) where
+
+$$
+T _ { k } ( { \bf L } ) = 2 { \bf L } T _ { k - 1 } [ { \bf L } ] - T _ { k - 2 } [ { \bf L } ] ,
+$$
+
+with the initial values defined as $T _ { 0 } [ \mathbf { L } ] = \mathbf { I }$ and $T _ { 1 } \left[ \mathbf { L } \right] = \mathbf { L }$ . We refer later to this method as ChebNet for performance comparison. Note that the Laplacian matrix can be seen as a differentiator operator. The assumption of symmetric $\mathbf { A }$ restricts the application to undirected graphs. Note that in Defferrard et al. (2016), Laplacian matrix polynomials with maximum order $K = 2 5$ is needed to approximate the convolution operation on the left-hand side in (4), which imposes the computational burden. While TAGCN only needs an adjacency matrix polynomials with maximum order 2 to achieve better performance as shown in the experiment part.
+
+In Kipf & Welling (2017), a graph convolutional network (GCN) was obtained by a first order approximation of (4). In particular, let $K = 1$ and make the further assumptions that $\lambda _ { \operatorname* { m a x } } = 2$ and $\theta _ { 0 } = \theta _ { 1 } = \theta$ . Then a simpler convolution operator that does not depend on the spectrum knowledge is obtained as
+
+$$
+\mathbf { U } g _ { \theta } \mathbf { U } ^ { T } \mathbf { x } \approx \sum _ { k = 0 } ^ { 1 } \theta _ { k } T _ { k } \left[ \frac { 2 } { \lambda _ { \operatorname* { m a x } } } \mathbf { L } - \mathbf { I } \right] \mathbf { x } _ { c } ^ { ( \ell ) } \approx \theta ( \mathbf { I } + \mathbf { D } ^ { - \frac { 1 } { 2 } } \mathbf { A } \mathbf { D } ^ { - \frac { 1 } { 2 } } ) \mathbf { x } _ { c } ^ { ( \ell ) } .
+$$
+
+Note that $\mathbf { I } + \mathbf { D } ^ { - \frac { 1 } { 2 } } \mathbf { A } \mathbf { D } ^ { - \frac { 1 } { 2 } }$ is a matrix with eigenvalues in $[ 0 , 2 ]$ . A renormalization trick is adopted here by letting $\widetilde { \mathbf { A } } = \mathbf { A } + \mathbf { I }$ and $\begin{array} { r } { \widetilde { \bf D } _ { i , i } = \sum _ { j } \widetilde { \bf A } _ { i , j } } \end{array}$ . Finally, the convolutional operator is approximated by
+
+$$
+\mathbf { U } g _ { \theta } \mathbf { U } ^ { T } \mathbf { x } \approx \theta \mathbf { \widetilde { D } } ^ { - \frac { 1 } { 2 } } \mathbf { \widetilde { A } } \mathbf { \widetilde { D } } ^ { - \frac { 1 } { 2 } } \mathbf { x } _ { c } ^ { ( \ell ) } = \theta \mathbf { \widehat { A } } ,
+$$
+
+where $\widehat { \mathbf { A } } = \widetilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } \widetilde { \mathbf { A } } \widetilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } }$ e e . It is interesting to observe that this method though obtained by simplifying the spectrum method has a better performance than the spectrum method (Defferrard et al., 2016). The reason may be because the simplified form is equivalent to propagating vertex features on the graph, which can be seen as a special case of our TAGCN method, though there are other important differences.
+
+Though our TAGCN is able to leverage information at a farther distance, it is not a simple extension of GCN Kipf & Welling (2017). First, the graph convolution in GCN is defined as a first order Chebyshev polynomial of the graph Laplacian matrix, which is an approximation to the graph convolution defined in the spectrum domain in Defferrard et al. (2016). In contrast, our graph convolution is rigorously defined as multiplication by polynomials of the graph adjacency matrix; this is not an approximation, rather, it simply is filtering with graph filters as defined and as being consistent with graph signal processing.
+
+Next, we show the difference between our work and the GCN method in Kipf & Welling (2017) when using 2nd order $\mathrm { K } { = } 2$ , 2 steps away from the central node) Chebyshev polynomials of Laplacian matrix. In the GCN paper Kipf & Welling (2017), it has been shown that $\begin{array} { r } { \sum _ { k = 0 } ^ { 1 } \theta _ { k } T _ { k } ( { \bf L } ) \approx \widehat { \bf A } } \end{array}$ as repeated in (6), and $T _ { 2 } [ \mathbf { L } ] = 2 \mathbf { L } ^ { 2 }$ by the definition of Chebyshev polynomial. Then, extending GCN to the second order Chebyshev polynomials (two steps away from a central node) can be obtained from the original definition in T. Kipfs GCN (Kipf & Welling, 2017, eqn (5)) as $\begin{array} { r } { \sum _ { k = 0 } ^ { 2 } \theta T _ { k } ( L ) = \widehat { \mathbf { A } } + 2 \bar { \mathbf { L } ^ { 2 } } - \mathbf { I } } \end{array}$ , which is different from our definition as in (2). Thus, it is evident that our method is not a simple extension of GCN. We apply graph convolution as proposed from basic principles in the graph signal processing, with no approximations involved, while both GCN in Kipf & Welling (2017) and Defferrard et al. (2016) Levie et al. (2017) are based on approximating the convolution defined in the spectrum domain. In our approach, the degree of freedom is the design of the graph filter-its degree and its coefficients. Ours is a principled approach and provides a generic methodology. The performance gains we obtain are the result of capturing the underlying graph structure with no approximation in the convolution operation.
+
+Simonovsky & Komodakis (2017) proposed the edge convolution network (ECC) to extend the convolution operator from regular grids to arbitrary graphs. The convolution operator is defined similarly to (6) as
+
+$$
+\mathbf { y } _ { f } ^ { ( \ell ) } ( i ) = \sum _ { j \in \mathcal { N } ( i ) } \frac { 1 } { | \mathcal { N } ( i ) | } \Theta _ { j , i } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) } ( j ) + b _ { f } ^ { ( \ell ) } ,
+$$
+
+$\Theta _ { j , i } ^ { ( \ell ) }$
+
+A mixture model network (MoNet) was proposed in Monti et al. (2017), with convolution defined as
+
+$$
+\mathbf { y } _ { f } ^ { ( \ell ) } ( i ) = \sum _ { f = 1 } ^ { F } \sum _ { j \in \mathcal { N } ( i ) } g _ { f } \kappa _ { f } \mathbf { x } _ { c } ^ { ( \ell ) } ( j ) ,
+$$
+
+where $\kappa _ { f }$ is a Gaussian kernel with $\begin{array} { r } { \kappa _ { f } = \exp \left\{ - \frac { 1 } { 2 } ( \mathbf { u } - \pmb { \mu } _ { f } ) ^ { T } \pmb { \Sigma } _ { f } ^ { - 1 } ( \mathbf { u } - \pmb { \mu } _ { f } ) \right\} } \end{array}$ and $g _ { f }$ is the weight coefficient for each Gaussian kernel $\kappa _ { f }$ . It is further assumed that $\Sigma _ { f }$ is a $2 \times 2$ diagonal matrix.
+
+GCN, ECC, and MoNet all design a propagation model on the graph; their differences are on the weightings used by each model.
+
+Table 1: Summary of number of weights need to be learned for the $\ell$ -th layer.   
+
+<table><tr><td>DCNN</td><td>ECC</td><td>ChebNet</td><td>GCN</td><td>MoNet</td><td>TAGCN</td></tr><tr><td>FeCe</td><td>FeCe</td><td>25FeCe</td><td>FeCe</td><td>4FeCe</td><td>2FeCe</td></tr></table>
+
+Atwood & Towsley (2016) proposed a diffusion CNN (DCNN) method that considers a diffusion process over the graph. The transition probability of a random walk on a graph is given by $\mathbf { P } =$ $\mathbf { b } ^ { - 1 } \mathbf { A }$ , which is equivalent to the normalized adjacency matrix.
+
+$$
+\begin{array} { r } { \mathbf { y } _ { c , f } ^ { ( \ell ) } = \mathbf { g } _ { c , f } ^ { ( \ell ) } \mathbf { P } ^ { f } \mathbf { x } _ { c } ^ { ( \ell ) } . } \end{array}
+$$
+
+By comparing the above methods with TAGCN in (3), it can be concluded that GCN, ECC and MoNet can be seen as a special case of TAGCN because in (3) the item with $k = 1$ can be seen as an information propagation term. However, the strength of the correlation between vertices that are taken into account in $\omega ( p _ { j , i } ^ { k } )$ in (3) can not be utilized in the GCN, ECC and MoNet methods. Further, the TAGCN method is a systematic way to design a set of fixed size filters that is adaptive to the input graph topology when performing convolution on the graph. Compared with existing spectrum methods (Bruna et al., 2014; Defferrard et al., 2016; Levie et al., 2017), TAGCN satisfies the convolution theorem as shown in the previous subsection and implemented in the vertex domain, which avoids performing costly and practically numerical unstable eigendecompositions.
+
+We further compare the number of weights that need to be learned in each hidden layer for these different methods in Table 1. As we show later, $K = 2$ is selected in our experiments using cross validation. However, for ChebNet in (Defferrard et al., 2016), it is suggested that one needs a $2 5 ^ { \mathrm { t h } }$ degree Chebyshev polynomial to provide a good approximation to the graph Laplacian spectrum. Thus we have a moderate number of weights to be learned. In the following, we show that our method achieves the best performance for each of those commonly used graph-structured data sets.
+
+# 4 EXPERIMENTS
+
+The proposed TAGCN is general and can be fit to the general graph CNN architectures for different tasks. In the experiments, we focus on the vertex semisupervised learning problem, where we have access to only a few labeled vertices, and the task is to classify the remaining unlabeled vertices. To compare the performance of TAGCN with that of existing methods, we extensively evaluate TAGCN on three graph-structured datasets, including the Cora, Citesser and Pubmed datasets. The datasets split and experiments setting closely follow the standard criteria in Yang et al. (2016).
+
+# 4.1 DATASETS
+
+TAGCN is evaluated on three data sets coming from citation networks: Citeseer, Cora, and Pubmed. We closely follow the data set split and experimental setup in Yang et al. (2016). Each data set consists of a certain classes of documents, yet only a few documents are labeled. The task is to classify the documents in the test set with these limited number of labels. In each data set, the vertices are the documents and the edges are the citation links. Each document is represented by sparse bagof-words feature vectors, and the citation links between documents are provided. Detailed statistics of these three data sets are summarized in Table 2. It shows the number of nodes and edges that corresponding to documents and citation links. Classes for the graph vertices shows the number of document classes in each data set. Also, the number of features at each vertex is given. Label rate denotes the number of labeled documents that are used for training divided by the total number of documents in each data set.
+
+Table 2: Dataset statistics, following Yang et al. (2016); Kipf & Welling (2017)   
+
+<table><tr><td>Dataset Size</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>Label rate</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Citeseer</td><td>3,327</td><td>4,732</td><td>6</td><td>3,703</td><td>0.036</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>0.052</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>0.003</td></tr></table>
+
+Table 3: Summary of results in terms of percentage classification accuracy with standard variance   
+
+<table><tr><td>Citeseer</td><td>Cora</td><td>Pubmed</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Planetoid 64.7</td><td>75.7</td><td>77.2</td></tr><tr><td>DCNN</td><td>76.8±0.6</td><td>73.0±0.5</td></tr><tr><td>ChebNet (K=2) 69.6</td><td>81.2</td><td>73.8</td></tr><tr><td>ChebNet (K=3) 69.8</td><td>79.5</td><td>74.4</td></tr><tr><td>GCN 70.3</td><td>81.5</td><td>79.0</td></tr><tr><td>MoNet =</td><td>81.69±0.5</td><td>78.81±0.4</td></tr><tr><td>TAGCN (K=2)</td><td>70.9± 0.9 82.5±0.7</td><td>81.1±0.4</td></tr></table>
+
+# 4.2 EXPERIMENTAL SETTINGS
+
+We construct a graph for each data set with nodes representing documents and undirected edges4 linking two papers if there is a citation relationship. We obtain the adjacency matrix $\bar { \bf A }$ with 0 and 1 components and further obtain the normalized matrix A.
+
+In the following experiments, we design a TAGCN with two hidden layers (obtained from cross validation) for the semi-supervised node classification. In each hidden layer, the proposed TAGCN is applied for convolution, followed by a ReLU activation. 16 hidden units (filters) are designed for each hidden layer, and dropout is applied after each hidden layer. The softmax activation function is applied to the output of the second hidden layer for the final classification. For ablation study, we evaluate the performance of TAGCN with different kernel size from 1 to 4. To investigate the performance for different number of parameters, we also design a TAGCN with 8 filters for each hidden layer and compare its classification accuracy with all the baselines and TAGCN with 16 filters. We train our model using Adam (Kinga & Ba, 2015) with a learning rate of 0.01 and early stopping with a window size of 20. Hyperparameters of the networks (filter size, dropout rate, and number of hidden layers) are selected by cross validation.
+
+To make a fair comparison, we closely follow the same split of training, validation, and testing sets as in Kipf & Welling (2017); Yang et al. (2016), i.e., 500 labeled examples for hyperparameters (filter size, dropout rate, and number of hidden layers) optimization and cross-entropy error is used for classification accuracy evaluation. The performance results of the proposed TAGCN method are an average over 100 runs with random initializations (Glorot & Bengio, 2010).
+
+# 4.3 QUANTITATIVE EVALUATIONS
+
+We compare the classification accuracy with other recently proposed graph CNN methods as well as a graph embedding method known as Planetoid Yang et al. (2016). The quantitative results are summarized in Table 3. Reported numbers denote classification accuracy in percent. Results for Planetoid, GCN, and ChebNet are taken from Kipf & Welling (2017), and results for DCNN and MoNet are taken from Monti et al. (2017). All the experiments for different methods are based on the same data statistics shown in Table 2. The datasets split and experiments settings closely follow the standard criteria in Yang et al. (2016); Kipf & Welling (2017). Table 3 shows that our method
+
+Table 4: TAGCN classification accuracy (ACC) with different parameters   
+
+<table><tr><td>Data Set</td><td>Filter Size</td><td>Filter Number</td><td>ACC</td></tr><tr><td rowspan="5">Citeseer</td><td>1</td><td>16</td><td>68.9</td></tr><tr><td>2</td><td>16</td><td>70.9</td></tr><tr><td>3</td><td>16</td><td>70.0</td></tr><tr><td>4</td><td>16</td><td>69.8</td></tr><tr><td>2</td><td>8</td><td>70.6</td></tr><tr><td rowspan="6">Cora</td><td>1</td><td>16</td><td>81.4</td></tr><tr><td>2</td><td>16</td><td>82.5</td></tr><tr><td>3</td><td>16</td><td>82.1</td></tr><tr><td>4</td><td>16</td><td>81.8</td></tr><tr><td>2</td><td>8</td><td>82.5</td></tr><tr><td>1</td><td>16</td><td>79.4</td></tr><tr><td rowspan="4">Pubmed</td><td>2</td><td>16</td><td>81.1</td></tr><tr><td>3</td><td>16</td><td>80.9</td></tr><tr><td>4</td><td>16</td><td>79.5</td></tr><tr><td>2</td><td>8</td><td>80.8</td></tr></table>
+
+outperforms all the recent state-of-the-art methods by obvious margins for all the three datasets.   
+These results verify the efficacy of the proposed TAGCN.
+
+For ablation study, we further compare the performance of different filter sizes from $K = 1$ to $K = 4$ in Table 4. It shows that the performances for filter size $K = 2$ are always better than that for other filter sizes. The value $K = 1$ gives the worst classification accuracy. This further validates that a local feature extraction is more important than just propagating features among neighbors on the graph. In Table 4, we also compare the performance of different number of filters, which reflects different number of network parameters. Note, we also choose filer size $K = 2$ and filter number $F _ { \ell } = 8$ that results in the same number of network parameters as that in GCN, MoNet, ECC and DCNN according to Table 1. It shows that the classification accuracy using 8 filters is comparable with that using 16 filters in each hidden layer for TAGCN. Moreover, TAGCN with 8 filters can still achieve higher accuracy than GCN, MoNet, ECC and DCNN methods. This proves that, even with a similar number of parameters or architecture, our method still exhibits superior performance than GCN.
+
+As we explained in Section 3, TAGCN in our paper is not simply extending GCN Kipf & Welling (2017) to $k$ -th order. Nevertheless, we implement ${ \bf A } ^ { 2 }$ and compare its performance with ours. For the data sets Pubmed, Cora, and Citeseer, the classification accuracies are 79.1(81.1), 81.7(82.5) and 70.8(70.9), where the numbers in parentheses are the results obtained with our method. Our method still achieves a noticeable performance advantage over ${ \bf A } ^ { 2 }$ for the Pubmed and Cora data; in particular, we note the significant performance gain with the Pubmed database that has the largest number of nodes among these three data sets.
+
+# 5 CONCLUSIONS
+
+We have defined a novel graph convolutional network that rearchitects the CNN architecture for graph-structured data. The proposed method, known as TAGCN, is adaptive to the graph topology as the filter scans the graph. Further, TAGCN inherits properties of the convolutional layer in classical CNN, i.e., local feature extraction and weight sharing. It can further extract the strength of correlation between vertices in the filtering region. On the other hand, by the convolution theorem, TAGCN that implements in the vertex domain offers implement in the spectrum domain unifying graph CNN in both the spectrum domain and the vertex domain. TAGCN is consistent with convolution in graph signal processing. These nice properties lead to a noticeable performance advantage in classification accuracy on different graph-structured datasets for semi-supervised graph vertex classification problems with low computational complexity.
+
+# 6 APPENDIX: SPECTRUM RESPONSE OF TAGCN
+
+In classical signal processing (Oppenheim & Schafer, 2009), the convolution in the time domain is equivalent to multiplication in the spectrum domain. This relationship is known as the convolution theorem. Sandryhaila & Moura (2013) showed that the graph filtering defined in the vertex domain satisfies the generalized convolution theorem naturally and can also interpret spectrum filtering for both directed and undirected graphs. Recent work (Bruna et al., 2014; Defferrard et al., 2016) used the convolution theorem for undirected graph-structured data and designed a spectrum graph filtering.
+
+![](images/1523d9fd4c75a2486b139f369d3766910ea55f641bd451f66d8d026e7b489b75.jpg)  
+Figure 3: Graph topology of a 1-D cyclic graph.
+
+Assume that the adjacency matrix A for a graph is diagonalizable, i.e., $\mathbf { A } = \mathbf { F } ^ { - 1 } \mathbf { J } \mathbf { F }$ with $\mathbf { J }$ a diagonal matrix. The components on the diagonal of $\mathbf { J }$ are eigenvalues of $\mathbf { A }$ , and the column vectors of $\mathbf { F } ^ { - 1 }$ are the right eigenvectors of $\mathbf { A }$ ; the row vectors of $\mathbf { F }$ are the left eigenvectors of A 5. By diagonalizing $\mathbf { A }$ in (2) for TAGCN, we obtain
+
+$$
+\mathbf { G } _ { c , f } ^ { ( \ell ) } \mathbf { x } _ { c } ^ { ( \ell ) } = \mathbf { F } ^ { - 1 } \left( \sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \ell ) } \mathbf { J } ^ { k } \right) \mathbf { F } \mathbf { x } _ { c } ^ { ( \ell ) } .
+$$
+
+The expression on the left-hand-side of the above equation represents the filtering/convolution on the vertex domain. Matrix $\mathbf { F }$ defines the graph Fourier transform (Sandryhaila $\&$ Moura, 2013; 2014), and $\mathbf { F } \mathbf { x } _ { c } ^ { ( \ell ) }$ is the input feature spectrum map, which is a linear mapping from the input feature on the vertex domain to the spectrum domain. The polynomial $\scriptstyle \sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \ell ) } \mathbf { J } ^ { k }$ is the spectrum of the graph filter. Relation (7), which is equation (27) in Sandryhaila & Moura (2013) generalizes the classical convolution theorem to graph-structured data: convolution/filtering on the vertex domain becomes multiplication in the spectrum domain. When the graph is in the 1D cyclic form, as shown in Fig. 3, the corresponding adjacency matrix is of the form
+
+$$
+\mathbf { A } = \left[ \begin{array} { c c c c } { \vrule } & { } & { } & { 1 } \\ { 1 } & { } & { } & { } \\ { \vrule } & { \ddots } & { } & { } \\ { } & { } & { 1 } & { } \end{array} \right] .
+$$
+
+The eigendecomposition of $\mathbf { A }$ is
+
+$$
+\mathbf { A } = \frac { 1 } { N } \mathrm { D F T } ^ { - 1 } \left[ \begin{array} { c c } { \begin{array} { c } { e ^ { - j \frac { 2 \pi 0 } { N } } } \\ { \ddots } \\ { \end{array} } } \\  \begin{array} { c } { \begin{array} { r l } { e ^ { - j \frac { 2 \pi ( N - 1 ) } { N } } } \end{array} } \end{array} \right] \mathrm { D F T } , \end{array}
+$$
+
+where DFT is the discrete Fourier transform matrix. The convolution operator defined in (2) is consistent with that in classical signal processing.
+
+# REFERENCES
+
+James Atwood and Don Towsley. Diffusion-convolutional neural networks. In Advances in Neural Information Processing Systems (NIPS). 2016.
+
+Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. In Internatonal Conference on Learning Representations (ICLR). 2014.
+
+Hanjun Dai, Bo Dai, and Le Song. Discriminative embeddings of latent variable models for structured data. In International Conference on Machine Learning (ICML). 2016.
+
+Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks ¨ on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, 2016.
+
+Joya Deri and Jose M. F. Moura. Spectral projector-based graph Fourier transforms. ´ IEEE Journal of Selected Topics in Signal Processing, 11(6):785–795, 2017.
+
+Jian Du, Shaodan Ma, Yik-Chung Wu, Soummya Kar, and Jose M. F. Moura. Convergence analysis ´ of distributed inference with vector-valued Gaussian belief propagation. arXiv:1611.02010, 2016.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. 2010.
+
+Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM International Conference on Knowledge Discovery and Data Mining (KDD). 2016.
+
+Richard Hammack, Wilfried Imrich, and Sandi Klavzar. ˇ Handbook of product graphs. CRC press, 2011.
+
+Diederik P. Kinga and Jimmy Ba. A method for stochastic optimization. In International Conference on Learning Representations (ICLR). 2015.
+
+Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In Internatonal Conference on Learning Representations (ICLR). 2017.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems (NIPS). 2012.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey E. Hinton. Deep learning. Nature, 521(7553):436–444, 2015.
+
+Ron Levie, Federico Monti, Xavier Bresson, and Michael M Bronstein. Cayleynets: Graph convolutional neural networks with complex rational spectral filters. arXiv preprint arXiv:1705.07664, 2017.
+
+Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M \` Bronstein. Geometric deep learning on graphs and manifolds using mixture model CNNs. In Conference on Computer Vision and Pattern Recognition (CVPR). 2017.
+
+Alan V. Oppenheim and Ronald W. Schafer. Discrete-time signal processing (3rd Edition). Prentice Hall, 2009.
+
+Aliaksei Sandryhaila and Jose M. F. Moura. Discrete signal processing on graphs. ´ IEEE Transactions on Signal Processing, 61(7), 2013.
+
+Aliaksei Sandryhaila and Jose M. F. Moura. Big data analysis with signal processing on graphs: ´ Representation and processing of massive data sets with irregular structure. IEEE Signal Processing Magazine, 31(5), 2014.
+
+David I Shuman, Sunil K Narang, Pascal Frossard, Antonio Ortega, and Pierre Vandergheynst. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30, 2013.
+
+Martin Simonovsky and Nikos Komodakis. Dynamic edge-conditioned filters in convolutional neural networks on graphs. In Conference on Computer Vision and Pattern Recognition (CVPR). 2017.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In Internatonal Conference on Learning Representations (ICLR). 2015.
+
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Conference on Computer Vision and Pattern Recognition (CVPR). 2015.
+
+Zhilin Yang, William Cohen, and Ruslan Salakhutdinov. Revisiting semi-supervised learning with graph embeddings. In Internatonal Conference on Learning Representations (ICLR). 2016.

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+    {
+        "type": "text",
+        "text": "TOPOLOGY ADAPTIVE GRAPH CONVOLUTIONAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolution acts as a local feature extractor in convolutional neural networks (CNNs). However, the convolution operation is not applicable when the input data is supported on an irregular graph such as with social networks, citation networks, or knowledge graphs. This paper proposes the topology adaptive graph convolutional network (TAGCN), a novel graph convolutional network that generalizes CNN architectures to graph-structured data and provides a systematic way to design a set of fixed-size learnable filters to perform convolutions on graphs. The topologies of these filters are adaptive to the topology of the graph when they scan the graph to perform convolution, replacing the square filter for the grid-structured data in traditional CNNs. The outputs are the weighted sum of these filters’ outputs, extraction of both vertex features and strength of correlation between vertices. It can be used with both directed and undirected graphs. The proposed TAGCN not only inherits the properties of convolutions in CNN for grid-structured data, but it is also consistent with convolution as defined in graph signal processing. Further, as no approximation to the convolution is needed, TAGCN exhibits better performance than existing graph-convolution-approximation methods on a number of data sets. As only the polynomials of degree two of the adjacency matrix are used, TAGCN is also computationally simpler than other recent methods. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            334,
+            560
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional neural network (CNN) architectures exhibit state-of-the-art performance on a variety of learning tasks dealing with 1D, 2D, and 3D grid-structured data such as acoustic signals, images, and videos, in which convolution serves as a feature extractor (LeCun et al., 2015). However, the (usual) convolution operation is not applicable when applying CNN to data that is supported on an arbitrary graph rather than on a regular grid structure, since the number of neighbors of each vertex on the graph varies, and it is difficult to design a fixed-size filter scanning over the graph-structured data for feature extraction. ",
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+    {
+        "type": "text",
+        "text": "Recently, there has been an increasing interest in graph CNNs (Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2017; Monti et al., 2017; Levie et al., 2017), attempting to generalize deep learning methods to graph-structured data, specifically focusing on the design of graph CNN . In this paper, we propose the topology adaptive graph convolutional network (TAGCN), a unified convolutional neural network to learn nonlinear representations for the graph-structured data. It slides a set of fixed-size learnable filters on the graph simultaneously, and the output is the weighted sum of these filters’ outputs, which extract both vertex features and strength of correlation between vertices. Each filter is adaptive to the topology of the local region on the graph where it is applied. TAGCN unifies filtering in both the spectrum and vertex domains; and applies to both directed and undirected graphs. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In general, the existing graph CNNs can be grouped into two types: spectral domain techniques and vertex domain techniques. In Bruna et al. (2014), CNNs have been generalized to graph-structured data, where convolution is achieved by a pointwise product in the spectrum domain according to the convolution theorem. Later, Defferrard et al. (2016) and Levie et al. (2017) proposed spectrum filtering based methods that utilize Chebyshev polynomials and Cayley polynomials, respectively. The assumption of symmetric adjacency matrix in these spectrum based methods restrict the application to undirected graphs. Kipf & Welling (2017) simplified this spectrum method and obtained a filter in the vertex domain, which achieves state-of-the-art performance. Other researchers (Atwood & Towsley, 2016; Monti et al., 2017) worked on designing feature propagation models in the vertex domain for graph CNNs. Yang et al. (2016); Dai et al. (2016); Grover & Leskovec (2016); Du et al. (2016) study transforming graph-structured data to embedding vectors for learning problems. Nevertheless, it still remains open how to extend CNNs from grid-structured data to arbitrary graph-structured data with local feature extraction capability. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "This paper proposes a modification to the graph convolution step in CNNs that is particularly relevant for graph structured data. Our proposed TAGCN is graph-based convolution and draws on techniques from graph signal processing. We define rigorously the graph convolution operation on the vertex domain as multiplication by polynomials of the graph adjacency matrix, which is consistent with the notion of convolution in graph signal processing. In graph signal processing, polynomials of the adjacency matrix are graph filters, extending to graph based data from the usual concept of filters in traditional time or image based signal processing. Thus, comparing ours with existing work on graph CNNs, our paper provides a solid theoretical foundation for our proposed convolution step instead of an ad-hoc approach to convolution in CNNs for graph structured data. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Further, our method avoids computing the spectrum of the graph Laplacian as in Bruna et al. (2014), or approximating the spectrum using high degree Chebyshev polynomials of the graph Laplacian matrix (in Defferrard et al. (2016), it is suggested that one needs a $2 5 ^ { \\mathrm { t h } }$ degree Chebyshev polynomial to provide a good approximation to the graph Laplacian spectrum) or using high degree Cayley polynomials of the graph Laplacian matrix (in Levie et al. (2017), $1 2 ^ { \\mathrm { t h } }$ degree Cayley polynomials are needed). We also clarify that the GCN method in Kipf & Welling (2017) is a first order approximation of the Chebyshev polynomials approximation in Defferrard et al. (2016), which is very different from our method. Our method has a much lower computational complexity than the complexity of the methods proposed in Bruna et al. (2014); Defferrard et al. (2016); Levie et al. (2017), since our method only uses polynomials of the adjacency matrix with maximum degree 2 as shown in our experiments. Finally, the method that we propose exhibits better performance than existing methods. Our contributions are summarized follows: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We propose a general $K$ -localized filter for graph convolution in the vertex domain to extract local features on a set of size-1 up to size- $K$ receptive fields. The topologies of these filters are adaptive to the topology of the graph as they scan the graph to perform convolution. It replaces the fixed square filters in traditional CNNs for the input gridstructured data volumes in traditional CNNs. Thus, our convolution definition that we use in the convolution step for the vertex domain is consistent with convolution in traditional CNNs. TAGCN is based on the graph signal processing and it is consistent with the convolution in graph signal processing. It applies to both directed and undirected graphs. Moreover, it has a much lower computational complexity compared with recent methods since it only needs polynomials of the adjacency matrix with maximum degree 2 compared with the $2 5 ^ { \\mathrm { { i } \\tilde { h } } }$ and $1 \\bar { 2 } ^ { \\mathrm { t h } }$ degree Laplacian matrix polynomials in Defferrard et al. (2016) and Levie et al. (2017). As no approximation to the convolution is needed in TAGCN, it achieves better performance compared with existing methods. We contrast TAGCN with recently proposed graph CNN including both spectrum filtering methods (Bruna et al., 2014; Defferrard et al., 2016) and vertex domain propagation methods (Kipf & Welling, 2017; Monti et al., 2017; Atwood & Towsley, 2016), evaluating their performances on three commonly used data sets for graph vertices classification. Our experimental tests show that TAGCN outperforms consistently all other approaches for each of these data sets. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 GRAPH POLYNOMIAL BASED CONVOLUTION ON GRAPHS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use boldface uppercase and lowercase letters to represent matrices and vectors, respectively. The information and their relationship on a graph $\\mathcal { G }$ can be represented by $\\mathcal { G } = ( \\nu , \\mathcal { E } , \\vec { \\mathbf { A } } )$ , where $\\nu$ is the set of vertices, $\\mathcal { E }$ is the set of edges, and $\\bar { \\bf A }$ is the weighted adjacency matrix of the graph; the graph can be weighted or unweighted, directed or undirected. We assume there is no isolated vertex in $\\mathcal { G }$ . If $\\mathcal { G }$ is a directed weighted graph, the weight $\\bar { \\mathbf { A } } _ { n , m }$ is on the directed edge from vertex $m$ to $n$ . The entry $\\bar { \\mathbf { A } } _ { n , m }$ reveals the dependency between node $n$ and $m$ and can take arbitrary real or complex values. The graph convolution is general and can be adapted to graph CNNs for particular tasks. In this paper, we focus on the vertex semisupervised learning problem, where we have access to limited labeled vertices, and the task is to classify the remaining unlabeled vertices. ",
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+    },
+    {
+        "type": "text",
+        "text": "2.1 GRAPH CONVOLUTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Without loss of generality, we demonstrate graph convolution on the $\\ell \\cdot$ -th hidden layer. The results apply to any other hidden layers. Suppose on the $\\ell \\cdot$ -th hidden layer, the input feature map for each vertex of the graph has $C _ { \\ell }$ features. We collect the $\\ell$ -th hidden layer input data on all vertices for the $c$ -th feature by the vector $\\mathbf { x } _ { c } ^ { ( \\ell ) } \\in \\mathbb { R } ^ { N _ { \\ell } }$ , where $c = 1 , 2 , . . . C _ { \\ell }$ and $N _ { \\ell }$ is the number of vertices1. The components of $\\mathbf { x } _ { c } ^ { ( \\ell ) }$ are indexed by vertices of the data graph representation $\\mathcal { G } = ( \\gamma , \\mathcal { E } , \\bar { \\mathbf { A } } ) ^ { 2 }$ . Let $\\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\in \\mathbb { R } ^ { N _ { \\ell } \\times N _ { \\ell } }$ denote the $f$ -th graph filter. The graph convolution is the matrix-vector product, i.e., $\\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) }$ . Then the $f$ -th output feature map is ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81da209ccec2cb6e4fb87531a42608dbe5ee936266397b9e2fcfec7696fe2ad4.jpg",
+        "text": "$$\n\\mathbf { y } _ { f } ^ { ( \\ell ) } = \\sum _ { c = 1 } ^ { C _ { \\ell } } \\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) } + b _ { f } \\mathbf { 1 } _ { N _ { \\ell } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $b _ { f } ^ { ( \\ell ) }$ is a learnable bias, and ${ \\bf 1 } _ { N _ { \\ell } }$ is the $N _ { \\ell }$ dimension vector of all ones. We design $\\mathbf { G } _ { c , f } ^ { ( \\ell ) }$ such that $\\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) }$ is a meaningful convolution on a graph with arbitrary topology. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the recent theory on graph signal processing (Sandryhaila & Moura, 2013), the graph shift is defined as a local operation that replaces a graph signal at a graph vertex by a linear weighted combination of the values of the graph signal at the neighboring vertices: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e199b3da9ac080e4dbde035ce740cf6ef476525125fdb5895ec78f9fde2b890.jpg",
+        "text": "$$\n\\tilde { \\mathbf { x } } _ { c } ^ { ( \\ell ) } = \\bar { \\mathbf { A } } \\mathbf { x } _ { c } ^ { ( \\ell ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            462,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The graph shift $\\bar { \\bf A }$ extends the time shift in traditional signal processing to graph-structured data. Following Sandryhaila & Moura (2013), a graph filter $\\mathbf { G } _ { c , f } ^ { \\overline { { ( \\ell ) } } }$ is shift-invariant, i.e., the shift $\\bar { \\bf A }$ and the filter $\\mathbf { G } _ { c , f } ^ { ( \\ell ) }$ commute, $\\bar { \\mathbf { A } } ( \\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) } ) = \\mathbf { G } _ { c , f } ^ { ( \\ell ) } ( \\bar { \\mathbf { A } } \\mathbf { x } _ { c } ^ { ( \\ell ) } )$ , if under appropriate assumption $\\mathbf { G } _ { c , f } ^ { ( \\ell ) }$ is a polynomial in A, ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6183aa49d889dfb4775ab70c480fbd66586b035892a1d6f1384300a082e003e.jpg",
+        "text": "$$\n\\mathbf { G } _ { c , f } ^ { ( \\ell ) } = \\sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \\ell ) } \\mathbf { A } ^ { k } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In (2), the g(\\`)c,f,k are the graph filter polynomial coefficients; the quantity $\\mathbf { A } = \\mathbf { D } ^ { - \\frac { 1 } { 2 } } \\bar { \\mathbf { A } } \\mathbf { D } ^ { - \\frac { 1 } { 2 } }$ is the normalized adjacency matrix of the graph, and $\\mathbf { D } = \\mathrm { d i a g } [ \\mathbf { d } ]$ with the $i$ th component being ${ \\bf d } ( i ) =$ $\\textstyle \\sum _ { j } \\mathbf { A } _ { i , j }$ .3 We adopt the normalized adjacency matrix to guarantee that all the eigenvalues of $\\mathbf { A }$ are inside the unit circle, and therefore $\\mathbf { G } _ { c , f } ^ { ( \\ell ) }$ is computationally stable. The next subsection shows we , and $K \\times C _ { \\ell }$   \ncoincides with GoogLeNet (Szegedy et al., 2015), in which a set of filters with different sizes are used in each convolutional layer. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following the CNN architecture, an additional nonlinear operation, e.g, rectified linear unit (ReLU) is used after every graph convolution operation: ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b29eaedab74124c7e060dad256bab54bf8aadc122e73522d4f0beb82a37a4af.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { x } _ { f } ^ { ( \\ell + 1 ) } = \\sigma \\left( \\mathbf { y } _ { f } ^ { ( \\ell ) } \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma ( \\cdot )$ denotes the ReLU activation function applied to the vertex values. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the following subsection, we demonstrate that convolution operator shown in (2) exhibits the nice properties of a square filter in traditional CNNs and generalize the convolution theorem from classical signal processing to signal processing on graphs (Sandryhaila & Moura, 2013). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 FILTER DESIGN FOR TAGCN CONVOLUTIONAL LAYER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this section, we would like to understand the proposed convolution as a feature extraction operator in traditional CNN rather than as propagating labeled data on the graph. Taking this point of view helps us to profit from the design knowledge/experience from traditional CNN and apply it to grid structured data. Our definition of weight of a path and the following filter size for graph convolution in this section make it possible to design a graph CNN architecture similar to GoogLeNet (Szegedy et al., 2015), in which a set of filters with different sizes are used in each convolutional layer. In fact, we found that a combination of size 1 and size 2 filters gives the best performance in all three data sets studied, which is a polynomial with maximum order 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In traditional CNN, a $K \\times K \\times C _ { \\ell }$ filter scans over the input grid-structured data for feature extraction. For image classification problems, the value $K$ varies for different CNN architectures and tasks to achieve better performance. For example, in VGG-Verydeep-16 CNN model (Simonyan & Zisserman, 2015), only $3 \\times 3 \\times C _ { \\ell }$ filters are used; in ImageNet CNN model (Krizhevsky et al., 2012), $1 1 \\times 1 1 \\times C _ { \\ell }$ filters are adopted; and in GoogLeNet (Szegedy et al., 2015), rather than using the same size filter in each convolutional layer, different size filters, for example, $1 \\times 1 \\times C _ { \\ell }$ , $3 \\times 3 \\times C _ { \\ell }$ and $5 \\times 5 \\times C _ { \\ell }$ filters, are concatenated in each convolution layer. Similarly, we propose a general $K$ -localized filter for graph CNN. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a graph-structured data, we cannot use a square filter window since the graph topology is no longer a grid. In the following, we demonstrate that the convolution operation $\\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) }$ with $\\mathbf { G } _ { c , f } ^ { ( \\ell ) }$ a polyn ial filt $\\begin{array} { r } { \\mathbf { G } _ { c , f } ^ { ( \\ell ) } = \\sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \\ell ) } \\mathbf { A } ^ { k } } \\end{array}$ is equivalent to using a set of filters with filter size from 1 $K$ $k$ -size filter, which is used for local feature extraction on the graph, is $k$ -localized in the vertex domain. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Define a path of length $m$ on a graph $\\mathcal { G }$ as a sequence $v = ( v _ { 0 } , v _ { 1 } , . . . , v _ { m } )$ of vertices $v _ { k } \\in \\mathcal V$ such that each step of the path $( v _ { k } , v _ { k + 1 } )$ corresponds to an (directed) edge of the graph, i.e., $( v _ { k } , v _ { k + 1 } ) \\in$ $\\mathcal { E }$ . Here one path may visit the same vertex or cross the same edge multiple times. The following adjacency matrix $\\mathbf { A }$ is one such example: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edf52ae448502e9680c41505f0962b65d98deae6ce8b065bb209abaa8f5c9767.jpg",
+        "text": "$$\n\\mathbf { A } = { \\left[ \\begin{array} { l l l l l l l l l } { 0 } & { 1 } & { 0 } & { 2 } & { 3 } & { 0 } & { 0 } & { \\cdots } \\\\ { 1 } & { 0 } & { 4 } & { 5 } & { 0 } & { 0 } & { 0 } & { \\cdots } \\\\ { 0 } & { 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { \\cdots } \\\\ { 1 } & { 1 } & { 0 } & { 0 } & { 6 } & { 0 } & { 0 } & { \\cdots } \\\\ { 1 } & { 0 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { \\cdots } \\\\ { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { 0 } & { 0 } & { \\cdots } \\\\ { 0 } & { 0 } & { 0 } & { 1 } & { 0 } & { 0 } & { 0 } & { \\cdots } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\cdots } \\end{array} \\right] } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since A is asymmetric, it represents a directed graph, given in Fig. 2. In this example, there are 6 different length 3-paths on the graph from vertex 2 to vertex 1, namely, $( 2 , 1 , 4 , 1 )$ , $( 2 , 1 , 2 , 1 )$ , $( 2 , 1 , 5 , 1 ) , ( 2 , \\bar { 3 } , 2 , 1 \\bar { ) } , ( 2 , 4 , 2 , 1 )$ , and $( 2 , 4 , 5 , 1 )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We further define the weight of a path to be the product of the edge weights along the path, i.e., $\\begin{array} { r } { \\phi ( p _ { 0 , m } ) = \\prod _ { k = 1 } ^ { m } \\mathbf { A } _ { v _ { k - 1 } , v _ { k } } } \\end{array}$ , where $p _ { 0 , m } = ( v _ { 0 } , v _ { 1 } , \\dots v _ { m } )$ . For example, the weight of the path $( 2 , 1 , 4 , 1 )$ is $1 \\times 1 \\times 2 = 2$ . Then, the $( i , j )$ th entry of $\\mathbf { A } ^ { k }$ in (2), denoted by $\\omega ( p _ { j , i } ^ { k } )$ , can be interpreted as the sum of the weights of all the length- $k$ paths from $j$ to $i$ , which is ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8da05f1454eded1f0fe410c8c032d560e04a9a693421eea102ce88b304dd93fb.jpg",
+        "text": "$$\n\\omega ( p _ { j , i } ^ { k } ) = \\sum _ { j \\in \\{ \\tilde { j } | \\tilde { j } \\mathrm { i s } k \\mathrm { p a t h s t o } i \\} } \\phi ( p _ { j , i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the above example, it can be easily verified that $\\mathbf { A } _ { 1 , 2 } ^ { 3 } = 1 8$ by summing up the weights of all the above six paths from vertex 2 to vertex 1 with length 3. Then, the $i$ th component of $\\mathbf { A } ^ { k } \\mathbf { x } _ { c } ^ { ( \\ell ) }$ is the weighted sum of the input features of each vertex $\\mathbf { x } _ { c } ^ { ( \\ell ) }$ that are length- $k$ paths away to vertex $i$ . Here, $k$ is defined as the filter size. The output feature map is a vector with each component given by the size- $k$ filter sliding on the graph following a fixed order of the vertex indices. ",
+        "bbox": [
+            173,
+            803,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The output at the $i$ -th component can be written explicitly as $\\begin{array} { r l } { \\sum _ { c = 1 } ^ { C _ { \\ell } } \\sum _ { j } g _ { c , f , k } ^ { ( \\ell ) } \\omega ( p _ { j , i } ^ { k } ) \\mathbf { x } _ { c } ^ { ( \\ell ) } ( j ) } & { { } } \\end{array}$ . This weighted sum is similar to the dot product for convolution for a grid-structured data in traditional ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb78b28841ad6e4469230c3a3d293d4e5717dd4c69b348648686bd534fceac2f.jpg",
+        "image_caption": [
+            "Figure 1: Graph convolution operation at each neuron in the graph convolution layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            302,
+            99,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "CNNs. Finnaly, the output feature map is a weighted sum of convolution results from filters with different sizes, which is ",
+        "bbox": [
+            171,
+            257,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fbceca0acaa750cb2fdcb56d29969aabab60d8cb68311377266be0033cba542.jpg",
+        "text": "$$\n\\mathbf { y } _ { f } ^ { ( \\ell ) } ( i ) = \\sum _ { k = 1 } ^ { K _ { \\ell } } \\sum _ { c = 1 } ^ { C _ { \\ell } } \\sum _ { \\substack { j \\in \\{ \\tilde { j } | \\tilde { j } \\mathrm { ~ i s ~ } k \\mathrm { ~ p a t h s ~ t o ~ } i \\} } } g _ { c , f , k } ^ { ( \\ell ) } \\omega ( p _ { j , i } ^ { k } ) \\mathbf { x } _ { c } ^ { ( \\ell ) } ( j ) + b _ { f } \\mathbf { 1 } _ { N _ { \\ell } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            291,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above equation shows that each neuron in the graph convolutional layer is connected only to a local region (local vertices and edges) in the vertex domain of the input data volume, which is adaptive to the graph topology. The strength of correlation is explicitly utilized in $\\omega ( p _ { j , i } ^ { k } )$ . Fig. 1 summarizes the graph convolution operation at each neuron. We refer to this method as topology adaptive graph convolutional network (TAGCN). ",
+        "bbox": [
+            174,
+            347,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 2, we show TAGCN with an example of 2-size filter sliding from vertex 1 (figure on the lefthand-side) to vertex 2 (figure on the right-hand-side). The filter is first placed at vertex 1. Since paths $( 1 , 2 , 1 ) \\ ( 5 , 4 , 1 )$ and so on (paths with red glow) are all 2-length paths to vertex 1, they are covered by this 2-size filter. Since paths $( 2 , 3 )$ and $( 7 , 3 )$ are not on any 2-length path to vertex 1, they are not covered by this filter. Further, when this 2-size filter moves to vertex 2, paths $( 1 , 5 )$ , $( 4 , 5 )$ and $( 6 , 5 )$ are no longer covered, but paths $( 2 , 3 )$ and $( 7 , 3 )$ are first time covered and contribute to the convolution with output at vertex 2. ",
+        "bbox": [
+            173,
+            422,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Further, ${ \\bf y } _ { f } ^ { ( \\ell ) } ( i )$ is the weighted sum of the input features of vertices in $\\mathbf { x } _ { c } ^ { ( \\ell ) }$ that are within $k$ -paths away to vertex $i$ , for $k = 0 , 1 , \\ldots K$ , with weights given by the products of components of $\\mathbf { A } ^ { k }$ and $g _ { c , f , k } ^ { ( \\ell ) ^ { \\top } }$ . Thus the output is the weighted sum of the feature map given by the filtered results from 1-size up to $K$ -size filters. It is evident that the vertex convolution on the graph using $K$ th order polynomials is $K$ -paths localized. Moreover, different vertices on the graph share $g _ { c , f , k } ^ { ( \\ell ) }$ . The above local convolution and weight sharing properties of the convolution (3) on a graph are very similar to those in traditional CNN. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Though the convolution operator defined in (2) is defined on the vertex domain, it can also be understood as a filter in the spectrum domain, and it is consistent with the definition of convolution in graph signal processing. We provide detailed discussion in the appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 COMPUTATION COMPLEXITY OF THE CONVOLUTION OPERATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            710,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Equation (7) shows that TAGCN is equivalent to performing graph convolution either on the vertex domain or on the spectrum domain. We next show that it is preferred to implement TAGCN in the vertex domain. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To obtain F, $\\mathbf { F } ^ { - 1 }$ and J, the eigendecomposition for a diagonalizable matrix is in general $\\mathcal { O } ( N ^ { 3 } )$ . If the graph is directed, A may only be block diagonalizable, needing the Jordan decomposition, which is sensitive to round-off errors. Additional matrix and vector multiplications may be needed to project the filtered out results from the spectrum domain back to the vertex domain. In contrast, filtering directly in the vertex domain that involves powers of the graph adjacency matrix with $N$ vertices and $M$ edges can be computed in $\\mathcal { O } ( M N )$ by performing a breadth-first search starting from each vertex to determine the distances to all other vertices Hammack et al. (2011). For a sparse graph, which is often the case in practical problems, $M \\ll N$ . Since the motivation for performing graph convolution is to detect local features for graph-structured data, the optimal $K$ usually is very small. The experiments section shows that a filter size $K = 2$ leads to the best performance is selected for the data sets and specific tasks tested. In summary, using polynomial filters in the vertex domain rather than in the spectrum domain saves significantly computational costs. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/25af37966a6ee9d9d3d18b6528a58b0bb43ad5dd6fa24bee254ee0eacd8239b5.jpg",
+        "image_caption": [
+            "Figure 2: An example of a directed graph with weights along directed edges corresponding to A. The parts with glow on left(right)-hand-side represent filters at different locations. The figure on the left-hand-side denotes the filtering/convolution starting from vertex 1, then the filter slides to vertex 2 as shown on the right-hand-side with filter topology adaptive to the new local region. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 RELATION WITH OTHER EXISTING FORMULATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show the connections and differences between the proposed TAGCN and existing methods. In general, there are two types of graph convolution operators for the CNN architecture. One defines the convolution in the spectrum domain, whose output feature map is the multiplication of the inverse Fourier transform matrix with the filtered results in the spectrum domain (Bruna et al., 2014; Defferrard et al., 2016; Levie et al., 2017). By doing further approximations based on this spectrum domain operator, a simplified convolution was obtained in Kipf & Welling (2017). The other defines convolution by a feature propagation model in the vertex domain such as MoNet in Monti et al. (2017) and the diffusion CNN (DCNN) in Atwood & Towsley (2016). We investigate in detail each alternative. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Bruna et al. (2014); Defferrard et al. (2016); Levie et al. (2017), the convolution operation was defined using the convolution theorem and filtering operation in the spectrum domain by computing the eigendecomposition of the normalized Laplacian matrix of the graph. The Laplacian matrix $\\mathbf { L }$ is defined as $\\mathbf { L } = \\mathbf { D } - \\mathbf { A }$ with the further assumption that $\\mathbf { A }$ is symmetric to guarantee that $\\mathbf { L }$ is positive semi-definite. The convolution defined by the multiplication in the spectrum domain is approximated by Defferrard et al. (2016) by ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02545a6e800f25022b99f167273700008a7eb0af8ff563e36de9a21930d22c27.jpg",
+        "text": "$$\n\\mathbf { U } g _ { \\theta } \\mathbf { U } ^ { T } \\mathbf { x } \\approx \\sum _ { k = 0 } ^ { K } \\theta _ { k } T _ { k } \\left[ \\frac { 2 } { \\lambda _ { \\operatorname* { m a x } } } \\mathbf { L } - \\mathbf { I } \\right] \\mathbf { x } _ { c } ^ { ( \\ell ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            700,
+            630,
+            744
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $T _ { k } \\left[ \\cdot \\right]$ is the $k$ th order matrix Chebyshev polynomial (Shuman et al., 2013) where ",
+        "bbox": [
+            171,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3981517aa5f3d948cd16c6eab8c12453d09f201851f0732bb96327f4ccc7fabd.jpg",
+        "text": "$$\nT _ { k } ( { \\bf L } ) = 2 { \\bf L } T _ { k - 1 } [ { \\bf L } ] - T _ { k - 2 } [ { \\bf L } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            768,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "with the initial values defined as $T _ { 0 } [ \\mathbf { L } ] = \\mathbf { I }$ and $T _ { 1 } \\left[ \\mathbf { L } \\right] = \\mathbf { L }$ . We refer later to this method as ChebNet for performance comparison. Note that the Laplacian matrix can be seen as a differentiator operator. The assumption of symmetric $\\mathbf { A }$ restricts the application to undirected graphs. Note that in Defferrard et al. (2016), Laplacian matrix polynomials with maximum order $K = 2 5$ is needed to approximate the convolution operation on the left-hand side in (4), which imposes the computational burden. While TAGCN only needs an adjacency matrix polynomials with maximum order 2 to achieve better performance as shown in the experiment part. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Kipf & Welling (2017), a graph convolutional network (GCN) was obtained by a first order approximation of (4). In particular, let $K = 1$ and make the further assumptions that $\\lambda _ { \\operatorname* { m a x } } = 2$ and $\\theta _ { 0 } = \\theta _ { 1 } = \\theta$ . Then a simpler convolution operator that does not depend on the spectrum knowledge is obtained as ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b4859a4f7385e8c17807993b9d46f9c1c07b5e52f469f6e70f11590720b25ac.jpg",
+        "text": "$$\n\\mathbf { U } g _ { \\theta } \\mathbf { U } ^ { T } \\mathbf { x } \\approx \\sum _ { k = 0 } ^ { 1 } \\theta _ { k } T _ { k } \\left[ \\frac { 2 } { \\lambda _ { \\operatorname* { m a x } } } \\mathbf { L } - \\mathbf { I } \\right] \\mathbf { x } _ { c } ^ { ( \\ell ) } \\approx \\theta ( \\mathbf { I } + \\mathbf { D } ^ { - \\frac { 1 } { 2 } } \\mathbf { A } \\mathbf { D } ^ { - \\frac { 1 } { 2 } } ) \\mathbf { x } _ { c } ^ { ( \\ell ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Note that $\\mathbf { I } + \\mathbf { D } ^ { - \\frac { 1 } { 2 } } \\mathbf { A } \\mathbf { D } ^ { - \\frac { 1 } { 2 } }$ is a matrix with eigenvalues in $[ 0 , 2 ]$ . A renormalization trick is adopted here by letting $\\widetilde { \\mathbf { A } } = \\mathbf { A } + \\mathbf { I }$ and $\\begin{array} { r } { \\widetilde { \\bf D } _ { i , i } = \\sum _ { j } \\widetilde { \\bf A } _ { i , j } } \\end{array}$ . Finally, the convolutional operator is approximated by ",
+        "bbox": [
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+            238
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82a04b6d0355dc79cccd7f66e9e798cc4d5f0b5204bf28671172f0c1c7741140.jpg",
+        "text": "$$\n\\mathbf { U } g _ { \\theta } \\mathbf { U } ^ { T } \\mathbf { x } \\approx \\theta \\mathbf { \\widetilde { D } } ^ { - \\frac { 1 } { 2 } } \\mathbf { \\widetilde { A } } \\mathbf { \\widetilde { D } } ^ { - \\frac { 1 } { 2 } } \\mathbf { x } _ { c } ^ { ( \\ell ) } = \\theta \\mathbf { \\widehat { A } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            238,
+            617,
+            262
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\widehat { \\mathbf { A } } = \\widetilde { \\mathbf { D } } ^ { - \\frac { 1 } { 2 } } \\widetilde { \\mathbf { A } } \\widetilde { \\mathbf { D } } ^ { - \\frac { 1 } { 2 } }$ e e . It is interesting to observe that this method though obtained by simplifying the spectrum method has a better performance than the spectrum method (Defferrard et al., 2016). The reason may be because the simplified form is equivalent to propagating vertex features on the graph, which can be seen as a special case of our TAGCN method, though there are other important differences. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Though our TAGCN is able to leverage information at a farther distance, it is not a simple extension of GCN Kipf & Welling (2017). First, the graph convolution in GCN is defined as a first order Chebyshev polynomial of the graph Laplacian matrix, which is an approximation to the graph convolution defined in the spectrum domain in Defferrard et al. (2016). In contrast, our graph convolution is rigorously defined as multiplication by polynomials of the graph adjacency matrix; this is not an approximation, rather, it simply is filtering with graph filters as defined and as being consistent with graph signal processing. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we show the difference between our work and the GCN method in Kipf & Welling (2017) when using 2nd order $\\mathrm { K } { = } 2$ , 2 steps away from the central node) Chebyshev polynomials of Laplacian matrix. In the GCN paper Kipf & Welling (2017), it has been shown that $\\begin{array} { r } { \\sum _ { k = 0 } ^ { 1 } \\theta _ { k } T _ { k } ( { \\bf L } ) \\approx \\widehat { \\bf A } } \\end{array}$ as repeated in (6), and $T _ { 2 } [ \\mathbf { L } ] = 2 \\mathbf { L } ^ { 2 }$ by the definition of Chebyshev polynomial. Then, extending GCN to the second order Chebyshev polynomials (two steps away from a central node) can be obtained from the original definition in T. Kipfs GCN (Kipf & Welling, 2017, eqn (5)) as $\\begin{array} { r } { \\sum _ { k = 0 } ^ { 2 } \\theta T _ { k } ( L ) = \\widehat { \\mathbf { A } } + 2 \\bar { \\mathbf { L } ^ { 2 } } - \\mathbf { I } } \\end{array}$ , which is different from our definition as in (2). Thus, it is evident that our method is not a simple extension of GCN. We apply graph convolution as proposed from basic principles in the graph signal processing, with no approximations involved, while both GCN in Kipf & Welling (2017) and Defferrard et al. (2016) Levie et al. (2017) are based on approximating the convolution defined in the spectrum domain. In our approach, the degree of freedom is the design of the graph filter-its degree and its coefficients. Ours is a principled approach and provides a generic methodology. The performance gains we obtain are the result of capturing the underlying graph structure with no approximation in the convolution operation. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Simonovsky & Komodakis (2017) proposed the edge convolution network (ECC) to extend the convolution operator from regular grids to arbitrary graphs. The convolution operator is defined similarly to (6) as ",
+        "bbox": [
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+            662,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35e8510a8c80a1e78b659b005da2f568f1f27139d90d8b0f2701bc26948d9eaa.jpg",
+        "text": "$$\n\\mathbf { y } _ { f } ^ { ( \\ell ) } ( i ) = \\sum _ { j \\in \\mathcal { N } ( i ) } \\frac { 1 } { | \\mathcal { N } ( i ) | } \\Theta _ { j , i } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) } ( j ) + b _ { f } ^ { ( \\ell ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "$\\Theta _ { j , i } ^ { ( \\ell ) }$ ",
+        "bbox": [
+            173,
+            751,
+            537,
+            768
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A mixture model network (MoNet) was proposed in Monti et al. (2017), with convolution defined as ",
+        "bbox": [
+            173,
+            773,
+            823,
+            801
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14dfbb8896473a580c1c320db444beb9ec7b1172ff8dd717012718424b1930df.jpg",
+        "text": "$$\n\\mathbf { y } _ { f } ^ { ( \\ell ) } ( i ) = \\sum _ { f = 1 } ^ { F } \\sum _ { j \\in \\mathcal { N } ( i ) } g _ { f } \\kappa _ { f } \\mathbf { x } _ { c } ^ { ( \\ell ) } ( j ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            799,
+            609,
+            845
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\kappa _ { f }$ is a Gaussian kernel with $\\begin{array} { r } { \\kappa _ { f } = \\exp \\left\\{ - \\frac { 1 } { 2 } ( \\mathbf { u } - \\pmb { \\mu } _ { f } ) ^ { T } \\pmb { \\Sigma } _ { f } ^ { - 1 } ( \\mathbf { u } - \\pmb { \\mu } _ { f } ) \\right\\} } \\end{array}$ and $g _ { f }$ is the weight coefficient for each Gaussian kernel $\\kappa _ { f }$ . It is further assumed that $\\Sigma _ { f }$ is a $2 \\times 2$ diagonal matrix. ",
+        "bbox": [
+            173,
+            852,
+            823,
+            890
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "GCN, ECC, and MoNet all design a propagation model on the graph; their differences are on the weightings used by each model. ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/c55f8aacae810073b1931d099addcfe28ac05dd83ad3d44eaf28d6e956871117.jpg",
+        "table_caption": [
+            "Table 1: Summary of number of weights need to be learned for the $\\ell$ -th layer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DCNN</td><td>ECC</td><td>ChebNet</td><td>GCN</td><td>MoNet</td><td>TAGCN</td></tr><tr><td>FeCe</td><td>FeCe</td><td>25FeCe</td><td>FeCe</td><td>4FeCe</td><td>2FeCe</td></tr></table>",
+        "bbox": [
+            295,
+            131,
+            699,
+            179
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Atwood & Towsley (2016) proposed a diffusion CNN (DCNN) method that considers a diffusion process over the graph. The transition probability of a random walk on a graph is given by $\\mathbf { P } =$ $\\mathbf { b } ^ { - 1 } \\mathbf { A }$ , which is equivalent to the normalized adjacency matrix. ",
+        "bbox": [
+            176,
+            222,
+            823,
+            263
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/546c2bc4eb34a279e67876eb615c142441698a7699a5eaa2d6559c45dcfa3366.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { y } _ { c , f } ^ { ( \\ell ) } = \\mathbf { g } _ { c , f } ^ { ( \\ell ) } \\mathbf { P } ^ { f } \\mathbf { x } _ { c } ^ { ( \\ell ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            284,
+            562,
+            306
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "By comparing the above methods with TAGCN in (3), it can be concluded that GCN, ECC and MoNet can be seen as a special case of TAGCN because in (3) the item with $k = 1$ can be seen as an information propagation term. However, the strength of the correlation between vertices that are taken into account in $\\omega ( p _ { j , i } ^ { k } )$ in (3) can not be utilized in the GCN, ECC and MoNet methods. Further, the TAGCN method is a systematic way to design a set of fixed size filters that is adaptive to the input graph topology when performing convolution on the graph. Compared with existing spectrum methods (Bruna et al., 2014; Defferrard et al., 2016; Levie et al., 2017), TAGCN satisfies the convolution theorem as shown in the previous subsection and implemented in the vertex domain, which avoids performing costly and practically numerical unstable eigendecompositions. ",
+        "bbox": [
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+            332,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further compare the number of weights that need to be learned in each hidden layer for these different methods in Table 1. As we show later, $K = 2$ is selected in our experiments using cross validation. However, for ChebNet in (Defferrard et al., 2016), it is suggested that one needs a $2 5 ^ { \\mathrm { t h } }$ degree Chebyshev polynomial to provide a good approximation to the graph Laplacian spectrum. Thus we have a moderate number of weights to be learned. In the following, we show that our method achieves the best performance for each of those commonly used graph-structured data sets. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            583,
+            326,
+            599
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The proposed TAGCN is general and can be fit to the general graph CNN architectures for different tasks. In the experiments, we focus on the vertex semisupervised learning problem, where we have access to only a few labeled vertices, and the task is to classify the remaining unlabeled vertices. To compare the performance of TAGCN with that of existing methods, we extensively evaluate TAGCN on three graph-structured datasets, including the Cora, Citesser and Pubmed datasets. The datasets split and experiments setting closely follow the standard criteria in Yang et al. (2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            739,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "TAGCN is evaluated on three data sets coming from citation networks: Citeseer, Cora, and Pubmed. We closely follow the data set split and experimental setup in Yang et al. (2016). Each data set consists of a certain classes of documents, yet only a few documents are labeled. The task is to classify the documents in the test set with these limited number of labels. In each data set, the vertices are the documents and the edges are the citation links. Each document is represented by sparse bagof-words feature vectors, and the citation links between documents are provided. Detailed statistics of these three data sets are summarized in Table 2. It shows the number of nodes and edges that corresponding to documents and citation links. Classes for the graph vertices shows the number of document classes in each data set. Also, the number of features at each vertex is given. Label rate denotes the number of labeled documents that are used for training divided by the total number of documents in each data set. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4a6a9bf868f6485138e51d1fac70f174150b72a732ce9453adab350edcfc81da.jpg",
+        "table_caption": [
+            "Table 2: Dataset statistics, following Yang et al. (2016); Kipf & Welling (2017) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset Size</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>Label rate</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Citeseer</td><td>3,327</td><td>4,732</td><td>6</td><td>3,703</td><td>0.036</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>0.052</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>0.003</td></tr></table>",
+        "bbox": [
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+            131,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/41eecc35ff7b73c43dbce9491df7314e348d7562b1c8995b6d00b118fe8eae8e.jpg",
+        "table_caption": [
+            "Table 3: Summary of results in terms of percentage classification accuracy with standard variance "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Citeseer</td><td>Cora</td><td>Pubmed</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Planetoid 64.7</td><td>75.7</td><td>77.2</td></tr><tr><td>DCNN</td><td>76.8±0.6</td><td>73.0±0.5</td></tr><tr><td>ChebNet (K=2) 69.6</td><td>81.2</td><td>73.8</td></tr><tr><td>ChebNet (K=3) 69.8</td><td>79.5</td><td>74.4</td></tr><tr><td>GCN 70.3</td><td>81.5</td><td>79.0</td></tr><tr><td>MoNet =</td><td>81.69±0.5</td><td>78.81±0.4</td></tr><tr><td>TAGCN (K=2)</td><td>70.9± 0.9 82.5±0.7</td><td>81.1±0.4</td></tr></table>",
+        "bbox": [
+            305,
+            256,
+            692,
+            386
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We construct a graph for each data set with nodes representing documents and undirected edges4 linking two papers if there is a citation relationship. We obtain the adjacency matrix $\\bar { \\bf A }$ with 0 and 1 components and further obtain the normalized matrix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the following experiments, we design a TAGCN with two hidden layers (obtained from cross validation) for the semi-supervised node classification. In each hidden layer, the proposed TAGCN is applied for convolution, followed by a ReLU activation. 16 hidden units (filters) are designed for each hidden layer, and dropout is applied after each hidden layer. The softmax activation function is applied to the output of the second hidden layer for the final classification. For ablation study, we evaluate the performance of TAGCN with different kernel size from 1 to 4. To investigate the performance for different number of parameters, we also design a TAGCN with 8 filters for each hidden layer and compare its classification accuracy with all the baselines and TAGCN with 16 filters. We train our model using Adam (Kinga & Ba, 2015) with a learning rate of 0.01 and early stopping with a window size of 20. Hyperparameters of the networks (filter size, dropout rate, and number of hidden layers) are selected by cross validation. ",
+        "bbox": [
+            173,
+            496,
+            825,
+            648
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To make a fair comparison, we closely follow the same split of training, validation, and testing sets as in Kipf & Welling (2017); Yang et al. (2016), i.e., 500 labeled examples for hyperparameters (filter size, dropout rate, and number of hidden layers) optimization and cross-entropy error is used for classification accuracy evaluation. The performance results of the proposed TAGCN method are an average over 100 runs with random initializations (Glorot & Bengio, 2010). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 QUANTITATIVE EVALUATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            746,
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+            758
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We compare the classification accuracy with other recently proposed graph CNN methods as well as a graph embedding method known as Planetoid Yang et al. (2016). The quantitative results are summarized in Table 3. Reported numbers denote classification accuracy in percent. Results for Planetoid, GCN, and ChebNet are taken from Kipf & Welling (2017), and results for DCNN and MoNet are taken from Monti et al. (2017). All the experiments for different methods are based on the same data statistics shown in Table 2. The datasets split and experiments settings closely follow the standard criteria in Yang et al. (2016); Kipf & Welling (2017). Table 3 shows that our method ",
+        "bbox": [
+            174,
+            772,
+            825,
+            869
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ceaed514cb724ab1ef1f790c27055ec60658c1f25b8e53a947dffd7aa7e3d1ac.jpg",
+        "table_caption": [
+            "Table 4: TAGCN classification accuracy (ACC) with different parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Data Set</td><td>Filter Size</td><td>Filter Number</td><td>ACC</td></tr><tr><td rowspan=\"5\">Citeseer</td><td>1</td><td>16</td><td>68.9</td></tr><tr><td>2</td><td>16</td><td>70.9</td></tr><tr><td>3</td><td>16</td><td>70.0</td></tr><tr><td>4</td><td>16</td><td>69.8</td></tr><tr><td>2</td><td>8</td><td>70.6</td></tr><tr><td rowspan=\"6\">Cora</td><td>1</td><td>16</td><td>81.4</td></tr><tr><td>2</td><td>16</td><td>82.5</td></tr><tr><td>3</td><td>16</td><td>82.1</td></tr><tr><td>4</td><td>16</td><td>81.8</td></tr><tr><td>2</td><td>8</td><td>82.5</td></tr><tr><td>1</td><td>16</td><td>79.4</td></tr><tr><td rowspan=\"4\">Pubmed</td><td>2</td><td>16</td><td>81.1</td></tr><tr><td>3</td><td>16</td><td>80.9</td></tr><tr><td>4</td><td>16</td><td>79.5</td></tr><tr><td>2</td><td>8</td><td>80.8</td></tr></table>",
+        "bbox": [
+            325,
+            131,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "outperforms all the recent state-of-the-art methods by obvious margins for all the three datasets.   \nThese results verify the efficacy of the proposed TAGCN. ",
+        "bbox": [
+            174,
+            411,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For ablation study, we further compare the performance of different filter sizes from $K = 1$ to $K = 4$ in Table 4. It shows that the performances for filter size $K = 2$ are always better than that for other filter sizes. The value $K = 1$ gives the worst classification accuracy. This further validates that a local feature extraction is more important than just propagating features among neighbors on the graph. In Table 4, we also compare the performance of different number of filters, which reflects different number of network parameters. Note, we also choose filer size $K = 2$ and filter number $F _ { \\ell } = 8$ that results in the same number of network parameters as that in GCN, MoNet, ECC and DCNN according to Table 1. It shows that the classification accuracy using 8 filters is comparable with that using 16 filters in each hidden layer for TAGCN. Moreover, TAGCN with 8 filters can still achieve higher accuracy than GCN, MoNet, ECC and DCNN methods. This proves that, even with a similar number of parameters or architecture, our method still exhibits superior performance than GCN. ",
+        "bbox": [
+            174,
+            446,
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+            613
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As we explained in Section 3, TAGCN in our paper is not simply extending GCN Kipf & Welling (2017) to $k$ -th order. Nevertheless, we implement ${ \\bf A } ^ { 2 }$ and compare its performance with ours. For the data sets Pubmed, Cora, and Citeseer, the classification accuracies are 79.1(81.1), 81.7(82.5) and 70.8(70.9), where the numbers in parentheses are the results obtained with our method. Our method still achieves a noticeable performance advantage over ${ \\bf A } ^ { 2 }$ for the Pubmed and Cora data; in particular, we note the significant performance gain with the Pubmed database that has the largest number of nodes among these three data sets. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            748,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have defined a novel graph convolutional network that rearchitects the CNN architecture for graph-structured data. The proposed method, known as TAGCN, is adaptive to the graph topology as the filter scans the graph. Further, TAGCN inherits properties of the convolutional layer in classical CNN, i.e., local feature extraction and weight sharing. It can further extract the strength of correlation between vertices in the filtering region. On the other hand, by the convolution theorem, TAGCN that implements in the vertex domain offers implement in the spectrum domain unifying graph CNN in both the spectrum domain and the vertex domain. TAGCN is consistent with convolution in graph signal processing. These nice properties lead to a noticeable performance advantage in classification accuracy on different graph-structured datasets for semi-supervised graph vertex classification problems with low computational complexity. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX: SPECTRUM RESPONSE OF TAGCN ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In classical signal processing (Oppenheim & Schafer, 2009), the convolution in the time domain is equivalent to multiplication in the spectrum domain. This relationship is known as the convolution theorem. Sandryhaila & Moura (2013) showed that the graph filtering defined in the vertex domain satisfies the generalized convolution theorem naturally and can also interpret spectrum filtering for both directed and undirected graphs. Recent work (Bruna et al., 2014; Defferrard et al., 2016) used the convolution theorem for undirected graph-structured data and designed a spectrum graph filtering. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/1523d9fd4c75a2486b139f369d3766910ea55f641bd451f66d8d026e7b489b75.jpg",
+        "image_caption": [
+            "Figure 3: Graph topology of a 1-D cyclic graph. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            366,
+            246,
+            629,
+            311
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Assume that the adjacency matrix A for a graph is diagonalizable, i.e., $\\mathbf { A } = \\mathbf { F } ^ { - 1 } \\mathbf { J } \\mathbf { F }$ with $\\mathbf { J }$ a diagonal matrix. The components on the diagonal of $\\mathbf { J }$ are eigenvalues of $\\mathbf { A }$ , and the column vectors of $\\mathbf { F } ^ { - 1 }$ are the right eigenvectors of $\\mathbf { A }$ ; the row vectors of $\\mathbf { F }$ are the left eigenvectors of A 5. By diagonalizing $\\mathbf { A }$ in (2) for TAGCN, we obtain ",
+        "bbox": [
+            174,
+            359,
+            825,
+            416
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cfa4df7ac0a0f548350db8439975b5665a5838f2f87a95596ed3e4c430cccb56.jpg",
+        "text": "$$\n\\mathbf { G } _ { c , f } ^ { ( \\ell ) } \\mathbf { x } _ { c } ^ { ( \\ell ) } = \\mathbf { F } ^ { - 1 } \\left( \\sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \\ell ) } \\mathbf { J } ^ { k } \\right) \\mathbf { F } \\mathbf { x } _ { c } ^ { ( \\ell ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            422,
+            627,
+            467
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The expression on the left-hand-side of the above equation represents the filtering/convolution on the vertex domain. Matrix $\\mathbf { F }$ defines the graph Fourier transform (Sandryhaila $\\&$ Moura, 2013; 2014), and $\\mathbf { F } \\mathbf { x } _ { c } ^ { ( \\ell ) }$ is the input feature spectrum map, which is a linear mapping from the input feature on the vertex domain to the spectrum domain. The polynomial $\\scriptstyle \\sum _ { k = 0 } ^ { K } g _ { c , f , k } ^ { ( \\ell ) } \\mathbf { J } ^ { k }$ is the spectrum of the graph filter. Relation (7), which is equation (27) in Sandryhaila & Moura (2013) generalizes the classical convolution theorem to graph-structured data: convolution/filtering on the vertex domain becomes multiplication in the spectrum domain. When the graph is in the 1D cyclic form, as shown in Fig. 3, the corresponding adjacency matrix is of the form ",
+        "bbox": [
+            173,
+            472,
+            826,
+            592
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/557483d73c2e1d3384739b6aa51e52dc476a62a17467da2607c57ef549ce2237.jpg",
+        "text": "$$\n\\mathbf { A } = \\left[ \\begin{array} { c c c c } { \\vrule } & { } & { } & { 1 } \\\\ { 1 } & { } & { } & { } \\\\ { \\vrule } & { \\ddots } & { } & { } \\\\ { } & { } & { 1 } & { } \\end{array} \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            598,
+            578,
+            666
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The eigendecomposition of $\\mathbf { A }$ is ",
+        "bbox": [
+            174,
+            672,
+            387,
+            688
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/953773be0ddad78473c2e47e9d432971e9692776f20ad692061009bede95805e.jpg",
+        "text": "$$\n\\mathbf { A } = \\frac { 1 } { N } \\mathrm { D F T } ^ { - 1 } \\left[ \\begin{array} { c c } { \\begin{array} { c } { e ^ { - j \\frac { 2 \\pi 0 } { N } } } \\\\ { \\ddots } \\\\ { \\end{array} } } \\\\  \\begin{array} { c } { \\begin{array} { r l } { e ^ { - j \\frac { 2 \\pi ( N - 1 ) } { N } } } \\end{array} } \\end{array} \\right] \\mathrm { D F T } , \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            693,
+            658,
+            755
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where DFT is the discrete Fourier transform matrix. The convolution operator defined in (2) is consistent with that in classical signal processing. ",
+        "bbox": [
+            174,
+            761,
+            826,
+            790
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+]

parse/train/HkfXMz-Ab/HkfXMz-Ab.md

@@ -0,0 +1,388 @@
+# NEURAL SKETCH LEARNING FOR CONDITIONAL PROGRAM GENERATION
+
+Vijayaraghavan Murali, Letao Qi, Swarat Chaudhuri, and Chris Jermaine Department of Computer Science   
+Rice University   
+Houston, TX 77005, USA.   
+{vijay, letao.qi, swarat, cmj4}@rice.edu
+
+# ABSTRACT
+
+We study the problem of generating source code in a strongly typed, Java-like programming language, given a label (for example a set of API calls or types) carrying a small amount of information about the code that is desired. The generated programs are expected to respect a “realistic” relationship between programs and labels, as exemplified by a corpus of labeled programs available during training.
+
+Two challenges in such conditional program generation are that the generated programs must satisfy a rich set of syntactic and semantic constraints, and that source code contains many low-level features that impede learning. We address these problems by training a neural generator not on code but on program sketches, or models of program syntax that abstract out names and operations that do not generalize across programs. During generation, we infer a posterior distribution over sketches, then concretize samples from this distribution into type-safe programs using combinatorial techniques. We implement our ideas in a system for generating API-heavy Java code, and show that it can often predict the entire body of a method given just a few API calls or data types that appear in the method.
+
+# 1 INTRODUCTION
+
+Neural networks have been successfully applied to many generative modeling tasks in the recent past (Oord et al., 2016; Ha & Eck, 2017; Vinyals et al., 2015). However, the use of these models in generating highly structured text remains relatively understudied. In this paper, we present a method, combining neural and combinatorial techniques, for the condition generation of an important category of such text: the source code of programs in Java-like programming languages.
+
+The specific problem we consider is one of supervised learning. During training, we are given a set of programs, each program annotated with a label, which may contain information such as the set of API calls or the types used in the code. Our goal is to learn a function $g$ such that for a test case of the form $( \mathsf { X } , \mathsf { P r o g } )$ (where Prog is a program and $\mathsf { X }$ is a label), $g ( { \sf X } )$ is a compilable, type-safe program that is equivalent to Prog.
+
+This problem has immediate applications in helping humans solve programming tasks (Hindle et al., 2012; Raychev et al., 2014). In the usage scenario that we envision, a human programmer uses a label to specify a small amount of information about a program that they have in mind. Based on this information, our generator seeks to produce a program equivalent to the “target” program, thus performing a particularly powerful form of code completion.
+
+Conditional program generation is a special case of program synthesis (Manna & Waldinger, 1971; Summers, 1977), the classic problem of generating a program given a constraint on its behavior. This problem has received significant interest in recent years (Alur et al., 2013; Gulwani et al., 2017). In particular, several neural approaches to program synthesis driven by input-output examples have emerged (Balog et al., 2017; Parisotto et al., 2016; Devlin et al., 2017). Fundamentally, these approaches are tasked with associating a program’s syntax with its semantics. As doing so in general is extremely hard, these methods choose to only generate programs in highly controlled domainspecific languages. For example, Balog et al. (2017) consider a functional language in which the only data types permitted are integers and integer arrays, control flow is linear, and there is a sum total of 15 library functions. Given a set of input-output examples, their method predicts a vector of binary attributes indicating the presence or absence of various tokens (library functions) in the target program, and uses this prediction to guide a combinatorial search for programs.
+
+In contrast, in conditional program generation, we are already given a set of tokens (for example library functions or types) that appear in a program or its metadata. Thus, we sidestep the problem of learning the semantics of the programming language from data. We ask: does this simpler setting permit the generation of programs from a much richer, Java-like language, with one has thousands of data types and API methods, rich control flow and exception handling, and a strong type system?
+
+While simpler than general program synthesis, this problem is still highly nontrivial. Perhaps the central issue is that to be acceptable to a compiler, a generated program must satisfy a rich set of structural and semantic constraints such as “do not use undeclared variables as arguments to a procedure call” or “only use API calls and variables in a type-safe way”. Learning such constraints automatically from data is hard. Moreover, as this is also a supervised learning problem, the generated programs also have to follow the patterns in the data while satisfying these constraints.
+
+We approach this problem with a combination of neural learning and type-guided combinatorial search (Feser et al., 2015). Our central idea is to learn not over source code, but over tree-structured syntactic models, or sketches, of programs. A sketch abstracts out low-level names and operations from a program, but retains information about the program’s control structure, the orders in which it invokes API methods, and the types of arguments and return values of these methods. We propose a particular kind of probabilistic encoder-decoder, called a Gaussian Encoder-Decoder or GED, to learn a distribution over sketches conditioned on labels. During synthesis, we sample sketches from this distribution, then flesh out these samples into type-safe programs using a combinatorial method for program synthesis. Doing so effectively is possible because our sketches are designed to contain rich information about control flow and types.
+
+We have implemented our approach in a system called BAYOU.1 We evaluate BAYOU in the generation of API-manipulating Android methods, using a corpus of about 150,000 methods drawn from an online repository. Our experiments show that BAYOU can often generate complex method bodies, including methods implementing tasks not encountered during training, given a few tokens as input.
+
+# 2 PROBLEM STATEMENT
+
+Now we define conditional program generation. Assume a universe $\mathbb { P }$ of programs and a universe $\mathbb { X }$ of labels. Also assume a set of training examples of the form $\{ ( \mathsf { X } _ { 1 } , \mathsf { P r o g } _ { 1 } ) , ( \mathsf { X } _ { 2 } , \mathsf { P r o g } _ { 2 } ) , . . . \}$ , where each ${ \sf X } _ { i }$ is a label and each $\mathsf { P r o g } _ { i }$ is a program. These examples are sampled from an unknown distribution $Q ( X , P r o g )$ , where $X$ and Prog range over labels and programs, respectively.2
+
+We assume an equivalence relation $E q v \subseteq \mathbb { P } \times \mathbb { P }$ over programs. If $( \mathsf { P r o g } _ { 1 } , \mathsf { P r o g } _ { 2 } ) \in E q v$ , then $\mathsf { P r o g } _ { 1 }$ and $\mathsf { P r o g } _ { 2 }$ are functionally equivalent. The definition of functional equivalence differs across applications, but in general it asserts that two programs are “just as good as” one another.
+
+The goal of conditional program generation is to use the training set to learn a function $g : \mathbb { X } \to \mathbb { P }$ such that the expected value $\mathbf { E } [ I ( ( g ( X ) , P r o g ) \in E q v ) ]$ is maximized. Here, $I$ is the indicator function, returning 1 if its boolean argument is true, and 0 otherwise. Informally, we are attempting to learn a function $g$ such that if we sample $( \mathsf { X } , \mathsf { P r o g } ) \sim Q ( \boldsymbol { X } , \boldsymbol { P r o g } ) ,$ , $g$ should be able to reconstitute a program that is functionally equivalent to Prog, using only the label $\mathsf { X }$ .
+
+# 2.1 INSTANTIATION
+
+In this paper, we consider a particular form of conditional program generation. We take the domain $\mathbb { P }$ to be the set of possible programs in a programming language called AML that captures the essence of API-heavy Java programs (see Appendix A for more details). AML includes complex control flow such as loops, if-then statements, and exceptions; access to Java API data types; and calls to Java API methods. AML is a strongly typed language, and by definition, $\mathbb { P }$ only includes programs that are type-safe.3 To define labels, we assume three finite sets: a set Calls of possible API calls in AML, a set Types of possible object types, and a set Keys of keywords, defined as words, such as “read” and “file”, that often appear in textual descriptions of what programs do. The space of possible labels is $\mathbb { X } = 2 ^ { C a l l s } \times 2 ^ { \dot { T } y p e s } \times 2 ^ { K e y s }$ (here $2 ^ { \hat { S } }$ is the power set of $S$ ).
+
+![](images/e7ea069f757c2db638f66a0e2a1f1f0115929bb8e347d8c3be550436b40a0459.jpg)  
+Figure 1: Programs generated by BAYOU with the API method name readLine as a label. Names of variables of type $\mathrm { T }$ whose values are obtained from the environment are of the form $\$ 1$ .
+
+Defining Eqv in practice is tricky. For example, a reasonable definition of Eqv is that $( \mathsf { P r o g _ { 1 } } , \mathsf { P r o g _ { 2 } } ) \in E q v$ iff $\mathsf { P r o g } _ { 1 }$ and $\mathsf { P r o g } _ { 2 }$ produce the same outputs on all inputs. But given the richness of AML, the problem of determining whether two AML programs always produce the same output is undecidable. As such, in practice we can only measure success indirectly, by checking whether the programs use the same control structures, and whether they can produce the same API call sequences. We will discuss this issue more in Section 6.
+
+# 2.2 EXAMPLE
+
+Consider the label ${ \sf X } = ( { \sf X } _ { C a l l s } , { \sf X } _ { T y p e s } , { \sf X } _ { K e y s } )$ where $\mathsf { X } _ { C a l l s } = \{ \tt r e a d L i n e \}$ and $\mathsf { X } _ { T y p e s }$ and $\mathsf { X } _ { K e y s }$ are empty. Figure 1(a) shows a program that our best learner stochastically returns given this input. As we see, this program indeed reads lines from a file, whose name is given by a special variable $\$ 5$ tring that the code takes as input. It also handles exceptions and closes the reader, even though these actions were not directly specified.
+
+Although the program in Figure 1-(a) matches the label well, failures do occur. Sometimes, the system generates a program as in Figure 1-(b), which uses an InputStreamReader rather than a FileReader. It is possible to rule out this program by adding to the label. Suppose we amend $\mathsf { X } _ { T y p e s }$ so that $\mathsf { X } _ { T y p e s } = \mathsf { \{ F i l e R e a d e r \} }$ . BAYOU now tends to only generate programs that use FileReader. The variations then arise from different ways of handling exceptions and constructing FileReader objects (some programs use a String argument, while others use a File object). Figure 7 in the appendix shows two other top-five programs returned on this input.
+
+# 3 TECHNICAL APPROACH
+
+![](images/1fc00e281a931b9e0ffdefcdb26fae7f2e1ce28cc6157456e03e2aadafb55831.jpg)  
+Figure 2: Bayes net for Prog , X, Y
+
+Our approach is to learn $g$ via maximum conditional likelihood estimation (CLE). That is, given a distribution family $P ( P r o g | X , \theta )$ for a parameter set $\theta$ , we choose $\begin{array} { r } { \theta ^ { * } = \arg \operatorname* { m a x } _ { \theta } \sum _ { i } \log P ( \mathsf { P r o g } _ { i } \mid \mathsf { X } _ { i } , \theta ) } \end{array}$ . Then, $g ( \mathsf { X } ) = \arg \operatorname* { m a x } _ { \mathsf { P r o g } } P ( \mathsf { P r o g } | \mathsf { X } , \theta ^ { * } )$ .
+
+The key innovation of our approach is that here, learning happens at a higher level of abstraction than $( \mathsf { X } _ { i } , \mathsf { P r o g } _ { i } )$ pairs. In practice, Java-like programs contain many low-level details (for example, variable names and intermediate results) that can obscure patterns in code. Further, they contain complicated semantic rules (for example, for type safety) that are difficult to learn from data. In contrast, these are relatively easy for a combinatorial, syntax-guided program synthesizer (Alur et al., 2013) to deal with. However, synthesizers have a notoriously difficult time figuring out the correct “shape” of a program (such as the placement of loops and conditionals), which we hypothesize should be relatively easy for a statistical learner.
+
+Specifically, our approach learns over sketches: tree-structured data that capture key facets of program syntax. A sketch Y does not contain low-level variable names and operations, but carries information about broadly shared facets of programs such as the types and API calls. During generation, a program synthesizer is used to generate programs from sketches produced by the learner.
+
+Let the universe of all sketches be denoted by Y. The sketch for a given program is computed by applying an abstraction function $\alpha : \mathbb { P }  \mathbb { Y }$ . We call a sketch Y satisfiable, and write $s a t ( \mathsf { Y } )$ , if $\mathbf { \bar { \alpha } } \mathbf { \bar { \alpha } } ^ { - 1 } ( \mathbf { \bar { Y } } ) \mathbf { \beta } \neq \mathbf { \alpha } \varnothing$ . The process of generating (type-safe) programs given a satisfiable sketch $\textsf { Y }$ is probabilistic, and captured by a concretization distribution $\hat { P } ( \hat { P } r o g \mid \bar { \mathsf { Y } } , s a t ( \mathsf { Y } ) )$ . We require that for all programs Prog and sketches Y such that $s a t ( \mathsf { Y } )$ , we have $P ( \mathsf { P r o g } \mid \mathsf { Y } ) \neq 0$ only if $\mathsf { Y } = \alpha ( \mathsf { P r o g } )$ .
+
+Importantly, the concretization distribution is fixed and chosen heuristically. The alternative of learning this distribution from source code poses difficulties: a single sketch can correspond to many programs that only differ in superficial details, and deciding which differences between programs are superficial and which are not requires knowledge about program semantics. In contrast, our heuristic approach utilizes known semantic properties of programming languages like ours — for example, that local variable names do not matter, and that some algebraic expressions are semantically equivalent. This knowledge allows us to limit the set of programs that we generate.
+
+Y ${ \begin{array} { r l } { : : = } & { \mathbf { s k i p } \mid \mathbf { c a l l } \mathbf { C } \mathbf { e x p } \mid \mathsf { Y } _ { 1 } ; \mathsf { Y } _ { 2 } \mid } \\ & { \mathbf { i f } \mathbf { C s e q } \mathbf { t h e n } \textsf { Y } _ { 1 } \mathbf { e l s e } \mathsf { Y } _ { 2 } \mid } \\ & { { \mathrm { ~ w h i l e ~ } } \mathbf { C s e q } \mathbf { d o } \mathsf { Y } _ { 1 } \mid \mathbf { t r y } \mathsf { Y } _ { 1 } \mathbf { \textsf { C a t c t } } } \\ { : = } & { \tau _ { 0 } . a ( \tau _ { 1 } , \ldots , \tau _ { k } ) } \\ { : : = } & { \mathbf { L i s t } { \mathrm { ~ o f ~ } } \mathbf { C } \mathbf { e x p } } \\ { : : = } & { \mathbf { c a t c h } ( \tau _ { 1 } ) \textsf { Y } _ { 1 } \ \ldots \mathbf { c a t c h } ( \tau _ { k } ) \textsf { Y } _ { k } } \end{array} }$ h   
+Cexp   
+Cseq   
+Catch
+
+Let us define a random variable $Y = \alpha ( { \cal P } { r o g } )$ . We assume that the variables $X$ , $Y$ and Prog are related as in the Bayes net in Figure 2. Specifically, given $Y$ , Prog is conditionally independent of $X$ . Further, let us assume a distribution family $P ( \boldsymbol { Y } | \boldsymbol { X } , \boldsymbol { \theta } )$ parameterized on $\theta$ .
+
+Let $\mathsf { Y } _ { i } = \alpha ( \mathsf { P r o g } _ { i } )$ , and note that $P ( \mathsf { P r o g } _ { i } | \mathsf { Y } ) \neq$ 0 only if ${ \textsf { Y } } = { \textsf { Y } } _ { i }$ . Our problem now simplifies to learning over sketches, i.e., finding
+
+$$
+\begin{array} { r c l } { \theta ^ { * } } & { = } & { \underset { \theta } { \arg \operatorname* { m a x } } \sum _ { i } \log \underset { \Upsilon : s a t ( \Upsilon ) } { \sum } P ( \mathsf { P r o g } _ { i } | \mathsf { Y } ) P ( \mathsf { Y } | \mathsf { X } _ { i } , \theta ) } \\ & { = } & { \underset { \theta } { \arg \operatorname* { m a x } } \sum _ { i } \log P ( \mathsf { P r o g } _ { i } | \mathsf { Y } _ { i } ) P ( \mathsf { Y } _ { i } | \mathsf { X } _ { i } , \theta ) = \underset { \theta } { \arg \operatorname* { m a x } } \sum _ { i } \log P ( \mathsf { Y } _ { i } | \mathsf { X } _ { i } , \theta ) . } \end{array}
+$$
+
+# 3.1 INSTANTIATION
+
+Figure 3 shows the full grammar for sketches in our implementation. Here, $\tau _ { 0 } , \tau _ { 1 } , \ldots$ range over a finite set of API data types that AML programs can use. A data type, akin to a Java class, is identified with a finite set of API method names (including constructors), and $a$ ranges over these names. Note that sketches do not contain constants or variable names.
+
+A full definition of the abstraction function for AML appears in Appendix B. As an example, API calls in AML have the syntax “call $e . a ( e _ { 1 } , \ldots , e _ { k } ) ^ { \prime }$ , where $a$ is an API method, the expression $e$ evaluates to the object on which the method is called, and the expressions $e _ { 1 } , \ldots , e _ { k }$ evaluate to the arguments of the method call. We abstract this call into an abstract method call “call $\tau . a ( \tau _ { 1 } , \ldots , \tau _ { k } ) ^ { \ast }$ , where $\tau$ is the type of $e$ and $\tau _ { i }$ is the type of $e _ { i }$ . The keywords skip, while, if-then-else, and trycatch preserve information about control flow and exception handling. Boolean conditions Cseq are replaced by abstract expressions: lists whose elements abstract the API calls in Cseq.
+
+# 4 LEARNING
+
+Now we describe our learning approach. Equation 1 leaves us with the problem of computing arg $\begin{array} { r } { \operatorname* { m a x } _ { \theta } \sum _ { i } \log P ( \Upsilon _ { i } | \Upsilon _ { i } , \theta ) } \end{array}$ , when each $\mathsf { X } _ { i }$ is a label and $\mathsf { Y } _ { i }$ is a sketch. Our answer is to utilize an encoder-decoder and introduce a real vector-valued latent variable $Z$ to stochastically link labels and sketches: $\begin{array} { r } { P ( \Upsilon | \Upsilon , \theta ) = \int _ { \mathsf Z \in \mathbb R ^ { m } } P ( \boldsymbol { \mathrm Z } | \mathsf X , \theta ) P ( \Upsilon | \boldsymbol { \mathrm Z } , \theta ) d \boldsymbol { \mathrm Z } } \end{array}$ .
+
+$P ( \boldsymbol { Y } | \boldsymbol { Z } , \boldsymbol { \theta } )$ is realized as a probabilistic decoder mapping a vector-valued variable to a distribution over trees. We describe this decoder in Appendix C. As for $P ( Z | \mathsf { X } , \theta )$ , this distribution can, in principle, be picked in any way we like. In practice, because both $P ( \boldsymbol { Y } | \boldsymbol { Z } , \boldsymbol { \theta } )$ and $P ( Z | \mathsf { X } , \theta )$ have neural components with numerous parameters, we wish this distribution to regularize the learner. To provide this regularization, we assume a Normal $( \vec { 0 } , \mathbf { I } )$ prior on $Z$ .
+
+Recall that our labels are of the form $\mathsf { X } = ( \mathsf { X } _ { C a l l s } , \mathsf { X } _ { T y p e s } , \mathsf { X } _ { K e y s } )$ , where $\mathsf { X } _ { C a l l s }$ , $\mathsf { X } _ { \substack { T y p e s } }$ , and $\mathsf { X } _ { K e y s }$ are sets. Assuming that the $j$ -th elements $\mathsf { X } _ { C a l l s , j } , \mathsf { X } _ { T y p e s , j } ,$ and $\mathsf { X } _ { K e y s , j }$ of these sets are generated independently, and assuming a function $f$ for encoding these elements, let:
+
+$$
+\begin{array} { r c l } { ^ { P } ( { \bf X } | { \bf Z } , \theta ) } & { = } & { \left( \prod _ { j } \mathrm { N o r m a l } ( f ( X _ { C a l l s , j } ) | Z , { \bf I } \sigma _ { C a l l s } ^ { 2 } ) \right) \left( \prod _ { j } \mathrm { N o r m a l } ( f ( X _ { T y p e s , j } ) | Z , { \bf I } \sigma _ { T y p e s } ^ { 2 } ) \right) } \\ & & { \left( \prod _ { j } \mathrm { N o r m a l } ( f ( X _ { K e y s , j } ) | Z , { \bf I } \sigma _ { K e y s } ^ { 2 } ) \right) . } \end{array}
+$$
+
+That is, the encoded value of each $\mathsf { X } _ { T y p e s , j }$ , $\mathsf { X } _ { C a l l s , j }$ or $\mathsf { X } _ { K e y s , j }$ is sampled from a high-dimensional Normal distribution conjugacy, we have: $f$ ith the set , where $\mathbb { R } ^ { m }$ then from Normal-Normal $\begin{array} { r } { P ( \boldsymbol { \mathsf { Z } } | \mathsf { X } ) = \operatorname { N o r m a l } \left( \boldsymbol { \mathsf { Z } } \mid \frac { \overline { { \mathsf { X } } } } { 1 + n } , \frac { 1 } { 1 + n } \mathbf { I } \right) } \end{array}$
+
+$$
+\overline { { \mathsf { X } } } = \left( \sigma _ { T y p e s } ^ { - 2 } \sum _ { j } f ( \mathsf { X } _ { T y p e s , j } ) \right) + \left( \sigma _ { C a l l s } ^ { - 2 } \sum _ { j } f ( \mathsf { X } _ { C a l l s , j } ) \right) + \left( \sigma _ { K e y s } ^ { - 2 } \sum _ { j } f ( \mathsf { X } _ { K e y s , j } ) \right)
+$$
+
+$n = n _ { T y p e s } \sigma _ { T y p e s } ^ { - 2 } + n _ { C a l l s } \sigma _ { C a l l s } ^ { - 2 } + n _ { K e y s } \sigma _ { K e y s } ^ { - 2 }$ . Here, $n _ { T y p e s }$ is the number of types supplied, and $n _ { C a l l s }$ and $n _ { K e y s }$ are defined similarly.
+
+Note that this particular $P ( Z | \mathsf { X } , \theta )$ only follows directly from the Normal $( \vec { 0 } , \mathbf { I } )$ prior on $Z$ and Normal likelihood $P ( X | Z , \dot { \theta } )$ if the encoding function $f$ is 1-1 and onto. However, even if $f$ is not 1-1 and onto (as will be the case if $f$ is implemented with a standard feed-forward neural network) we can still use this probabilistic encoder, and in practice we still tend to see the benefits of the regularizing prior on Z, with $P ( Z )$ distributed approximately according to a unit Normal. We call this type of encoder-decoder, with a single, Normally-distributed latent variable $Z$ linking the input and output, a Gaussian encoder-decoder, or GED for short.
+
+Now that we have chosen $P ( X | Z , \theta )$ and $P ( \boldsymbol { Y } | \boldsymbol { Z } , \boldsymbol { \theta } )$ , we must choose $\theta$ to perform CLE. Note that:
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { i } \log P ( \mathsf { Y } _ { i } | \mathsf { X } _ { i } , \theta ) = \sum _ { i } \log \int _ { \mathbb { Z } \in \mathbb { R } ^ { m } } P ( \mathbb { Z } | \mathsf { X } _ { i } , \theta ) P ( \mathsf { Y } _ { i } | \mathbb { Z } , \theta ) d \mathbb { Z } = \sum _ { i } \log \mathbf { E } _ { \mathbb { Z } \sim P ( \mathbb { Z } | \mathsf { X } _ { i } , \theta ) } [ P ( \mathsf { Y } _ { i } | \mathbb { Z } , \theta ) ] } \\ & { \qquad \ge \displaystyle \sum _ { i } \mathbf { E } _ { \mathbb { Z } \sim P ( \mathbb { Z } | \mathsf { X } _ { i } , \theta ) } [ \log P ( \mathsf { Y } _ { i } | \mathbb { Z } , \theta ) ] = \mathcal { L } ( \theta ) . } \end{array}
+$$
+
+where the $\geq$ holds due to Jensen’s inequality. Hence, $\mathcal { L } ( \boldsymbol { \theta } )$ serves as a lower bound on the loglikelihood, and so we can compute $\theta ^ { * } = \arg \operatorname* { m a x } _ { \theta } \mathcal { L } ( \theta )$ as a proxy for the CLE. We maximize this lower bound using stochastic gradient ascent; as $P ( Z | \mathsf { X } _ { i } , \theta )$ is Normal, we can use the reparameterization trick common in variational auto-encoders (Kingma & Welling, 2014) while doing so. The parameter set $\theta$ contains all of the parameters of the encoding function $f$ as well as $\sigma _ { T y p e s }$ , $\sigma _ { C a l l s }$ , and $\sigma _ { K e y s }$ , and the parameters used in the decoding distribution funciton $P ( \boldsymbol { Y } | \boldsymbol { Z } , \boldsymbol { \theta } )$ .
+
+# 5 COMBINATORIAL CONCRETIZATION
+
+The final step in our algorithm is to “concretize” sketches into programs, following the distribution $P ( P r o g | \mathsf { Y } )$ . Our method of doing so is a type-directed, stochastic search procedure that builds on combinatorial methods for program synthesis (Schkufza et al., 2016; Feser et al., 2015).
+
+Given a sketch Y, our procedure performs a random walk in a space of partially concretized sketches (PCSs). A PCS is a term obtained by replacing some of the abstract method calls and expressions in a sketch by AML method calls and AML expressions. For example, the term $^ {  } x _ { 1 } . a ( x _ { 2 } ) ; \tau _ { 1 } . b ( \tau _ { 2 } ) ^ { \prime }$ ,
+
+which sequential composes an abstract method call to $b$ and a “concrete” method call to $a$ , is a PCS.   
+The state of the procedure at the $i$ -th point of the walk is a $\mathrm { P C S } \mathsf { H } _ { i }$ . The initial state is Y.
+
+Each state H has a set of neighbors $N e x t ( \mathsf { H } )$ . This set consists of all $\mathsf { P C S - s H ^ { \prime } }$ that are obtained by concretizing a single abstract method call or expression in H, using variable names in a way that is consistent with the types of all API methods and declared variables in H.
+
+The $( i + 1 )$ -th state in a walk is a sample from a predefined, heuristically chosen distribution $P ( \mathsf { H } _ { i + 1 } \mid \mathsf { H } _ { i } , )$ . The only requirement on this distribution is that it assigns nonzero probability to a state iff it belongs to ${ N e x t } ( \mathsf { H } _ { i } )$ . In practice, our implementation of this distribution prioritizes programs that are simpler. The random walk ends when it reaches a state $\mathsf { H } ^ { * }$ that has no neighbors. If $\mathsf { H } ^ { * }$ is fully concrete (that is, an AML program), then the walk is successful and $\mathsf { H } ^ { * }$ is returned as a sample. If not, the current walk is rejected, and a fresh walk is started from the initial state.
+
+Recall that the concretization distribution $P ( P r o g | \mathsf { Y } )$ is only defined for sketches Y that are satisfiable. Our concretization procedure does not assume that its input Y is satisfiable. However, if $\textsf { Y }$ is not satisfiable, all random walks that it performs end with rejection, causing it to never terminate.
+
+While the worst-case complexity of this procedure is exponential in the generated programs, it performs well in practice because of our chosen language of sketches. For instance, our search does not need to discover the high-level structure of programs. Also, sketches specify the types of method arguments and return values, and this significantly limits the search space.
+
+# 6 EXPERIMENTS
+
+Now we present an empirical evaluation of the effectiveness of our method. The experiments we describe utilize data from an online repository of about 1500 Android apps (and, 2017). We decompiled the APKs using JADX (Skylot, 2017) to generate their source code. Analyzing about 100 million lines of code that were generated, we extracted 150,000 methods that used Android APIs or the Java library. We then pre-processed all method bodies to translate the code from Java to AML, preserving names of relevant API calls and data types as well as the high-level control flow. Hereafter, when we say “program” we refer to an AML program.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Min</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Median</td><td rowspan=1 colspan=1>Vocab</td></tr><tr><td rowspan=1 colspan=1>Xcalls</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2584</td></tr><tr><td rowspan=1 colspan=1>XTypes</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>1521</td></tr><tr><td rowspan=1 colspan=1>XKeys</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>993</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>48</td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>5098</td></tr></table>
+
+From each program, we extracted the sets $\mathsf { X } _ { C a l l s }$ , $\mathsf { X } _ { \substack { T y p e s } }$ and $\mathsf { X } _ { K e y s }$ as well as a sketch Y. Lacking separate natural language dscriptions for programs, we defined keywords to be words obtained by splitting the names of the API types and calls that the program uses, based on camel case. For instance, the keywords obtained from the API call readLine are “read” and “line”. As API method and types in Java tend to be carefully named, these words often contain rich information about what programs do. Figure 4 gives some statistics on the sizes of the labels in the data. From the extracted data, we randomly selected 10,000 programs to be in the testing and validation data each.
+
+# 6.1 IMPLEMENTATION AND TRAINING
+
+We implemented our approach in our tool called BAYOU, using TensorFlow (Abadi et al., 2015) to implement the GED neural model, and the Eclipse IDE for the abstraction from Java to the language of sketches and the combinatorial concretization.
+
+In all our experiments we performed cross-validation through grid search and picked the best performing model. Our hyper-parameters for training the model are as follows. We used 64, 32 and 64 units in the encoder for API calls, types and keywords, respectively, and 128 units in the decoder. The latent space was 32-dimensional. We used a mini-batch size of 50, a learning rate of 0.0006 for the Adam gradient-descent optimizer (Kingma & Ba, 2014), and ran the training for 50 epochs.
+
+The training was performed on an AWS “p2.xlarge” machine with an NVIDIA K80 GPU with 12GB GPU memory. As each sketch was broken down into a set of production paths, the total number of data points fed to the model was around 700,000 per epoch. Training took 10 hours to complete.
+
+![](images/b0b174ba51e84ce61cc65ea64c0a37f4ee025a70fdcb45fa611a8924a00384fc.jpg)  
+Figure 5: 2-dimensional projection of latent space
+
+To visualize clustering in the 32-dimensional latent space, we provided labels $\mathsf { X }$ from the testing data and sampled Z from $P ( Z | \mathsf X )$ , and then used it to sample a sketch from $P ( \boldsymbol { Y } | \boldsymbol { Z } )$ . We then used t-SNE (Maaten & Hinton, 2008) to reduce the dimensionality of Z to 2-dimensions, and labeled each point with the API used in the sketch Y. Figure 5 shows this 2-dimensional space, where each label has been coded with a different color. It is immediately apparent from the plot that the model has learned to cluster the latent space neatly according to different APIs. Some APIs such as java.io have several modes, and we noticed separately that each mode corresponds to different usage scenarios of the API, such as reading versus writing in this case.
+
+# 6.3 ACCURACY
+
+To evaluate prediction accuracy, we provided labels from the testing data to our model, sampled sketches from the distribution $P ( \boldsymbol { Y } | \mathsf { X } )$ and concretized each sketch into an AML program using our combinatorial search. We then measured the number of test programs for which a program that is equivalent to the expected one appeared in the top-10 results from the model.
+
+As there is no universal metric to measure program equivalence (in fact, it is an undecidable problem in general), we used several metrics to approximate the notion of equivalence. We defined the following metrics on the top-10 programs predicted by the model:
+
+M1. This binary metric measures whether the expected program appeared in a syntactically equivalent form in the results. Of course, an impediment to measuring this is that the names of variables used in the expected and predicted programs may not match. It is neither reasonable nor useful for any model of code to learn the exact variable names in the training data. Therefore, in performing this equivalence check, we abstract away the variable names and compare the rest of the program’s Abstract Syntax Tree (AST) instead.   
+M2. This metric measures the minimum Jaccard distance between the sets of sequences of API calls made by the expected and predicted programs. It is a measure of how close to the original program were we able to get in terms of sequences of API calls.   
+M3. Similar to metric M2, this metric measures the minimum Jaccard distance between the sets of API calls in the expected and predicted programs.   
+M4. This metric computes the minimum absolute difference between the number of statements in the expected and sampled programs, as a ratio of that in the former.   
+M5. Similar to metric M4, this metric computes the minumum absolute difference between the number of control structures in the expected and sampled programs, as a ratio of that in the former. Examples of control structures are branches, loops, and try-catch statements.
+
+# 6.4 PARTIAL OBSERVABILITY
+
+To evaluate our model’s ability to predict programs given a small amount of information about its code, we varied the fraction of the set of API calls, types, and keywords provided as input from the testing data. We experimented with $7 5 \%$ , $50 \%$ and $2 5 \%$ observability in the testing data; the median number of items in a label in these cases were 9, 6, and 2, respectively.
+
+<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=1 colspan=4>InputLabel Observability</td></tr><tr><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>75%</td><td rowspan=1 colspan=1>50%</td><td rowspan=1 colspan=1>25%</td></tr><tr><td rowspan=2 colspan=1>GED-AMLGsNN-AMLGED-SkGsNN-Sk</td><td rowspan=1 colspan=1>0.13</td><td rowspan=1 colspan=1>0.09</td><td rowspan=1 colspan=1>0.07</td><td rowspan=2 colspan=1>0.020.010.210.18</td></tr><tr><td rowspan=1 colspan=1>0.070.590.57</td><td rowspan=1 colspan=1>0.040.510.48</td><td rowspan=1 colspan=1>0.030.440.41</td></tr></table>
+
+(a) M1. Proportion of test programs for which the expected AST appeared in the top-10 results.
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Input Label Observability</td></tr><tr><td>100%</td><td>75%</td><td>50%</td><td>25%</td></tr><tr><td>GED-AML</td><td>0.82</td><td>0.87</td><td>0.89</td><td>0.97</td></tr><tr><td>GsNN-AML</td><td>0.88</td><td>0.92</td><td>0.93</td><td>0.98</td></tr><tr><td>GED-Sk</td><td>0.34</td><td>0.43</td><td>0.50</td><td>0.76</td></tr><tr><td>GsNN-Sk</td><td>0.36</td><td>0.46</td><td>0.53</td><td>0.78</td></tr></table>
+
+(b) M2. Average minimum Jaccard distance on the set of sequences of API methods called in the test program vs the top-10 results.
+
+<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=1 colspan=4>Input Label Observability</td></tr><tr><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>75%</td><td rowspan=1 colspan=1>50%</td><td rowspan=1 colspan=1>25%</td></tr><tr><td rowspan=4 colspan=1>GED-AMLGsNN-AMLGED-SkGsNN-Sk</td><td rowspan=1 colspan=1>0.52</td><td rowspan=1 colspan=1>0.58</td><td rowspan=1 colspan=1>0.61</td><td rowspan=2 colspan=1>0.770.83</td></tr><tr><td rowspan=1 colspan=1>0.59</td><td rowspan=1 colspan=1>0.64</td><td rowspan=1 colspan=1>0.68</td></tr><tr><td rowspan=1 colspan=1>0.11</td><td rowspan=1 colspan=1>0.17</td><td rowspan=1 colspan=1>0.22</td><td rowspan=2 colspan=1>0.500.52</td></tr><tr><td rowspan=1 colspan=1>0.13</td><td rowspan=1 colspan=1>0.19</td><td rowspan=1 colspan=1>0.25</td></tr></table>
+
+(c) M3. Average minimum Jaccard distance on the set of API methods called in the test program vs the top-10 results.
+
+<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=1 colspan=4>Input Label Observability</td></tr><tr><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>75%</td><td rowspan=1 colspan=1>50%</td><td rowspan=1 colspan=1>25%</td></tr><tr><td rowspan=4 colspan=1>GED-AMLGsNN-AMLGED-SkGsNN-Sk</td><td rowspan=1 colspan=1>0.49</td><td rowspan=1 colspan=1>0.47</td><td rowspan=1 colspan=1>0.46</td><td rowspan=2 colspan=1>0.460.53</td></tr><tr><td rowspan=1 colspan=1>0.52</td><td rowspan=1 colspan=1>0.49</td><td rowspan=1 colspan=1>0.49</td></tr><tr><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.06</td><td rowspan=1 colspan=1>0.06</td><td rowspan=2 colspan=1>0.090.09</td></tr><tr><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.06</td><td rowspan=1 colspan=1>0.06</td></tr></table>
+
+(d) M4. Average minimum difference between the number of statements in the test program vs the top-10 results.
+
+(e) M5. Average minimum difference between the number of control structures in the test program vs the top-10 results.
+
+Figure 6: Accuracy of different models on testing data. GED-AML and GSNN-AML are baseline models trained over AML ASTs, GED-Sk and GSNN-Sk are models trained over sketches.   
+
+<table><tr><td rowspan="3">Model</td><td colspan="3">Input Label Observability</td></tr><tr><td>100%</td><td>75%</td><td>50%</td><td>25%</td></tr><tr><td>GED-AML</td><td>0.31</td><td>0.30</td><td>0.30</td><td>0.34</td></tr><tr><td>GsNN-AML</td><td>0.32</td><td>0.31</td><td>0.32</td><td>0.39</td></tr><tr><td>GED-Sk</td><td>0.03</td><td>0.03</td><td>0.03</td><td>0.04</td></tr><tr><td>GsNN-Sk</td><td>0.03</td><td>0.03</td><td>0.03</td><td>0.03</td></tr></table>
+
+<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=1 colspan=5>Metric</td></tr><tr><td rowspan=1 colspan=1>M1</td><td rowspan=1 colspan=1>M2</td><td rowspan=1 colspan=1>M3</td><td rowspan=1 colspan=1>M4</td><td rowspan=1 colspan=1>M5</td></tr><tr><td rowspan=3 colspan=1>GED-AMLGsNN-AMLGED-SkGsNN-Sk</td><td rowspan=1 colspan=1>0.02</td><td rowspan=1 colspan=1>0.97</td><td rowspan=1 colspan=1>0.71</td><td rowspan=1 colspan=1>0.50</td><td rowspan=2 colspan=1>0.370.370.04</td></tr><tr><td rowspan=2 colspan=1>0.010.230.20</td><td rowspan=1 colspan=1>0.980.70</td><td rowspan=1 colspan=1>0.740.30</td><td rowspan=1 colspan=1>0.510.08</td><td rowspan=1 colspan=1>0.370.04</td></tr><tr><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>0.33</td><td rowspan=1 colspan=1>0.08</td><td rowspan=1 colspan=1>0.04</td></tr></table>
+
+(f) Metrics for $5 0 \%$ obsevability evaluated only on unseen data
+
+# 6.5 COMPETING MODELS
+
+In order to compare our model with state-of-the-art conditional generative models, we implemented the Gaussian Stochastic Neural Network (GSNN) presented by (Sohn et al., 2015), using the same tree-structured decoder as the GED. There are two main differences: (i) the GSNN’s decoder is also conditioned directly on the input label $\mathsf { X }$ in addition to Z, which we accomplish by concatenating its initial state with the encoding of $\mathsf { X } ,$ , (ii) the GSNN loss function has an additional KL-divergence term weighted by a hyper-parameter $\beta$ . We subjected the GSNN to the same training and crossvalidation process as our model. In the end, we selected a model that happened to have very similar hyper-parameters as ours, with $\beta = 0 . 0 0 1$ .
+
+# 6.6 EVALUATING SKETCHES
+
+In order to evaluate the effect of sketch learning for program generation, we implemented and compared with a model that learns directly over programs. Specifically, the neural network structure is exactly the same as ours, except that instead of being trained on production paths in the sketches, the model is trained on production paths in the ASTs of the AML programs. We selected a model that had more units in the decoder (256) compared to our model (128), as the AML grammar is more complex than the grammar of sketches. We also implemented a similar GSNN model to train over AML ASTs directly.
+
+Figure 6 shows the collated results of this evaluation, where each entry computes the average of the corresponding metric over the 10000 test programs. It takes our model about 8 seconds, on average, to generate and rank 10 programs.
+
+When testing models that were trained on AML ASTs, namely the GED-AML and GSNN-AML models, we observed that out of a total of 87,486 AML ASTs sampled from the two models, 2525 (or $3 \%$ ) ASTs were not even well-formed, i.e., they would not pass a parser, and hence had to be discarded from the metrics. This number is 0 for the GED-Sk and GSNN-Sk models, meaning that all AML ASTs that were obtained by concretizing sketches were well-formed.
+
+In general, one can observe that the GED-Sk model performs best overall, with GSNN-Sk a reasonable alternative. We hypothesize that the reason GED-Sk performs slightly better is the regularizing prior on $Z$ ; since the GSNN has a direct link from $X$ to $Y$ , it can choose to ignore this regularization. We would classify both these models as suitable for conditional program generation. However, the other two models GED-AML and GSNN-AML perform quite worse, showing that sketch learning is key in addressing the problem of conditional program generation.
+
+# 6.7 GENERALIZATION
+
+To evaluate how well our model generalizes to unseen data, we gather a subset of the testing data whose data points, consisting of label-sketch pairs $( \mathsf { X } , \mathsf { Y } )$ , never occurred in the training data. We then evaluate the same metrics in Figure 6(a)-(e), but due to space reasons we focus on the $50 \%$ observability column. Figure 6(f) shows the results of this evaluation on the subset of 5126 (out of 10000) unseen test data points. The metrics exhibit a similar trend, showing that the models based on sketch learning are able to generalize much better than the baseline models, and that the GED-Sk model performs the best.
+
+# 7 RELATED WORK
+
+Unconditional, corpus-driven generation of programs has been studied before (Maddison & Tarlow, 2014; Allamanis & Sutton, 2014; Bielik et al., 2016), as has the generation of code snippets conditioned on a context into which the snippet is merged (Nguyen et al., 2013; Raychev et al., 2014; Nguyen & Nguyen, 2015). These prior efforts often use models like $n$ -grams (Nguyen et al., 2013) and recurrent neural networks (Raychev et al., 2014) that are primarily suited to the generation of straight-line programs; almost universally, they cannot guarantee semantic properties of generated programs. Among prominent exceptions, Maddison & Tarlow (2014) use log-bilinear tree-traversal models, a class of probabilistic pushdown automata, for program generation. Bielik et al. (2016) study a generalization of probabilistic grammars known as probabilistic higher-order grammars. Like our work, these papers address the generation of programs that satisfy rich constraints such as the type-safe use of names. In principle, one could replace our decoder and the combinatorial concretizer, which together form an unconditional program generator, with one of these models. However, given our experiments, doing so is unlikely to lead to good performance in the end-to-end problem of conditional program generation.
+
+There is a line of existing work considering the generation of programs from text (Yin & Neubig, 2017; Ling et al., 2016; Rabinovich et al., 2017). These papers use decoders similar to the one used in BAYOU, and since they are solving the text-to-code problem, they utilize attention mechanisms not found in BAYOU. Those attention mechanisms could be particularly useful were BAYOU extended to handle natural language evidence. The fundamental difference between these works and BAYOU, however, is the level of abstraction at which learning takes place. These papers attempt to translate text directly into code, whereas BAYOU uses neural methods to produce higher-level sketches that are translated into program code using symbolic methods. This two-step code generation process is central to BAYOU. It ensures key semantic properties of the generated code (such as type safety) and by abstracting away from the learner many lower-level details, it may make learning easier. We have given experimental evidence that this approach can give better results than translating directly into code.
+
+Kusner et al. (2017) propose a variational autoencoder for context-free grammars. As an autoencoder, this model is generative, but it is not a conditional model such as ours. In their application of synthesizing molecular structures, given a particular molecular structure, their model can be used to search the latent space for similar valid structures. In our setting, however, we are not given a sketch but only a label for the sketch, and our task is learn a conditional model that can predict a whole sketch given a label.
+
+Conditional program generation is closely related to program synthesis (Gulwani et al., 2017), the problem of producing programs that satisfy a given semantic specification. The programming language community has studied this problem thoroughly using the tools of combinatorial search and symbolic reasoning (Alur et al., 2013; Solar-Lezama et al., 2006; Gulwani, 2011; Feser et al., 2015). A common tactic in this literature is to put syntactic limitations on the space of feasible programs (Alur et al., 2013). This is done either by adding a human-provided sketch to a problem instance (Solar-Lezama et al., 2006), or by restricting synthesis to a narrow DSL (Gulwani, 2011; Polozov & Gulwani, 2015).
+
+A recent body of work has developed neural approaches to program synthesis. Terpret (Gaunt et al., 2016) and Neural Forth (Riedel et al., 2016) use neural learning over a set of user-provided examples to complete a user-provided sketch. In neuro-symbolic synthesis (Parisotto et al., 2016) and RobustFill (Devlin et al., 2017), a neural architecture is used to encode a set of input-output examples and decode the resulting representation into a Flashfill program. DeepCoder (Balog et al., 2017) uses neural techniques to speed up the synthesis of Flashfill (Gulwani et al., 2015) programs.
+
+These efforts differ from ours in goals as well as methods. Our problem is simpler, as it is conditioned on syntactic, rather than semantic, facets of programs. This allows us to generate programs in a complex programming language over a large number of data types and API methods, without needing a human-provided sketch. The key methodological difference between our work and symbolic program synthesis lies in our use of data, which allows us to generalize from a very small amount of specification. Unlike our approach, most neural approaches to program synthesis do not combine learning and combinatorial techniques. The prominent exception is Deepcoder (Balog et al., 2017), whose relationship with our work was discussed in Section 1.
+
+# 8 CONCLUSION
+
+We have given a method for generating type-safe programs in a Java-like language, given a label containing a small amount of information about a program’s code or metadata. Our main idea is to learn a model that can predict sketches of programs relevant to a label. The predicted sketches are concretized into code using combinatorial techniques. We have implemented our ideas in BAYOU, a system for the generation of API-heavy code. Our experiments indicate that the system can often generate complex method bodies from just a few tokens, and that learning at the level of sketches is key to performing such generation effectively.
+
+An important distinction between our work and classical program synthesis is that our generator is conditioned on uncertain, syntactic information about the target program, as opposed to hard constraints on the program’s semantics. Of course, the programs that we generate are type-safe, and therefore guaranteed to satisfy certain semantic constraints. However, these constraints are invariant across generation tasks; in contrast, traditional program synthesis permits instance-specific semantic constraints. Future work will seek to condition program generation on syntactic labels as well as semantic constraints. As mentioned earlier, learning correlations between the syntax and semantics of programs written in complex languages is difficult. However, the approach of first generating and then concretizing a sketch could reduce this difficulty: sketches could be generated using a limited amount of semantic information, and the concretizer could use logic-based techniques (Alur et al., 2013; Gulwani et al., 2017) to ensure that the programs synthesized from these sketches match the semantic constraints exactly. A key challenge here would be to calibrate the amount of semantic information on which sketch generation is conditioned.
+
+Acknowledgements This research was supported by DARPA MUSE award #FA8750-14-2-0270 and a Google Research Award.
+
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+
+![](images/f261c81be2773be917b399563df52f4856c4fc822b5e08b6898388a29597573b.jpg)  
+Figure 7: Programs generated in a typical run of BAYOU, given the API method name readLine and the type FileReader.   
+Figure 8: Grammar for AML
+
+# A THE AML LANGUAGE
+
+Prog $: : =$ skip | Prog1; $\mathsf { P r o g } _ { 2 }$ | call Call | let $x = \mathsf { C a l l } \mid$ if $\mathsf { E x p }$ then $\mathsf { P r o g } _ { 1 }$ else $\mathsf { P r o g } _ { 2 }$ | while $\mathsf { E x p }$ do $\mathsf { P r o g } _ { 1 }$ | try $\mathsf { P r o g } _ { 1 }$ Catch   
+Exp $\begin{array} { r l } { \mathrm { : : } = } & { { } \mathsf { S e x p } \mid \mathsf { C a l l } \mid \mathsf { l e t } x = \mathsf { C a l l } : \mathsf { E x p } _ { 1 } } \\ { \mathrm { : } = } & { { } c \mid x } \\ { \mathrm { : } = } & { { } \mathsf { S e x p } _ { 0 } . a ( \mathsf { S e x p } _ { 1 } , \ldots , \mathsf { S e x p } _ { k } ) } \\ { \mathrm { \cdot } \mathrm { : } = } & { { } \mathbf { c a t c h } ( x _ { 1 } ) \mathsf { P r o g } _ { 1 } \mathrm { ~ . ~ . ~ . ~ } \mathbf { c a t c h } ( x _ { k } ) \mathsf { P r o g } _ { k } } \end{array}$   
+Sexp   
+Call   
+Catc
+
+AML is a core language that is designed to capture the essence of API usage in Java-like languages. Now we present this language.
+
+AML uses a finite set of API data types. A type is identified with a finite set of $A P I$ method names (including constructors); the type for which this set is empty is said to be void. Each method name $a$ is associated with a type signature $( \tau _ { 1 } , \dots , \tau _ { k } ) \ \to \ \tau _ { 0 }$ , where $\tau _ { 1 } , \ldots , \tau _ { k }$ are the method’s input types and $\tau _ { 0 }$ is its return type. A method for which $\tau _ { 0 }$ is void is interpreted to not return a value.
+
+Finally, we assume predefined universes of constants and variable names.
+
+The grammar for AML is as in Figure 8. Here, $x , x _ { 1 } , \ldots$ are variable names, $c$ is a constant, and $a$ is a method name. The syntax for programs Prog includes method calls, loops, branches, statement sequencing, and exception handling. We use variables to feed the output of one method into another, and the keyword let to store the return value of a call in a fresh variable. $\mathsf { E x p }$ stands for (objectvalued) expressions, which include constants, variables, method calls, and let-expressions such as “let $x = { \mathsf { C a l l } } : { \mathsf { E x p } } ^ { , , }$ , which stores the return value of a call in a fresh variable $x$ , then uses this binding to evaluate the expression $\mathsf { E x p }$ . (Arithmetic and relational operators are assumed to be encompassed by API methods.)
+
+The operational semantics and type system for AML are standard, and consequently, we do not describe these in detail.
+
+# B ABSTRACTING AML PROGRAMS INTO SKETCHES
+
+We define the abstraction function $\alpha$ for the AML language in Figure 9.
+
+# C NEURAL NETWORK DETAILS
+
+In this section we present the details of the neural networks used by BAYOU.
+
+# C.1 THE ENCODER
+
+The task of the neural encoder is to implement the encoding function $f$ for labels, which accepts an element from a label, say $\mathsf { X } _ { C a l l s , i }$ as input and maps it into a vector in $d$ -dimensional space, where $d$ is the dimensionality of the latent space of $Z$ . To achieve this, we first convert each element $\mathsf { X } _ { C a l l s , i }$ into its one-hot vector representation, denoted $\mathsf { X } _ { C a l l s , i } ^ { \prime }$ . Then, let $h$ be the number of neural hidden
+
+$$
+\begin{array} { r l } { \alpha ( \mathrm { s i b p } ) } & { = \ \mathrm { s i b p } } \\ { \alpha ( \mathrm { c u l l ~ S e v } _ { 0 } , a ( \mathrm { S e v } _ { 1 } , \ldots , \ \mathrm { S e v } _ { b } ) ) } & { = \ \mathrm { e x i l p } _ { \eta , 0 } , a ( \tau _ { 1 } , \ldots , \tau _ { k } ) \mathrm { ~ w h e r e ~ } \tau _ { 1 } \mathrm { ~ i s ~ h e ~ t y p e ~ o f ~ S e p } } \\ { \alpha ( \mathrm { P o g } _ { 1 } , \ldots , \ \mathrm { S e v } _ { b } ) } & { = \ \alpha ( \mathrm { P o g } _ { 1 } ) ; \alpha ( \mathrm { P o g } _ { 2 } ) } & { = \ \alpha ( \mathrm { P o g } _ { 2 } ) } \\ { \alpha ( \mathrm { t e t } _ { x } = \mathrm { S e v } _ { 0 } , a ( \mathrm { S e v } _ { 0 } , \ldots , \ \mathrm { S e v } _ { b } ) ) } & { = \ \mathrm { c u l l ~ } _ { \eta , 0 } , a ( \tau _ { 1 } , \ldots , \ \tau _ { k } ) \mathrm { ~ w h e r e ~ } \tau _ { 1 } \mathrm { ~ i s ~ h e ~ t y p e ~ o f ~ S e p } } \\ { \alpha ( \mathrm { f i e t } _ { x } ) \mathrm { ~ t h e n ~ P o g ~  { F o g } _ { 2 } ~  { \alpha ~ l ~ } = ~ i f ~ } \alpha ( \mathrm { E v } _ { 0 } ) \mathrm { ~ t h e n ~ \alpha ( P o g _ { 1 } ) ~ e k e ~ } \alpha ( \mathrm { P r o g } _ { 2 } ) } \\ { \alpha ( \mathrm { P u l l ~ E e v } _ { 0 } ) \mathrm { ~ t h e ~ t h e ~ t y p e ~ o f ~  { \alpha ~ g } _ { 1 } ~ = ~ \mathrm { w h i l e ~ } \alpha ( \mathrm { C o n d } ) ~ } \mathrm { d o ~ } \alpha ( \mathrm { P r o g } _ { 1 } ) \mathrm { ~ e x i s ~ o f ~ } } \\ { \alpha ( \mathrm { P u l l ~ P o g ~ e a t e h } ( \tau _ { 1 } ) \mathrm { ~ P r o g } _ { 1 } , \ldots \ \mathrm { e a t e h } ( \tau _ { k } ) \mathrm { ~ P r o g } _ { k } ) } & { = \ \mathrm { t r y ~ c o s } ( \mathrm { P r o g } _ { 1 } ) } \\ & { \qquad \mathrm { c u c h } ( \mathrm { P r o g } _ { 1 } ) \ \alpha ( \mathrm { P o g } _ { 1 } ) \ldots \ \mathrm { c a t e h } ( \tau _ { k } ) \mathrm { ~ a n ~ } } \\ &  \qquad \mathrm { w h e r e ~ } \tau _ { i } \ \end{array}
+$$
+
+![](images/07fd8d83d7eb189b373d5909a43d38a4e9957261900127582edbe642461e6a88.jpg)  
+Figure 9: The abstraction function $\alpha$ .   
+Figure 10: Tree representation of the sketch in Figure 7(a)
+
+units in the encoder for API calls, and let $\mathbf { W } _ { h } \in \mathbb { R } ^ { | C a l l s | \times h }$ , $\mathbf { b } _ { h } \in \mathbb { R } ^ { h }$ , $\mathbf { W } _ { d } \in \mathbb { R } ^ { h \times d }$ , $\mathbf { b } _ { d } \in \mathbb { R } ^ { d }$ be real-valued weight and bias matrices of the neural network. The encoding function $f ( { \sf X } _ { C a l l s , i } )$ can be defined as follows:
+
+$$
+f ( \mathsf { X } _ { C a l l s , i } ) = \mathsf { t a n h } ( ( \boldsymbol { \mathbf { W } } _ { h } \boldsymbol { \mathbf { \ell } } _ { h } \boldsymbol { \mathbf { \ell } } _ { i } ^ { \prime } + \boldsymbol { \mathbf { \ell } } _ { h } ) \boldsymbol { \mathbf { \ell } } _ { \star } \boldsymbol { \mathbf { W } } _ { d } + b _ { d } )
+$$
+
+where tanh is a non-linearity defined as $\begin{array} { r } { \mathtt { t a n h } ( x ) = \frac { 1 - e ^ { - 2 x } } { 1 + e ^ { - 2 x } } } \end{array}$ 1−e−2x1+e−2x . This would map any given API call into a $d$ -dimensional real-valued vector. The values of entries in the matrices ${ \mathbf W } _ { h } , b _ { h } , { \mathbf W } _ { d }$ and $b _ { d }$ will be learned during training. The encoder for types can be defined analogously, with its own set of matrices and hidden state.
+
+# C.2 THE DECODER
+
+The task of the neural decoder is to implement the sampler for $\mathsf { Y } \sim P ( Y | Z )$ . This is implemented recursively via repeated samples of production rules $\mathsf { Y } _ { i }$ in the grammar of sketches, drawn as $\mathsf { Y } _ { i } \sim$ $P ( Y _ { i } | \mathbf { Y } _ { i - 1 } , Z )$ , where $\mathbf { Y } _ { i - 1 } = \mathsf { Y } _ { 1 } , \ldots , \mathsf { Y } _ { i - 1 }$ . The generation of each $\mathsf { Y } _ { i }$ requires the generation of a new “path” from a series of previous “paths”, where each path corresponds to a series of production rules fired in the grammar.
+
+As a sketch is tree-structured, we use a top-down tree-structured recurrent neural network similar to Zhang et al. (2016), which we elaborate in this section. First, similar to the notion of a “dependency path” in Zhang et al. (2016), we define a production path as a sequence of pairs ${ \langle ( v _ { 1 } , \bar { e } _ { 1 } ) , ( v _ { 2 } , \bar { e } _ { 2 } ) , \ldots , ( v _ { k } , e _ { k } ) \rangle }$ where $v _ { i }$ is a node in the sketch (i.e., a term in the grammar) and $e _ { i }$ is the type of edge that connects $v _ { i }$ with $v _ { i + 1 }$ . Our representation has two types of edges: sibling and child. A sibling edge connects two nodes at the same level of the tree and under the same parent node (i.e., two terms in the RHS of the same rule). A child edge connects a node with another that is
+
+$$
+y _ { i } = \{ \begin{array} { l l } { \mathsf { s o f t m a x } ( \mathbf { W } _ { y } ^ { c } \cdot h _ { i } + \mathbf { b } _ { y } ^ { c } ) } & { \mathrm { ~ i f ~ } e _ { i } = c h i l d } \\ { \mathsf { s o f t m a x } ( \mathbf { W } _ { y } ^ { s } \cdot h _ { i } + \mathbf { b } _ { y } ^ { s } ) } & { \mathrm { ~ i f ~ } e _ { i } = s i b l i n g } \end{array}
+$$
+
+$$
+\begin{array} { l l } { { \displaystyle h _ { i } ^ { c } = { \mathbf W } _ { h } ^ { c } \cdot h _ { i - 1 } + { \mathbf b } _ { h } ^ { c } + { \mathbf W } _ { v } ^ { c } \cdot v _ { i } ^ { \prime } + { \mathbf b } _ { v } ^ { c } } } & { { \mathrm { ~ w h e r e ~ } } } \\ { { \displaystyle h _ { i } ^ { s } = { \mathbf W } _ { h } ^ { s } \cdot h _ { i - 1 } + { \mathbf b } _ { h } ^ { s } + { \mathbf W } _ { v } ^ { s } \cdot v _ { i } ^ { \prime } + { \mathbf b } _ { v } ^ { s } } } & { { \mathrm { ~ t a n h } ( x ) = \frac { 1 - e ^ { - 2 x } } { 1 + e ^ { - 2 x } } \mathrm { a n d } } } \\ { { \displaystyle h _ { i } = \left\{ \begin{array} { l l } { { \displaystyle \mathrm { t a n h } ( h _ { i } ^ { c } ) } } & { { \mathrm { i f ~ } e _ { i } = c h i l d } } \\ { { \displaystyle \mathrm { t a n h } ( h _ { i } ^ { s } ) } } & { { \mathrm { i f ~ } e _ { i } = s i b l i n g } } \end{array} \right. } }  & { { \displaystyle \mathrm { s o f t m a x } ( { \mathbf x } ) _ { j } = \frac { e ^ { x _ { j } } } { \sum _ { k = 1 } ^ { K } e ^ { x _ { k } } } \mathrm { f o r ~ } j \in 1 . . . K } } \end{array}
+$$
+
+one level deeper in the tree (i.e., the LHS with a term in the RHS of a rule). We consider a sequence of API calls connected by sequential composition as siblings. The root of the entire tree is a special node named root, and so the first pair in all production paths is (root, child). The last edge in a production path is irrelevant (·) as it does not connect the node to any subsequent nodes.
+
+As an example, consider the sketch in Figure 7(a), whose representation as a tree for the decoder is shown in Figure 10. For brevity, we use $s$ and $c$ for sibling and child edges respectively, abbreviate some classnames with uppercase letters in their name, and omit the first pair (root, c) that occurs in all paths. There are four production paths in the tree of this sketch:
+
+$$
+\begin{array} { r l } & { 1 . \mathrm { ( } \operatorname { f r y } , c \big ) , \big ( \mathbb { F } \mathrm { R } . \mathrm { n e } \ ( \mathrm { S t r i n g } ) , s \big ) , \big ( \mathbb { B } \mathrm { R } . \mathrm { n e } \ ( \mathrm { F } \mathbb { R } ) , s \big ) , \big ( \mathrm { w h i l e } , c \big ) , \big ( \mathbb { B } \mathrm { R } . \mathrm { r e a d } \mathrm { L i n e } \ ( \mathrm { \Lambda } ) , c \big ) , \big ( \mathrm { s } \mathbf { k i p } , \cdot \big ) } \\ & { 2 . \mathrm { ( } \operatorname { f r y } , c \big ) , \big ( \mathbb { F } \mathrm { R } . \mathrm { n e } \ ( \mathrm { S t r i n g } ) , s \big ) , \big ( \mathbb { B } \mathrm { R } . \mathrm { n e } \ ( \mathrm { F } \mathbb { R } ) , s \big ) , \big ( \mathrm { w h i l e } , s \big ) , \big ( \mathbb { B } \mathrm { R } . \mathrm { c l o s e } \ ( \mathrm { \Lambda } ) , \cdot \big ) } \\ & { 3 . \mathrm { ( } \operatorname { f r y } , s \big ) , \big ( \mathrm { c a t c h } , c \big ) , \big ( \mathrm { F i r F E x c e p t i o n } , c \big ) , \big ( \mathrm { T } . \mathrm { p r i n t s t a c k T r a c e } \ ( \mathrm { \Lambda } ) , \cdot \big ) } \\ & { 4 . \mathrm { ( } \operatorname { f r y } , s \big ) , \big ( \mathrm { c a t c h } , s \big ) , \big ( \mathrm { c a t c h } , c \big ) , \big ( \mathrm { I o E x c e p t i o n } , c \big ) , \big ( \mathrm { T } . \mathrm { p r i n t s t a c k T r a c e } \ ( \mathrm { \Lambda } ) , \cdot \big ) } \end{array}
+$$
+
+Now, given a Z and a sequence of pairs $\mathbf Y _ { i } = \langle ( v _ { 1 } , e _ { 1 } ) , \dots , ( v _ { i } , e _ { i } ) \rangle$ along a production path, the next node in the path is assumed to be dependent solely on Z and $\mathbf { Y } _ { i }$ . Therefore, a single inference step of the decoder computes the probability $P ( v _ { i + 1 } | \mathbf { Y } _ { i } , Z )$ . To do this, the decoder uses two RNNs, one for each type of edge $c$ and $s$ , that act on the production pairs in $\mathbf { Y } _ { i }$ . First, all nodes $v _ { i }$ are converted into their one-hot vector encoding, denoted $\boldsymbol { v } _ { i } ^ { \prime }$ .
+
+Let $h$ be the number of hidden units in the decoder, and $| G |$ be the size of the decoder’s output vocabulary, i.e., the total number of terminals and non-terminals in the grammar of sketches. Let $\mathbf { W } _ { h } ^ { e } \in \mathbb { R } ^ { \bar { h } \times h }$ and ${ \bf b } _ { h } ^ { e } \in \mathbb { R } ^ { d }$ be the decoder’s hidden state weight and bias matrices, $\mathbf { W } _ { v } ^ { e } \in \mathbb { R } ^ { | G | \times h }$ and $\mathbf { b } _ { v } ^ { e } \in \mathbb { R } ^ { h }$ be the input weight and bias matrices, and $\mathbf { W } _ { y } ^ { e } \in \mathbb { R } ^ { h \times | G | }$ and $\mathbf { b } _ { y } ^ { e } \in \mathbb { R } ^ { | G | }$ be the output weight and bias matrices, where $e$ is the type of edge: either $c$ (child) or $s$ (sibling). We also use “lifting” matrices $\mathbf { W } _ { l } \in \mathbb { R } ^ { d \times h }$ and $\mathbf { b } _ { l } \in \check { \mathbb { R } } ^ { \dot { h } }$ , to lift the $d$ -dimensional vector Z onto the (typically) higher-dimensional hidden state space $h$ of the decoder.
+
+Let $h _ { i }$ and $y _ { i }$ be the hidden state and output of the network at time point $i$ . We compute these quantities as given in Figure 11, where tanh is a non-linear activation function that converts any given value to a value between -1 and 1, and softmax converts a given $K$ -sized vector of arbitrary values to another $K$ -sized vector of values in the range [0, 1] that sum to 1—essentially a probability distribution.
+
+The type of edge at time $i$ decides which RNN to choose to update the (shared) hidden state $h _ { i }$ and the output $y _ { i }$ . Training consists of learning values for the entries in all the W and b matrices. During training, $\boldsymbol { v } _ { i } ^ { \prime }$ , $e _ { i }$ and the target output are known from the data point, and so we optimize a standard cross-entropy loss function (over all $i$ ) between the output $y _ { i }$ and the target output. During inference, $P ( v _ { i + 1 } | \mathbf { Y } _ { i } , Z )$ is simply the probability distribution $y _ { i }$ , the result of the softmax.
+
+A sketch is obtained by starting with the root node pair $( v _ { 1 } , e _ { 1 } ) = ( \mathsf { r o o t } , c h i l d )$ , recursively applying Equation 2 to get the output distribution $y _ { i }$ , sampling a value for $v _ { i + 1 }$ from $y _ { i }$ , and growing the tree by adding the sampled node to it. The edge $e _ { i + 1 }$ is provided as $c$ or $s$ depending on the $v _ { i + 1 }$ that was sampled. If only one type of edge is feasible (for instance, if the node is a terminal in the grammar, only a sibling edge is possible with the next node), then only that edge is provided. If both edges are feasible, then both possibilities are recursively explored, growing the tree in both directions.
+
+![](images/aeb8ed09e6813c21e4d1ec1fe670d6dba9ae0fbba1a342da0bda3f295649ce30.jpg)  
+Figure 12: 2-dimensional projection of latent space of the GSNN-Sk model
+
+Remarks. In our implementation, we generate trees in a depth-first fashion, by exploring a child edge before a sibling edge if both are possible. If a node has two children, a neural encoding of the nodes that were generated on the left is carried onto the right sub-tree so that the generation of this tree can leverage additional information about its previously generated sibling. We refer the reader to Section 2.4 of Zhang et al. (2016) for more details.
+
+# D ADDITIONAL EVALUATION
+
+In this section, we provide results of additional experimental evaluation.
+
+# D.1 CLUSTERING
+
+Similar to the visualization of the 2-dimensional latent space in Figure 5, we also plotted the latent space of the GSNN-Sk model trained on sketches. Figure 12 shows this plot. We observed that the latent space is clustered, relatively, more densely than that of our model (keep in mind that the plot colors are different when comparing them).
+
+# D.2 QUALITATIVE EVALUATION
+
+To give a sense of the quality of the end-to-end generation, we present and discuss a few usage scenarios for our system, BAYOU. In each scenario, we started with a set of API calls, types or keywords as labels that indicate what we (as the user) would like the generated code to perform. We then pick a single program in the top-5 results returned by BAYOU and discuss it. Figure 13 shows three such example usage scenarios.
+
+In the first scenario, we would like the system to generate a program to write something to a file by calling write using the type FileWriter. With this label, we invoked BAYOU and it returned with a program that actually accomplishes the task. Note that even though we only specified FileWriter, the program uses it to feed a BufferedWriter to write to a file. This is an interesting pattern learned from data, that file reads and writes in Java often take place in a buffered manner. Also note that the program correctly flushes the buffer before closing it, even though none of this was explicitly specified in the input.
+
+In the second scenario, we would like the generated program to set the title and message of an Android dialog. This time we provide no API calls or types but only keywords. With this, BAYOU generated a program that first builds an Android dialog box using the helper class AlertDialog.Builder, and does set its title and message. In addition, the program also adds a button to the dialog box – another interesting pattern learned from data that dialog boxes in Android often have a button, typically to close the dialog. Finally it shows the dialog with these items.
+
+![](images/f53c37b7ce8c9cec5c40154bdf61c2522ec063b34bdbe8db2d32b89026dd5e94.jpg)  
+Figure 13: Qualitative usage scenarios of BAYOU.
+
+In the final scenario, we would like BAYOU to generate code to start preview mode in the phone’s camera. We provided simply the API call startPreview as input. With this, the system was automatically able to recognize that we are interested in the camera API, and generate a program that accomplishes the task. Note that the program first obtains the camera parameters, and sets the preview display size (the int arguments are the width and height) before starting the preview. We confirmed from the Android Camera API documentation that this is recommended practice, and the model appears to have learned this automatically from data.

parse/train/Sx-mvOvnmJj/Sx-mvOvnmJj.md

@@ -0,0 +1,286 @@
+# IMPROVING CALIBRATION FOR LONG-TAILED RECOG-NITION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Deep neural networks often perform poorly when training datasets are heavily classimbalanced. Recently, two-stage methods greatly improve the performances by decoupling representation learning and classifier learning. In this paper, we discover that networks trained on long-tailed datasets are more prone to miscalibrated and over-confident. The two-stage models suffer the same issue as well. We design two novel methods to improve calibration and performance in such scenarios. Motivated by the predicted probability distributions of classes are highly related to the numbers of class instances, we propose a label-aware smoothing to deal with the different degrees of over-confidence for different classes and improve classifier learning. Noting that there is a dataset bias between these two stages because of different samplers, we further propose a shifted batch normalization to solve the dataset bias in the decoupling framework. Through extensive experiments, we also observe that mixup can remedy over-confidence and improve representation learning but has a negative or negligible effect on classifier learning. Our proposed methods set new records on multiple popular long-tailed recognition benchmarks including LT CIFAR 10/100, ImageNet-LT, Places-LT, and iNaturalist 2018.
+
+# 1 INTRODUCTION
+
+With numerous available large-scale and high-quality datasets such as ImageNet (Russakovsky et al., 2015), COCO (Lin et al., 2014), and Places (Zhou et al., 2017), deep convolutional neural networks (CNNs) have made notable breakthroughs in various computer vision tasks such as image recognition (Krizhevsky et al., 2012; He et al., 2016), object detection (Ren et al., 2015) and semantic segmentation (Cordts et al., 2016). These delicate datasets are usually artificially balanced with respect to the number of instances for each object/class. However, in real-world applications, data often follows an unexpected long-tailed distribution, where the numbers of instances for different classes are seriously imbalanced. When training CNNs on such long-tailed datasets, the performances extremely degrade. Motivated by this phenomenon, a number of works have recently emerged that try to explore long-tailed recognition.
+
+Recently, many two-stage approaches have achieved significant improvement comparing with onestage methods. Concretely, DRS and DRW (Cao et al., 2019) first train CNNs in a normal way in Stage-1. DRS finetunes CNNs on datasets with class-balanced resampling while DRW finetunes CNNs by assigning different weights to different classes in Stage-2. Zhou et al. (2020) proposed BBN with one-stage to simulate the process of DRS by dynamically combining the instance-balanced sampler and the reverse-balanced sampler. Kang et al. (2020) proposed two-stage decoupling models, cRT and LWS, to further boost the performance: Decoupling models freeze the backbone and just finetune the classifier with class-balanced resampling in Stage-2.
+
+Confidence calibration (Niculescu-Mizil & Caruana, 2005; Guo et al., 2017) – the problem of predicting probability estimates representative of the true correctness likelihood – is important for recognition models in many applications (Bojarski et al., 2016; Jiang et al., 2012). In this study, we discover that networks trained on long-tailed datasets are more miscalibrated and over-confident: We draw the reliability diagrams with 15 bins in Fig. 1, which compares the plain model trained on the original CIFAR-100 dataset, the plain model, cRT, and LWS trained on long-tailed CIFAR-100 with imbalanced factor (IF) 100. We observe that networks trained on long-tailed datasets have higher expected calibration errors (ECEs). The two-stage models, cRT and LWS, suffer over-confidence as well. Moreover, Fig. 7 and Fig. 8 (the first two plots) in Appendix B depict that this phenomenon also commonly exists on other long-tailed datasets such as LT CIFAR-10 and ImageNet-LT.
+
+![](images/5e44aa4f49a7033716a9652c8793a4caf195ce8a53f64c89a6c341469c58c17a.jpg)  
+Figure 1: Reliability diagrams of ResNet-32. From left to right: the plain model trained on the original CIFAR-100 dataset, the plain model, cRT, and LWS trained on long-tailed CIFAR-100 with $\mathrm { { I F } = 1 0 0 }$ .
+
+Another issue is that two-stage decoupling methods ignore the dataset bias or domain shift (QuioneroCandela et al., 2009) between these two stages. Concretely, two-stage models are first trained on the instanced-balanced dataset $\mathcal { D } _ { \mathrm { I } }$ in Stage-1. Then, models are trained on the class-balanced dataset $\mathcal { D } _ { \mathrm { C } }$ in Stage-2. Obviously, $P _ { \mathcal { D } _ { \mathrm { I } } } ( { \pmb x } , y ) \neq P _ { \mathcal { D } _ { \mathrm { C } } } ( { \pmb x } , y )$ , the distributions of the dataset with different sampling manners are inconsistent. Motivated by the transfer learning methods (Li et al., 2018; Wang et al., 2019), we focus on the batch normalization (Ioffe & Szegedy, 2015) layer to deal with the dataset bias problem.
+
+In this work, we propose a Mixup Shifted Label-Aware Smoothing model (MiSLAS) to effectively solve the above issues. Our key contributions are as follows: (i) We discover that models trained on long-tailed datasets are much more miscalibrated and over-confident than them trained on balanced datasets. The two-stage models suffer the same problem as well. (ii) We find that mixup can remedy over-confidence and have a positive effect on representation learning but a negative or negligible effect on classifier learning. To further enhance classifier learning and calibration, we propose a label-aware smoothing to handle the different degrees of over-confidence for different classes. (iii) We are the first to note the dataset bias or domain shift in two-stage resampling methods for long-tailed recognition. To deal with the dataset bias in the decoupling framework, we propose shift learning on the batch normalization layer, which can greatly improve the performance.
+
+We extensively validate our MiSLAS on multiple long-tailed recognition benchmark datasets, i.e., LT CIFAR-10, LT CIFAR-100, ImageNet-LT, Places-LT, and iNaturalist 2018. Experimental results manifest that the effectiveness and our method yields new state-of-the-art.
+
+# 2 RELATED WORKS
+
+Re-sampling and re-weighting. There are two groups of re-sampling strategies: over-sampling the tail-class images (Shen et al., 2016; Buda et al., 2018; Byrd & Lipton, 2019) and under-sampling the head-class images (Japkowicz & Stephen, 2002; Buda et al., 2018). Over-sampling is regularly useful on large datasets and often suffers from heavy over-fitting to tail classes especially on small datasets. For under-sampling, it discards a large portion of data, which inevitably causes degradation of the generalization ability of deep models. Re-weighting (Huang et al., 2016; Wang et al., 2017) is another prominent strategy. It assigns different weights for classes and even instances. The vanilla re-weighting method gives class weights in reverse proportion to the number of samples of classes. However, with large-scale data, re-weighting makes the deep models difficult to optimize during training. Cui et al. (2019) relieved the problem using the effective numbers to calculate the class weights. Another line of work is to adaptively re-weight each instance, e.g., Focal loss (Lin et al., 2017) assigned smaller weights for well-classified samples.
+
+Network calibration and regularization. Calibrated confidence is significant for classification models in many applications. The calibration of modern neural networks is first discussed in Guo et al. (2017). The authors discovered that model capacity, normalization, and regularization have strong effects on network calibration. mixup (Zhang et al., 2018) is a regularization technique that is proposed to train with interpolations of inputs and labels. mixup inspires several follow-ups like manifold mixup (Verma et al., 2019), CutMix (Yun et al., 2019), and Remix (Chou et al., 2020) that have shown significant improvement over mixup. Thulasidasan et al. (2019) found that CNNs trained with mixup are significantly better calibrated. Label smoothing (Szegedy et al., 2016) is another regularization technique that encourages the model to be less over-confident. Unlike cross-entropy computes loss upon the ground truth labels, label smoothing computes loss upon a soft version of the label, which can relieve the over-fitting and increase calibration and reliability (Müller et al., 2019).
+
+Table 1: Top-1 accuracy of the decoupling models (cRT and LWS) for ResNet families trained on the ImageNetLT dataset. We vary the augmentation strategies (with or without mixup $\alpha = 0 . 2$ ) on both two stages.   
+
+<table><tr><td rowspan="2">Training setup for two stages</td><td colspan="2">ResNet-10</td><td colspan="2">ResNet-50</td><td colspan="2">ResNet-101</td><td colspan="2">ResNet-152</td></tr><tr><td>cRT LWS|</td><td></td><td>cRT LWS|</td><td></td><td>cRT</td><td>LWS</td><td>cRT1</td><td>LWS</td></tr><tr><td>Stage-1 (no mixup) Stage-1 (mixup)</td><td>36.8 35.7</td><td>36.8 35.7</td><td>45.8 45.6</td><td>45.8 45.6</td><td>47.3 47.7</td><td>47.3 47.7</td><td>48.7 48.4</td><td>48.7 48.4</td></tr><tr><td>Stage-1 (no mixup) + Stage-2 (no mixup) Stage-1 (no mixup) + Stage-2 (mixup)</td><td>43.3 43.0</td><td>43.5 43.3</td><td>50.3 50.2</td><td>51.2 51.1</td><td>51.4 51.4</td><td>52.3 52.2</td><td>52.7 52.8</td><td>53.8 53.6</td></tr><tr><td>Stage-1 (mixup) + Stage-2 (no mixup) Stage-1(mixup) + Stage-2 (mixup)</td><td>43.4 43.3</td><td>42.9 42.8</td><td>51.7 51.6</td><td>52.0 51.9</td><td>53.1 53.0</td><td>53.5 53.5</td><td>54.2 54.1</td><td>54.6 54.5</td></tr></table>
+
+![](images/dfadadbac16edfdf51bd971589ab03899f3826ea4b154206dcc6bd81b814a482.jpg)  
+Figure 2: Classifier weight norms for the ImageNet-LT validation set when classes are sorted by descending values of $N _ { j }$ . Left: weight norms of cRT with or without mixup. Right: weight norms of LWS with or without mixup. (light shade: true norm, dark lines: smooth version)
+
+Two-stage methods. Cao et al. (2019) first proposed deferred re-weighting (DRW) and deferred re-sampling (DRS) that are superior to conventional one-stage methods: Stage-2, starting from better features, adjusts the decision boundary and locally fine-tunes the features. Recently, Kang et al. (2020) and Zhou et al. (2020) concluded that although class re-balance strategies matter when jointly training representation and classifier, instance-balanced sampling gives more general representations. Based on this observation, Kang et al. (2020) achieved state-of-the-art results by decomposing representation and classifier learning, i.e., first train the deep models with instance-balanced sampling, then fine-tune the classifier with class-balanced sampling while keeping parameters of representation learning fixed. Similarly, Zhou et al. (2020) integrated mixup training into the proposed cumulative learning strategy with which they bridged the representation learning and classifier re-balancing. The cumulative learning strategy requires dual samplers: instance-balanced and reversed instance-balanced sampler.
+
+# 3 MAIN APPROACH
+
+# 3.1 IMPROVING CALIBRATION AND REPRESENTATION LEARNING BY MIXUP
+
+For the two-stage learning framework, Kang et al. (2020) and Zhou et al. (2020) found that instancebalanced sampling gives the most generalizable representations among other sampling methods. Thulasidasan et al. (2019) found that networks trained with mixup are better calibrated. When using instance-balanced sampling, to further improve the representation generalization and relieve over-confidence, we explore the effect of mixup in the two-stage decoupling framework.
+
+Here, we train two two-stage models, i.e. cRT and LWS, on ImageNet-LT for 180 epochs in Stage-1 and finetune for 10 epochs in Stage-2, respectively. We vary the training setup (with/without mixup $\alpha = 0 . 2$ ) for both two stages. Top-1 accuracy results of these variants are listed in Table 1. From it, we conclude that: (i) When applying mixup, the performance improvements of Stage-1 are ignorable but the performances of Stage-2 are greatly enhanced for both cRT and LWS. (ii) Applying additional mixup in Stage-2 has no obvious improvement or even damages the performance, which means that mixup encourages representation learning but has a negative or negligible effect on classifier learning.
+
+![](images/dfc97570c161303a3e0406ec2a403b6e2fedf1e4397e7348a43db447fc2d713d.jpg)  
+Figure 3: Violin plot of predicted probability distributions for different parts of classes, head (more than 100 images), medium (20 to 100 images), and tail (less than 20 images) on LT CIFAR-100, $\mathrm { { I F } = 1 0 0 }$ . The upper half part in light blue: LWS (cross-entropy). The bottom half part in deep blue: LWS (label-aware smoothing).
+
+We also draw the final classifier weight norms of these variants in Fig. 2. We show the $L _ { 2 }$ norms of the weight vectors for all classes, as well as the training data distribution sorted in a descending manner concerning the number of instances. We observe that when applying mixup (orange line), the weight norms of the tail classes uniformly tend to become larger and the weight norms of the head classes are decreased, which means mixup may be more friendly to the tail classes.
+
+The analysis of calibration for networks whether adding mixup will be discussed in our experiment part (Sec. 4.2). Due to the poor and unsatisfied enhancement of mixup for classifier learning, we further propose a label-aware smoothing to improve both the calibration and classifier learning.
+
+# 3.2 IMPROVING CALIBRATION AND CLASSIFIER LEARNING BY LABEL-AWARE SMOOTHING
+
+As discussed in the introduction part and Sec. 3.1, two-stage models suffer serious over-confidence and there is no significant improvement for classifier learning when adding additional mixup. In this subsection, we try to analyze and deal with these two issues. Suppose that the weight of the classifier is $\pmb { W } \in \mathbb { R } ^ { M \times K }$ , where $M$ is the number of features and $K$ is the number of classes. The cross-entropy encourages the whole network to be over-confident on the head classes: Concretely, the cross-entropy loss after the softmax activation is $l ( y , p ) = - \log ( p _ { y } ) = - { \pmb w } _ { y } ^ { \top } { \pmb x } + \log ( \sum \exp ( { \pmb w } _ { i } ^ { \top } { \pmb x } ) )$ , where $y \in \{ 1 , 2 , . . . , K \}$ is the label, $\pmb { x } \in \mathbb { R } ^ { M }$ is the feature vector send to classifier and ${ \pmb w } _ { i }$ is the $i$ -th column vector of $W$ . The optimal solution is $\pmb { w } _ { y _ { . } } ^ { * } \top \pmb { x } = \operatorname { i n f }$ while keeping others ${ \pmb w } _ { i } ^ { \top } { \pmb x }$ , $i \neq y$ , small enough. Because the head classes contain much more training examples, the network makes the weight norm $\| \pmb { w } \|$ of the head classes become larger to near the optimal solution as much as possible, which results that their predicted probabilities mainly concentrate near 1.0 (see Fig. 3, the upper half part showing in light blue). Another fact we can get from Fig. 3 is that the distributions of predicted probability are severely related to the instance numbers. Unlike balanced recognition, we claim that applying different strategies for different classes is extremely necessary for the long-tailed problem.
+
+Here, we propose a label-aware smoothing to solve the over-confidence in cross-entropy and the different distributions of predicted probability issue. The mathematical computation of label-aware smoothing is:
+
+$$
+l ( q , p ) = - \sum _ { i = 1 } ^ { K } q _ { i } \log p _ { i } , q _ { i } = \left\{ \begin{array} { l l } { 1 - \epsilon _ { y } = 1 - f ( N _ { y } ) , } & { i = y , } \\ { \frac { \epsilon _ { y } } { K - 1 } = \frac { f ( N _ { y } ) } { K - 1 } , } & { \mathrm { o t h e r w i s e } , } \end{array} \right.
+$$
+
+where $\epsilon _ { y }$ is a small label smoothing factor for Class- $_ y$ and relates to its class number $N _ { y }$ . Now the optimal solution becomes:
+
+$$
+{ \pmb w } _ { i } ^ { * } ^ { \top } { \pmb x } = \left\{ \begin{array} { l l } { \log \left( \frac { ( K - 1 ) ( 1 - \epsilon _ { y } ) } { \epsilon _ { y } } \right) + c , } & { i = y , } \\ { c , } & { \mathrm { o t h e r w } } \end{array} \right.
+$$
+
+where $c$ can be an arbitrary real number. Comparing with the infinite optimal solution in cross-entropy, the label-aware smoothing encourages a finite output, which can get more generalized results and remedy over-fitting. We suppose the labels of the long-tailed dataset are assigned in a descending manner concerning the number of instances, i.e., $N _ { 1 } \geq N _ { 2 } \geq . . . \geq N _ { K }$ . Because the head classes contain more various and diverse examples, the predicted probabilities are more promising than them of tail classes. Thus, we suggest classes with larger instance numbers should be penalized larger label smoothing factors, that is, the related function $f ( N _ { y } )$ should be negatively correlated to $N _ { y }$ . We define three types of related function $f ( N _ { y } )$ :
+
+$$
+\epsilon _ { y } = f ( N _ { y } ) = \left\{ \begin{array} { l l } { { \mathrm { ( C o n c a v e ) } } } & { { \epsilon _ { K } + \left( \epsilon _ { 1 } - \epsilon _ { K } \right) \sin \left[ \frac { \pi ( N _ { y } - N _ { K } ) } { 2 ( N _ { 1 } - N _ { K } ) } \right] , \qquad y = 1 , 2 , . . . , K , } } \\ { { } } & { { } } \\ { { \mathrm { ( L i n e a r ) } } } & { { \epsilon _ { K } + \left( \epsilon _ { 1 } - \epsilon _ { K } \right) \frac { N _ { y } - N _ { K } } { N _ { 1 } - N _ { K } } , \qquad y = 1 , 2 , . . . , K , } } \\ { { } } & { { } } \\ { { \mathrm { ( C o n v e x ) } } } & { { \epsilon _ { 1 } + \left( \epsilon _ { 1 } - \epsilon _ { K } \right) \sin \left[ \frac { 3 \pi } { 2 } + \frac { \pi ( N _ { y } - N _ { K } ) } { 2 ( N _ { 1 } - N _ { K } ) } \right] , \quad y = 1 , 2 , . . . , K , } } \end{array} \right.
+$$
+
+where $\epsilon _ { 1 }$ and $\epsilon _ { K }$ are two hyperparameters. If we set $\epsilon _ { 1 } \geq \epsilon _ { K }$ , then we can get $\epsilon _ { 1 } \geq \epsilon _ { 2 } \geq . . . \geq \epsilon _ { K }$ . It means that if the instance number $N _ { y }$ for Class- $y$ is larger, label-aware smoothing will allocate a larger smoothing factor and lower the fitting probability to relieve the over-confidence because the head and medium classes are more likely to be over-confident than the tail classes (see Fig. 3).
+
+As the form of label-aware smoothing is more complicated than cross-entropy, we propose a more generalized classifier learning framework to fit it. Here we give a quick review about cRT and LWS: cRT tries to learn a new classifier weight, which contains $K M$ learnable parameters. LWS is restricted to learn the weight scaling vector $\bar { \mathbf { \boldsymbol { s } } } \in \mathbb { R } ^ { K }$ , which contains only $K$ learnable parameters. By contrast, cRT has more learnable parameters. It means cRT has a more powerful representation ability. LWS tends to obtain better validation losses and performances on large-scale datasets (refer to the experiment part in Kang et al. (2020)). It means LWS has a better generalization property. To combine the advantages of both cRT and LWS, we redesign the classifier framework in Stage-2:
+
+$$
+z = \mathrm { d i a g } ( s ) \left( a { \cal W } + \Delta { \cal W } \right) ^ { \top } x .
+$$
+
+In Eqn. (3), we fix the original classifier weight $W$ in Stage-2. If we make the learnable scaling vector $\pmb { s }$ fixed, set $\mathbf { \delta } _ { \mathbf { { s } } } = \mathbf { 1 } \ \mathbf { \delta } _ { \mathbf { { l } } }$ , $a = 0$ , and just learn the new classifier weight $\Delta W \in \mathbb { R } ^ { M \times K }$ , Eqn. (4) will degrade to cRT. Because LWS fixes the original classifier weights $W$ and only learns the scaling $\pmb { s }$ , Eqn. (4) will degrade to LWS if we set $a = 1$ and $\Delta W = \mathbf { 0 }$ . In most cases, LWS generally achieves better results than cRT on large scale datasets. Thus, we let $\pmb { s }$ learnable and set $a = 1$ . We also make $\Delta \mathbf { W }$ learnable to improve the representation ability but optimize $\Delta \mathbf { W }$ by a different learning rate. $\Delta \mathbf { W }$ can be viewed as doing a shift transformation on $W$ . This transformation can change the direction of the original weight vector $\pmb { w }$ in $W$ , which is what LWS cannot do.
+
+# 3.3 SHIFT LEARNING ON BATCH NORMALIZATION
+
+In the two-stage training framework, models are first trained with instance-balanced sampling in Stage-1 and then trained with class-balanced sampling in Stage-2. Since the framework involves two samplers, or two datasets: the instance-balanced dataset $\mathcal { D } _ { \mathrm { I } }$ and the class-balanced dataset $\mathcal { D } _ { \mathrm { C } }$ , we can regard this two-stage training framework as a derivative of transfer learning approaches. However, if we view the two-stage decoupling training framework from the transfer learning perspective, fixing the backbone part and just fine-tuning the classifier in Stage-2 will be clearly unreasonable, especially for the batch normalization (BN) layers.
+
+Concretely, we suppose that the input of the network is $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , the input feature of some BN layer is $g ( \pmb { x } _ { i } )$ , and the mini-batch size is $m$ . The running mean and the running variance of Channel- $j$ for these two stages are:
+
+$$
+\begin{array} { r l } & { \pmb { \mu } _ { \mathrm { I } } ^ { ( j ) } = \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } g ( { \pmb x } _ { i } ) ^ { ( j ) } , \quad \pmb { \sigma } _ { \mathrm { I } } ^ { 2 ^ { ( j ) } } = \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \left[ g ( { \pmb x } _ { i } ) ^ { ( j ) } - \pmb { \mu } _ { \mathrm { I } } ^ { ( j ) } \right] ^ { 2 } , \quad \pmb { x } _ { i } \sim P _ { { \mathcal { D } } _ { \mathrm { I } } } ( { \pmb x } , y ) , } \\ & { \pmb { \mu } _ { \mathrm { C } } ^ { ( j ) } = \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } g ( { \pmb x } _ { i } ) ^ { ( j ) } , \quad \pmb { \sigma } _ { \mathrm { C } } ^ { 2 ^ { ( j ) } } = \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \left[ g ( { \pmb x } _ { i } ) ^ { ( j ) } - \pmb { \mu } _ { \mathrm { C } } ^ { ( j ) } \right] ^ { 2 } , \quad \pmb { x } _ { i } \sim P _ { { \mathcal { D } } _ { \mathrm { C } } } ( { \pmb x } , y ) . } \end{array}
+$$
+
+![](images/19b86e3a88f558bf61a12c4d7e37ede85b460b9f37eb25257a310659fcc06882.jpg)  
+Figure 4: Reliability diagrams of ResNet-32 trained on LT CIFAR-100, $\mathrm { { I F } = 1 0 0 }$ . From left to right: cRT with mixup, LWS with mixup, LWS with mixup and shifted BN, and MiSLAS. It is better to look together with Fig. 1.
+
+Due to the different sampling strategies, the composition ratios of the head, medium, and tail classes are also totally different, which leads to $P _ { \mathcal { D } _ { \mathrm { I } } } ( \pmb { x } , y ) \neq P _ { \mathcal { D } _ { \mathrm { C } } } ( \pmb { x } , y )$ . Calculated by Eqn. (5) and (6), there exist some biases in $\pmb { \mu }$ and $\pmb { \sigma }$ under two sampling strategies, i.e., $\mu _ { \mathrm { I } } \neq \mu _ { \mathrm { C } }$ , and $\sigma _ { \mathrm { I } } ^ { 2 } \neq \sigma _ { \mathrm { C } } ^ { 2 }$ Thus, it is clearly infeasible for the decoupling framework that BN shares mean and variance across datasets with two sampling strategies. Motivated by AdaBN (Li et al., 2018) and TransNorm (Wang et al., 2019), we unfreeze the update procedures of the running mean $\pmb { \mu }$ and running variance $\sigma$ but fix the learnable linear transformation parameters $_ { \pmb { \alpha } }$ and $\beta$ for a better normalization in Stage-2.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+Our experimental setup including the implementation details and evaluation protocol mainly follows Cao et al. (2019) for LT CIFAR-10 and LT CIFAR-100, and Kang et al. (2020) for ImageNet-LT, Places-LT, and iNuturalist 2018. Please see Appendix A for further details.
+
+# 4.2 ABLATION STUDY
+
+Improving calibration. Here we show the reliability diagrams with 15 bins of our methods in Fig. 4. Comparing with Fig. 1 in the introduction part, both the mixup and label-aware smoothing can not only largely enhance the network calibration (even lower ECEs than them on balanced datasets) but also greatly improve the performance for long-tailed recognition. The similar trends can also be found on LT CIFAR-10, ImageNet-LT, and Places-LT (please see the figures in Appendix B for detail), which proves the powerful effects of the proposed method on calibration. According to all experiment results, training networks on imbalanced datasets leads to more severe over-confidence. Since the conventional mixup and label-smoothing both contain the operation of softening the ground truth labels, which may suggest that training with hard labels is likely to be another contributing factor leading to network over-confidence.
+
+Further analysis of label-aware smoothing. In our label-aware smoothing, there are two hyperparameters in Eqn. (3), $\epsilon _ { 1 }$ and $\epsilon _ { K }$ , which control the penalties of classes. In recognition system, if the predicted probability of some Class- $y$ is larger than 0.5, the classifier will classify the input to Class- $y$ . Thus, to ensure reasonability, we limit $0 \le \epsilon _ { K } \le \epsilon _ { 1 } \le 0 . 5$ . Here we conduct a comparing experiment for varying $\epsilon _ { 1 }$ and $\epsilon _ { K }$ both from 0.0 to 0.5 on LT CIFAR-100 with imbalanced factor 100. We plot the performance matrix upon $\epsilon _ { 1 }$ and $\epsilon _ { k }$ in Fig. 5 for all possible variants. From it, the classification accuracy can be further improved by $0 . 9 \%$ comparing with the conventional cross-entropy $( \epsilon _ { 1 } = 0$ , $\epsilon _ { K } = 0$ , green square) when we pick $\epsilon _ { 1 } = 0 . 4$ , $\epsilon _ { K } = 0 . 1$ (orange square) for label-aware smoothing. A more surprising improvement (growing by $3 . 3 \%$ ) can be found on LT CIFAR-10 (see Appendix D.1 for detail). We also find that the concave related function $f ( \cdot )$ in Eqn. (3) achieves the best performance but the gain is quite limited (refer Appendix D.2 for detail).
+
+To visualize the change in predicted probability distributions, we train two LWS models, one with cross-entropy and the other with label-aware smoothing on long-tailed CIFAR-100 with imbalanced factor 100. The cross-entropy-based distributions of the head, medium, and tail classes are showing in the upper half part of Fig. 3 in light blue. The label-aware smoothing-based distributions are showing in the bottom half part in deep blue. We observe that the over-confidence of head and medium classes relieve greatly, and the whole distribution of the tail classes slightly moves right (a larger mean) when using label-aware smoothing. This empirical visualization is consistent with our analysis mentioned in Sec. 3.2.
+
+Table 2: Ablation study for all proposed modules on long-tailed CIFAR-100, $\scriptstyle { \mathrm { I F } = 1 0 0 }$ . MU: applying mixup just in Stage-1. SL: shift learning on batch normalization. LAS: label-aware smoothing.   
+
+<table><tr><td colspan="3">Module</td><td colspan="2">LT CIFAR-100</td></tr><tr><td>MU</td><td>SL</td><td>LAS</td><td>100 50</td><td>10</td></tr><tr><td>区</td><td>区</td><td>区</td><td>41.2</td><td>46.0 58.5</td></tr><tr><td>回</td><td>区</td><td>区</td><td>44.2 50.6</td><td>62.2</td></tr><tr><td>回</td><td>回</td><td>区</td><td>45.3 51.4</td><td>62.8</td></tr><tr><td>回</td><td>回</td><td>回</td><td>47.0</td><td>52.3 63.0</td></tr></table>
+
+![](images/0315c7dbd655f6f6197a72a630ed74cf36e29310af7ed324d9922d4ddb62b979.jpg)
+
+![](images/aa2933fd6bdf9f8326ec3fc973f8a1fb1eb3117c88fb6fbb3ecb56edd5a3dd36.jpg)  
+Figure 5: Ablation study of two hyperparameters $\epsilon _ { 1 }$ and $\epsilon _ { K }$ in label-aware smoothing.   
+Figure 6: Visualization of the changes in the running mean $\pmb { \mu }$ and variance $\sigma ^ { 2 }$ . The ResNet-32 based model is trained on LT CIFAR-100 with imbalanced factor 100. Left: $\pmb { \mu }$ and $\sigma ^ { 2 }$ in the first BN of ResNet-32, which contains 16 channels. Right: $\pmb { \mu }$ and $\sigma ^ { 2 }$ in the last BN of ResNet-32, which contains 64 channels.
+
+Further analysis of shift learning. In this part, we conduct an empirical experiment to show the effectiveness and reasonability of shift learning on BN. We train the LWS model on long-tailed CIFAR-100 with imbalanced factor 100. After 10 epochs finetuning in Stage-2, the model trained with BN shifting achieves accuracy at $4 5 . 3 \%$ , which is $1 . 1 \%$ higher than it without BN shifting. We also draw a visualization of the change in BN. As shown in Fig. 6, we see that there indeed exist biases in $\pmb { \mu }$ and $\sigma ^ { 2 }$ between the dataset using different sampling strategies. Due to the composition ratios of the head classes, medium classes and tail classes are different in terms of different sampling strategies, the statistic running mean $\pmb { \mu }$ and running variance $\sigma ^ { 2 }$ are certainly different. We also find some interesting phenomenons need for future exploration: (i) The changes in the running variance $\sigma ^ { 2 }$ are larger than the changes in the running mean $\pmb { \mu }$ . (ii) The changes of $\pmb { \mu }$ and $\sigma ^ { 2 }$ in deep BN layers are quite smaller than them in shallow BN layers.
+
+Overall, Table 2 shows the ablation investigation on the effects of mixup (adding mixup in Stage1, MU), shift learning on batch normalization (SL), and label-aware smoothing (LAS). From it, each proposed module can further improve the performances on long-tailed CIFAR-100 for all commonly-used imbalanced factors, which firmly demonstrates the effectiveness.
+
+# 4.3 COMPARISON WITH THE STATE-OF-THE-ART
+
+In this subsection, we compare the proposed method against previous one-stage methods, such as Range Loss (Zhang et al., 2017), LDAM Loss (Cao et al., 2019), FSLwF (Gidaris & Komodakis, 2018), and OLTR (Liu et al., 2019), and against previous two-stage methods, such as DRS-like, DRW-like (Cao et al., 2019), LFME (Xiang & Ding, 2020), cRT, and LWS (Kang et al., 2020). For fair comparisons, we also add mixup on the LWS and cRT models. Remix (Chou et al., 2020) is a recently proposed augmentation method for long-tail recognition. Because BBN (Zhou et al., 2020) has double samplers and is trained in a mixup-like manner, we directly compare our method with it.
+
+Table 3: Top-1 accuracy $( \% )$ for ResNet-32 models trained on long tailed CIFAR-10 and CIFAR-100.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Long-tailed CIFAR-10</td><td colspan="3">Long-tailed CIFAR-100</td></tr><tr><td>100</td><td>50</td><td>10</td><td>100</td><td>50</td><td>10</td></tr><tr><td>CE</td><td>70.4</td><td>74.8</td><td>86.4</td><td>38.4</td><td>43.9</td><td>55.8</td></tr><tr><td>mixup</td><td>73.1</td><td>77.8</td><td>87.1</td><td>39.6</td><td>45.0</td><td>58.2</td></tr><tr><td>LDAM+DRW</td><td>77.1</td><td>81.1</td><td>88.4</td><td>42.1</td><td>46.7</td><td>58.8</td></tr><tr><td>BBN(include mixup)</td><td>79.9</td><td>82.2</td><td>88.4</td><td>42.6</td><td>47.1</td><td>59.2</td></tr><tr><td>Remix+DRW(30 epochs)</td><td>79.8</td><td>=</td><td>89.1</td><td>46.8</td><td>-</td><td>61.3</td></tr><tr><td>cRT+mixup</td><td>79.1</td><td>84.2</td><td>89.8</td><td>45.1</td><td>50.9</td><td>62.1</td></tr><tr><td>LWS+mixup</td><td>76.3</td><td>82.6</td><td>89.6</td><td>44.2</td><td>50.6</td><td>62.2</td></tr><tr><td>MiSLAS</td><td>82.1</td><td>85.8</td><td>89.9</td><td>47.0</td><td>52.3</td><td>63.0</td></tr></table>
+
+Table 4: Top-1 accuracy $( \% )$ on ImageNet-LT (left), iNaturalist 2018 (center) and Place-LT (right).   
+
+<table><tr><td>Method</td><td>ResNet-50</td></tr><tr><td>CE</td><td>44.6</td></tr><tr><td>CE+DRW</td><td>48.5</td></tr><tr><td>Focal+DRW</td><td>47.9</td></tr><tr><td>LDAM+DRW</td><td>48.8</td></tr><tr><td>CRT+mixup LWS+mixup MiSLAS</td><td>51.7 52.0 52.7</td></tr></table>
+
+(a) ImageNet-LT (b) iNaturalist 2018 (c) Place-LT
+
+<table><tr><td rowspan=1 colspan=4>Method</td><td rowspan=1 colspan=1>ResNet-50</td></tr><tr><td rowspan=1 colspan=4>CB-FocalLDAM+DRW</td><td rowspan=1 colspan=1>61.168.0</td></tr><tr><td rowspan=1 colspan=4>BBN(include mixup)</td><td rowspan=1 colspan=1>69.6</td></tr><tr><td rowspan=1 colspan=4>Remix+DRW</td><td rowspan=1 colspan=1>70.5</td></tr><tr><td rowspan=1 colspan=3>cRT+mixup</td><td rowspan=1 colspan=1>up</td><td rowspan=1 colspan=1>70.2</td></tr><tr><td rowspan=2 colspan=4>LWS+mixupMiSLAS</td><td rowspan=1 colspan=1>mixup</td></tr><tr><td rowspan=1 colspan=1>71.6</td></tr></table>
+
+<table><tr><td>Method</td><td>ResNet-152</td></tr><tr><td>Range Loss</td><td>35.1</td></tr><tr><td>FSLwF</td><td>34.9</td></tr><tr><td>OLTR OLTR+LFME</td><td>35.9 36.2</td></tr><tr><td></td><td>38.3</td></tr><tr><td>cRT+mixup LWS+mixup</td><td>39.7</td></tr><tr><td>MiSLAS</td><td>40.4</td></tr></table>
+
+Experimental results on CIFAR-LT. We conduct extensive experiments on long-tailed CIFAR-10 and CIFAR-100 with imbalanced factors of 10, 50, and 100, which is the same as the previous setting (Cao et al., 2019; Zhou et al., 2020). The experimental results are summarized in Table 3. Compared with previous methods ( $^ +$ mixup, one/two-stage), our MiSLAS outperforms all previous methods by a large margin. Moreover, this superiority of the proposed method holds for all imbalanced factors on both long-tailed CIFAR-10 and CIFAR-100.
+
+Experimental results on ImageNet-LT, iNaturalist 2018, and Place-LT. We further verify the effectiveness of our method on three large-scale imbalanced datasets, i.e., ImageNet-LT, iNaturalist 2018, and Place-LT. Table 4 lists experimental results on ImageNet-LT (left), iNaturalist 2018 (center), and Places-LT (right). Notably, our MiSLAS still outperforms all competing approaches and sets new state-of-the-art records for all three large-scale long-tailed benchmarks. More detailed results about the split class accuracies and different backbones on these three datasets are listed in Appendix C.
+
+# 5 CONCLUSION
+
+In this paper, we discover that models trained on long-tailed datasets are more miscalibrated and over-confident than them trained on balanced datasets. The two-stage models suffer the same issue as well. To relieve over-confidence, we propose two solutions: (i) We find that mixup can remedy over-confidence and have a positive effect on representation learning but a negative or negligible effect on classifier learning. (ii) To further improve classifier learning and calibration, we propose label-aware smoothing to handle the different degrees of over-confidence for different classes. We are the first to note the dataset bias or domain shift in two-stage resampling methods for long-tailed recognition. To solve the dataset bias producing by different re-sampling in the decoupling framework, we propose shift learning on the batch normalization layer and this novel model can greatly improve the performance. Extensive quantitative and qualitative experiments on multiple benchmark datasets show that our MiSLAS achieves superior performances over the state-of-the-art methods.
+
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+
+# A EXPERIMENT SETUP
+
+# A.1 DATASETS EXPLANATION
+
+CIFAR-10-LT and CIFAR-100-LT. CIFAR-10 and CIFAR-100 both have 60,000 images 50,000 for training and 10,000 for validation with 10 categories and 100 categories, respectively. For a fair comparison, we use the long-tailed versions of CIFAR datasets with the same setting as those used in Cao et al. (2019). They control the degrees of data imbalance with an imbalanced factor $\beta$ . $\begin{array} { r } { \beta = \frac { N _ { \mathrm { m a x } } } { N _ { \mathrm { m i n } } } } \end{array}$ , where $N _ { \mathrm { m a x } }$ and $N _ { \mathrm { m i n } }$ are the numbers of training samples for the most frequent class and the least frequent class. Following Cao et al. (2019) and Zhou et al. (2020), we conduct experiments with imbalanced factors 100, 50, and 10.
+
+ImageNet-LT and Places-LT. ImageNet-LT and Places-LT were proposed by Liu et al. (2019). ImageNet-LT is a long-tailed version of the large-scale object classification dataset ImageNet (Russakovsky et al., 2015) by sampling a subset following the Pareto distribution with power value $\alpha = 6$ . It contains 115.8K images from 1,000 categories, with class cardinality ranging from 5 to 1,280. Places-LT is a long-tailed version of the large-scale scene classification dataset Places (Zhou et al., 2017). It consists of 184.5K images from 365 categories with class cardinality ranging from 5 to 4,980.
+
+iNaturalist 2018. iNaturalist 2018 (Van Horn et al., 2018) is one species classification dataset, which is on a large scale and suffers from extremely imbalanced label distributions. It is composed of 437.5K images from 8,142 categories. In addition to the extreme imbalance, the iNaturalist 2018 dataset also confronts the fine-grained problem.
+
+# A.2 EVALUATION PROTOCOL
+
+Following Liu et al. (2019) and Kang et al. (2020), we report the commonly used top-1 accuracy over all classes on the balanced test/validation datasets, denoted as All. In more detail, further report accuracy on three splits of the set of classes: Head-Many (more than 100 images), Med.-Medium (20 to 100 images) and Tail-Few (less than 20 images).
+
+# A.3 IMPLEMENTATION DETAILS
+
+For all experiments, we use the SGD optimizer with momentum 0.9 to optimize networks.
+
+For long-tailed CIFAR, we mainly follow Cao et al. (2019). We train all MiSLAS models with the ResNet-32 backbone on one GPU and use the multistep learning rate schedule, which decreases the learning rate by 0.1 at the $1 6 0 ^ { \mathrm { t h } }$ epoch and the $1 8 0 ^ { \mathrm { t h } }$ epochs in Stage-1.
+
+For ImageNet-LT, Place-LT, and iNaturalist 2018, We mainly follow Kang et al. (2020) and use the cosine learning rate schedule (Loshchilov & Hutter, 2017) to train all MiSLAS models with the ResNet-10/50/101/152 backbones on 4 GPUs.
+
+Table 5: Detailed experiment settings on five benchmark datasets. LR: learning rate, BS: batch size, WD: weight decay, and LRS: learning rate schedule, $\Delta \mathbf { W }$ : learning rate ratio of $\Delta \mathbf { W }$ .   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">Common</td><td colspan="2">Stage-1</td><td colspan="5">Stage-2</td></tr><tr><td>LR</td><td>BS</td><td>WD</td><td>Epochs</td><td>LRS</td><td>Epochs</td><td>LRS</td><td>@1</td><td>EK</td><td>△W</td></tr><tr><td>LT CIFAR-10</td><td>0.1</td><td>128</td><td>2e-4</td><td>200</td><td>multi.</td><td>10</td><td>cosine</td><td>0.3</td><td>0.0</td><td>0.5x</td></tr><tr><td>LT CIFAR-100</td><td>0.1</td><td>128</td><td>2e-4</td><td>200</td><td>multi.</td><td>10</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.2x</td></tr><tr><td>ImageNet-LT</td><td>0.1</td><td>256</td><td>5e-4</td><td>180</td><td>cosine</td><td>10</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.05x</td></tr><tr><td>Places-LT</td><td>0.1</td><td>256</td><td>5e-4</td><td>90</td><td>cosine</td><td>10</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.05x</td></tr><tr><td>iNaturalist&#x27;18</td><td>0.1</td><td>256</td><td>1e-4</td><td>200</td><td>cosine</td><td>30</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.05x</td></tr></table>
+
+# B CALIBRATION
+
+![](images/eacc92a146d5ac8a9a51356e182064f8bab077b1209b16054aeef90c0ea78cb2.jpg)  
+Figure 7: Reliability diagrams on CIFAR10 with 15 bins. From left to right: plain ResNet-32 model trained on the original CIFAR-10 dataset, plain model, cRT, LWS, and MiSLAS trained on long-tailed CIFAR-10 with imbalanced factor 100.
+
+![](images/ec26f5f2647f296366c96bb51460bb73017ab9b16d3e9ad67012cc02bd8b43d6.jpg)  
+Figure 8: Reliability diagrams on ImageNet with 15 bins. From left to right: plain ResNet-50 model trained on the original ImageNet dataset, plain model, cRT, LWS, and MiSLAS trained on ImageNet-LT.
+
+![](images/4b1d456880deb636b722c684adcb5b082238c3880ce0191066e72f548d599faf.jpg)
+
+Figure 9: Reliability diagrams of ResNet-152 trained on Places-LT with 15 bins. From left to right: cRT, LWS, cRT with mixup, LWS with mixup, and MiSLAS.
+
+# C MORE DETAILED RESULTS ON IMAGENET-LT, PLACES-LT AND INATURALIST 2018
+
+Table 6: Comprehensive accuracy results on ImageNet-LT with different backbone networks (ResNet50, ResNet-101 & ResNet-152) and training 180 epochs.   
+
+<table><tr><td>Backbone</td><td>Method</td><td>Many</td><td>Medium</td><td>Few</td><td>All</td></tr><tr><td rowspan="4">ResNet-50</td><td>cRT</td><td>62.5</td><td>47.4</td><td>29.5</td><td>50.3</td></tr><tr><td>LWS</td><td>61.8</td><td>48.6</td><td>33.5</td><td>51.2</td></tr><tr><td>cRT+mixup</td><td>63.9</td><td>49.1</td><td>30.2</td><td>51.7</td></tr><tr><td>LWS+mixup MiSLAS</td><td>62.9 61.7</td><td>49.8 51.3</td><td>31.6 35.8</td><td>52.0 52.7</td></tr><tr><td rowspan="4">ResNet-101</td><td>cRT LWS</td><td>63.8 63.1</td><td>48.5</td><td>30.0</td><td>51.4</td></tr><tr><td>cRT+mixup</td><td>65.2</td><td>49.9</td><td>33.8</td><td>52.3</td></tr><tr><td>LWS+mixup</td><td>64.5</td><td>50.6</td><td>31.6</td><td>53.1</td></tr><tr><td>MiSLAS</td><td>64.3</td><td>51.2 52.1</td><td>34.1 35.8</td><td>53.5 54.1</td></tr><tr><td rowspan="4">ResNet-152</td><td>cRT LWS</td><td>64.9</td><td>50.4</td><td>30.6</td><td>52.7</td></tr><tr><td></td><td>64.1</td><td>51.8</td><td>35.5</td><td>53.8</td></tr><tr><td>cRT+mixup</td><td>66.5</td><td>51.6</td><td>32.8</td><td>54.2</td></tr><tr><td>LWS+mixup MiSLAS</td><td>66.1 65.4</td><td>52.2 53.2</td><td>34.5 37.1</td><td>54.6 55.2</td></tr></table>
+
+Table 7: Comprehensive accuracy results on iNaturalist 2018 with ResNet-50 and training 200 epochs.   
+
+<table><tr><td>Backbone</td><td>Method</td><td>Many</td><td>Medium</td><td>Few</td><td>All</td></tr><tr><td rowspan="5">ResNet-50</td><td>cRT</td><td>73.2</td><td>68.8</td><td>66.1</td><td>68.2</td></tr><tr><td>T-normalized</td><td>71.1</td><td>68.9</td><td>69.3</td><td>69.3</td></tr><tr><td>LWS</td><td>71.0</td><td>69.8</td><td>68.8</td><td>69.5</td></tr><tr><td>cRT+mixup</td><td>74.2</td><td>71.1</td><td>68.2</td><td>70.2</td></tr><tr><td>LWS+mixup</td><td>72.8</td><td>71.6</td><td>69.8</td><td>70.9</td></tr><tr><td></td><td>MiSLAS</td><td>73.2</td><td>72.4</td><td>70.4</td><td>71.6</td></tr></table>
+
+Table 8: Detailed accuracy results on Places-LT, starting from an ImageNet pre-trained ResNet-152.   
+
+<table><tr><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Many      Medium      Few       All</td></tr><tr><td rowspan=10 colspan=1>ResNet-152</td><td rowspan=1 colspan=1>Lifted Loss</td><td rowspan=1 colspan=1>41.1         35.4        24.0      35.2</td></tr><tr><td rowspan=1 colspan=1>Focal Loss</td><td rowspan=1 colspan=1>41.1         34.8        22.4      34.6</td></tr><tr><td rowspan=1 colspan=1>Range Loss</td><td rowspan=1 colspan=1>41.1         35.4        23.2      35.1</td></tr><tr><td rowspan=1 colspan=1>FSLwFOLTR</td><td rowspan=1 colspan=1>43.9         29.9        29.5      34.944.7         37.0        25.3      35.9</td></tr><tr><td rowspan=1 colspan=1>OLTR+LFME</td><td rowspan=1 colspan=1>39.3         39.6        24.2      36.2</td></tr><tr><td rowspan=2 colspan=1>cRTT-normalizedLWS</td><td rowspan=1 colspan=1>42.0         37.6        24.9      36.7</td></tr><tr><td rowspan=1 colspan=1>37.8         40.7        31.8      37.940.6         39.1        28.6      37.6</td></tr><tr><td rowspan=3 colspan=1>cRT+mixupLWS+mixupMiSLAS</td><td rowspan=1 colspan=1>44.1         38.5        27.1      38.1</td></tr><tr><td rowspan=1 colspan=1>41.7         41.3        33.1      39.7</td></tr><tr><td rowspan=1 colspan=1>39.6         43.3        36.1      40.4</td></tr></table>
+
+# D ABLATION STUDY OF LABEL-AWARE SMOOTHING
+
+![](images/16ab8345b0b49db5c1f6d862e06d90e6e500f84f1e06457336244442b23313d5.jpg)  
+D.1 MORE RESULTS ABOUT THE HYPERPARAMETERS $\epsilon _ { 1 }$ AND $\epsilon _ { K }$   
+Figure 10: Ablation study of two hyperparameters $\epsilon _ { 1 }$ and $\epsilon _ { K }$ in label-aware smoothing. Our labelaware smoothing (orange square) outperforms cross-entropy (green square) by a large margin on both long-tailed CIFAR-10 (left) and long-tailed CIFAR-100 (right).
+
+# D.2 FORM OF THE RELATED FUNCTION $f ( \cdot )$
+
+As discussed in Sec. 3.2 and Sec. 4.2, the form of the related function $f ( \cdot )$ may play a significant role for the final model performance. We draw the visualization of Eqn. (3) at the left part of Fig. 11. For the LT CIFAR-100 dataset with balanced factor 100, $N _ { 1 } = 5 0 0$ and $N _ { 1 0 0 } = 5$ . Based on the ablation study results of $\epsilon _ { 1 }$ and $\epsilon _ { K }$ mentioned in Sec. 4.2 and above, we set $\epsilon _ { 1 } = 0 . 4$ and $\epsilon _ { 1 0 0 } = 0 . 1 $ here. After fintuning for 10 epochs in Stage-2, the accuracy of the concave model is the best. We also design a power-like related function, which can be written as:
+
+$$
+\epsilon _ { y } = f ( N _ { y } ) = \epsilon _ { K } + ( \epsilon _ { 1 } - \epsilon _ { K } ) \bigg ( { \frac { N _ { y } - N _ { K } } { N _ { 1 } - N _ { K } } } \bigg ) ^ { p } , \qquad y = 1 , 2 , . . . , K ,
+$$
+
+where $p$ is a hyperparameter to control the shape of the related function. For example, we will get concave related function if we set $p < 1$ and we will get convex related function if we set $p > 1$ . The visualization of Eqn. (7) is shown at the right part of Fig. 11. However, comparing the accuracies of all variants, the influence of the related function form is quite limited for the final performance (just growing by $0 . 3 \%$ ). Because the concave related function in Eqn. (3) achieves the best performance among all variants, we choose it as the default setting of the related function $f ( \cdot )$ for other experiments.
+
+![](images/98f172c1a250bbc32349248542a822455a2e8cc42d80596499181b72ec7f5323.jpg)  
+Figure 11: Function visualization and accuracy of Eqn. (3) (left) and Eqn. (7) (right).

parse/train/Sx-mvOvnmJj/Sx-mvOvnmJj_content_list.json

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+    {
+        "type": "text",
+        "text": "IMPROVING CALIBRATION FOR LONG-TAILED RECOG-NITION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks often perform poorly when training datasets are heavily classimbalanced. Recently, two-stage methods greatly improve the performances by decoupling representation learning and classifier learning. In this paper, we discover that networks trained on long-tailed datasets are more prone to miscalibrated and over-confident. The two-stage models suffer the same issue as well. We design two novel methods to improve calibration and performance in such scenarios. Motivated by the predicted probability distributions of classes are highly related to the numbers of class instances, we propose a label-aware smoothing to deal with the different degrees of over-confidence for different classes and improve classifier learning. Noting that there is a dataset bias between these two stages because of different samplers, we further propose a shifted batch normalization to solve the dataset bias in the decoupling framework. Through extensive experiments, we also observe that mixup can remedy over-confidence and improve representation learning but has a negative or negligible effect on classifier learning. Our proposed methods set new records on multiple popular long-tailed recognition benchmarks including LT CIFAR 10/100, ImageNet-LT, Places-LT, and iNaturalist 2018. ",
+        "bbox": [
+            233,
+            267,
+            766,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            334,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "With numerous available large-scale and high-quality datasets such as ImageNet (Russakovsky et al., 2015), COCO (Lin et al., 2014), and Places (Zhou et al., 2017), deep convolutional neural networks (CNNs) have made notable breakthroughs in various computer vision tasks such as image recognition (Krizhevsky et al., 2012; He et al., 2016), object detection (Ren et al., 2015) and semantic segmentation (Cordts et al., 2016). These delicate datasets are usually artificially balanced with respect to the number of instances for each object/class. However, in real-world applications, data often follows an unexpected long-tailed distribution, where the numbers of instances for different classes are seriously imbalanced. When training CNNs on such long-tailed datasets, the performances extremely degrade. Motivated by this phenomenon, a number of works have recently emerged that try to explore long-tailed recognition. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, many two-stage approaches have achieved significant improvement comparing with onestage methods. Concretely, DRS and DRW (Cao et al., 2019) first train CNNs in a normal way in Stage-1. DRS finetunes CNNs on datasets with class-balanced resampling while DRW finetunes CNNs by assigning different weights to different classes in Stage-2. Zhou et al. (2020) proposed BBN with one-stage to simulate the process of DRS by dynamically combining the instance-balanced sampler and the reverse-balanced sampler. Kang et al. (2020) proposed two-stage decoupling models, cRT and LWS, to further boost the performance: Decoupling models freeze the backbone and just finetune the classifier with class-balanced resampling in Stage-2. ",
+        "bbox": [
+            174,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Confidence calibration (Niculescu-Mizil & Caruana, 2005; Guo et al., 2017) – the problem of predicting probability estimates representative of the true correctness likelihood – is important for recognition models in many applications (Bojarski et al., 2016; Jiang et al., 2012). In this study, we discover that networks trained on long-tailed datasets are more miscalibrated and over-confident: We draw the reliability diagrams with 15 bins in Fig. 1, which compares the plain model trained on the original CIFAR-100 dataset, the plain model, cRT, and LWS trained on long-tailed CIFAR-100 with imbalanced factor (IF) 100. We observe that networks trained on long-tailed datasets have higher expected calibration errors (ECEs). The two-stage models, cRT and LWS, suffer over-confidence as well. Moreover, Fig. 7 and Fig. 8 (the first two plots) in Appendix B depict that this phenomenon also commonly exists on other long-tailed datasets such as LT CIFAR-10 and ImageNet-LT. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e44aa4f49a7033716a9652c8793a4caf195ce8a53f64c89a6c341469c58c17a.jpg",
+        "image_caption": [
+            "Figure 1: Reliability diagrams of ResNet-32. From left to right: the plain model trained on the original CIFAR-100 dataset, the plain model, cRT, and LWS trained on long-tailed CIFAR-100 with $\\mathrm { { I F } = 1 0 0 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            818,
+            239
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            281,
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+            309
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another issue is that two-stage decoupling methods ignore the dataset bias or domain shift (QuioneroCandela et al., 2009) between these two stages. Concretely, two-stage models are first trained on the instanced-balanced dataset $\\mathcal { D } _ { \\mathrm { I } }$ in Stage-1. Then, models are trained on the class-balanced dataset $\\mathcal { D } _ { \\mathrm { C } }$ in Stage-2. Obviously, $P _ { \\mathcal { D } _ { \\mathrm { I } } } ( { \\pmb x } , y ) \\neq P _ { \\mathcal { D } _ { \\mathrm { C } } } ( { \\pmb x } , y )$ , the distributions of the dataset with different sampling manners are inconsistent. Motivated by the transfer learning methods (Li et al., 2018; Wang et al., 2019), we focus on the batch normalization (Ioffe & Szegedy, 2015) layer to deal with the dataset bias problem. ",
+        "bbox": [
+            174,
+            316,
+            825,
+            414
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a Mixup Shifted Label-Aware Smoothing model (MiSLAS) to effectively solve the above issues. Our key contributions are as follows: (i) We discover that models trained on long-tailed datasets are much more miscalibrated and over-confident than them trained on balanced datasets. The two-stage models suffer the same problem as well. (ii) We find that mixup can remedy over-confidence and have a positive effect on representation learning but a negative or negligible effect on classifier learning. To further enhance classifier learning and calibration, we propose a label-aware smoothing to handle the different degrees of over-confidence for different classes. (iii) We are the first to note the dataset bias or domain shift in two-stage resampling methods for long-tailed recognition. To deal with the dataset bias in the decoupling framework, we propose shift learning on the batch normalization layer, which can greatly improve the performance. ",
+        "bbox": [
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+            420,
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+            560
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We extensively validate our MiSLAS on multiple long-tailed recognition benchmark datasets, i.e., LT CIFAR-10, LT CIFAR-100, ImageNet-LT, Places-LT, and iNaturalist 2018. Experimental results manifest that the effectiveness and our method yields new state-of-the-art. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            630,
+            354,
+            646
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Re-sampling and re-weighting. There are two groups of re-sampling strategies: over-sampling the tail-class images (Shen et al., 2016; Buda et al., 2018; Byrd & Lipton, 2019) and under-sampling the head-class images (Japkowicz & Stephen, 2002; Buda et al., 2018). Over-sampling is regularly useful on large datasets and often suffers from heavy over-fitting to tail classes especially on small datasets. For under-sampling, it discards a large portion of data, which inevitably causes degradation of the generalization ability of deep models. Re-weighting (Huang et al., 2016; Wang et al., 2017) is another prominent strategy. It assigns different weights for classes and even instances. The vanilla re-weighting method gives class weights in reverse proportion to the number of samples of classes. However, with large-scale data, re-weighting makes the deep models difficult to optimize during training. Cui et al. (2019) relieved the problem using the effective numbers to calculate the class weights. Another line of work is to adaptively re-weight each instance, e.g., Focal loss (Lin et al., 2017) assigned smaller weights for well-classified samples. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Network calibration and regularization. Calibrated confidence is significant for classification models in many applications. The calibration of modern neural networks is first discussed in Guo et al. (2017). The authors discovered that model capacity, normalization, and regularization have strong effects on network calibration. mixup (Zhang et al., 2018) is a regularization technique that is proposed to train with interpolations of inputs and labels. mixup inspires several follow-ups like manifold mixup (Verma et al., 2019), CutMix (Yun et al., 2019), and Remix (Chou et al., 2020) that have shown significant improvement over mixup. Thulasidasan et al. (2019) found that CNNs trained with mixup are significantly better calibrated. Label smoothing (Szegedy et al., 2016) is another regularization technique that encourages the model to be less over-confident. Unlike cross-entropy computes loss upon the ground truth labels, label smoothing computes loss upon a soft version of the label, which can relieve the over-fitting and increase calibration and reliability (Müller et al., 2019). ",
+        "bbox": [
+            174,
+            840,
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+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/f963db073ee460c2b5c41f6812b7f887854d5166890b8d68e958900f6889d749.jpg",
+        "table_caption": [
+            "Table 1: Top-1 accuracy of the decoupling models (cRT and LWS) for ResNet families trained on the ImageNetLT dataset. We vary the augmentation strategies (with or without mixup $\\alpha = 0 . 2$ ) on both two stages. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Training setup for two stages</td><td colspan=\"2\">ResNet-10</td><td colspan=\"2\">ResNet-50</td><td colspan=\"2\">ResNet-101</td><td colspan=\"2\">ResNet-152</td></tr><tr><td>cRT LWS|</td><td></td><td>cRT LWS|</td><td></td><td>cRT</td><td>LWS</td><td>cRT1</td><td>LWS</td></tr><tr><td>Stage-1 (no mixup) Stage-1 (mixup)</td><td>36.8 35.7</td><td>36.8 35.7</td><td>45.8 45.6</td><td>45.8 45.6</td><td>47.3 47.7</td><td>47.3 47.7</td><td>48.7 48.4</td><td>48.7 48.4</td></tr><tr><td>Stage-1 (no mixup) + Stage-2 (no mixup) Stage-1 (no mixup) + Stage-2 (mixup)</td><td>43.3 43.0</td><td>43.5 43.3</td><td>50.3 50.2</td><td>51.2 51.1</td><td>51.4 51.4</td><td>52.3 52.2</td><td>52.7 52.8</td><td>53.8 53.6</td></tr><tr><td>Stage-1 (mixup) + Stage-2 (no mixup) Stage-1(mixup) + Stage-2 (mixup)</td><td>43.4 43.3</td><td>42.9 42.8</td><td>51.7 51.6</td><td>52.0 51.9</td><td>53.1 53.0</td><td>53.5 53.5</td><td>54.2 54.1</td><td>54.6 54.5</td></tr></table>",
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+        "type": "image",
+        "img_path": "images/dfadadbac16edfdf51bd971589ab03899f3826ea4b154206dcc6bd81b814a482.jpg",
+        "image_caption": [
+            "Figure 2: Classifier weight norms for the ImageNet-LT validation set when classes are sorted by descending values of $N _ { j }$ . Left: weight norms of cRT with or without mixup. Right: weight norms of LWS with or without mixup. (light shade: true norm, dark lines: smooth version) "
+        ],
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+        "type": "text",
+        "text": "Two-stage methods. Cao et al. (2019) first proposed deferred re-weighting (DRW) and deferred re-sampling (DRS) that are superior to conventional one-stage methods: Stage-2, starting from better features, adjusts the decision boundary and locally fine-tunes the features. Recently, Kang et al. (2020) and Zhou et al. (2020) concluded that although class re-balance strategies matter when jointly training representation and classifier, instance-balanced sampling gives more general representations. Based on this observation, Kang et al. (2020) achieved state-of-the-art results by decomposing representation and classifier learning, i.e., first train the deep models with instance-balanced sampling, then fine-tune the classifier with class-balanced sampling while keeping parameters of representation learning fixed. Similarly, Zhou et al. (2020) integrated mixup training into the proposed cumulative learning strategy with which they bridged the representation learning and classifier re-balancing. The cumulative learning strategy requires dual samplers: instance-balanced and reversed instance-balanced sampler. ",
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+    {
+        "type": "text",
+        "text": "3 MAIN APPROACH",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 IMPROVING CALIBRATION AND REPRESENTATION LEARNING BY MIXUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the two-stage learning framework, Kang et al. (2020) and Zhou et al. (2020) found that instancebalanced sampling gives the most generalizable representations among other sampling methods. Thulasidasan et al. (2019) found that networks trained with mixup are better calibrated. When using instance-balanced sampling, to further improve the representation generalization and relieve over-confidence, we explore the effect of mixup in the two-stage decoupling framework. ",
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+        "type": "text",
+        "text": "Here, we train two two-stage models, i.e. cRT and LWS, on ImageNet-LT for 180 epochs in Stage-1 and finetune for 10 epochs in Stage-2, respectively. We vary the training setup (with/without mixup $\\alpha = 0 . 2$ ) for both two stages. Top-1 accuracy results of these variants are listed in Table 1. From it, we conclude that: (i) When applying mixup, the performance improvements of Stage-1 are ignorable but the performances of Stage-2 are greatly enhanced for both cRT and LWS. (ii) Applying additional mixup in Stage-2 has no obvious improvement or even damages the performance, which means that mixup encourages representation learning but has a negative or negligible effect on classifier learning. ",
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+        "image_caption": [
+            "Figure 3: Violin plot of predicted probability distributions for different parts of classes, head (more than 100 images), medium (20 to 100 images), and tail (less than 20 images) on LT CIFAR-100, $\\mathrm { { I F } = 1 0 0 }$ . The upper half part in light blue: LWS (cross-entropy). The bottom half part in deep blue: LWS (label-aware smoothing). "
+        ],
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+    {
+        "type": "text",
+        "text": "We also draw the final classifier weight norms of these variants in Fig. 2. We show the $L _ { 2 }$ norms of the weight vectors for all classes, as well as the training data distribution sorted in a descending manner concerning the number of instances. We observe that when applying mixup (orange line), the weight norms of the tail classes uniformly tend to become larger and the weight norms of the head classes are decreased, which means mixup may be more friendly to the tail classes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The analysis of calibration for networks whether adding mixup will be discussed in our experiment part (Sec. 4.2). Due to the poor and unsatisfied enhancement of mixup for classifier learning, we further propose a label-aware smoothing to improve both the calibration and classifier learning. ",
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+    {
+        "type": "text",
+        "text": "3.2 IMPROVING CALIBRATION AND CLASSIFIER LEARNING BY LABEL-AWARE SMOOTHING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As discussed in the introduction part and Sec. 3.1, two-stage models suffer serious over-confidence and there is no significant improvement for classifier learning when adding additional mixup. In this subsection, we try to analyze and deal with these two issues. Suppose that the weight of the classifier is $\\pmb { W } \\in \\mathbb { R } ^ { M \\times K }$ , where $M$ is the number of features and $K$ is the number of classes. The cross-entropy encourages the whole network to be over-confident on the head classes: Concretely, the cross-entropy loss after the softmax activation is $l ( y , p ) = - \\log ( p _ { y } ) = - { \\pmb w } _ { y } ^ { \\top } { \\pmb x } + \\log ( \\sum \\exp ( { \\pmb w } _ { i } ^ { \\top } { \\pmb x } ) )$ , where $y \\in \\{ 1 , 2 , . . . , K \\}$ is the label, $\\pmb { x } \\in \\mathbb { R } ^ { M }$ is the feature vector send to classifier and ${ \\pmb w } _ { i }$ is the $i$ -th column vector of $W$ . The optimal solution is $\\pmb { w } _ { y _ { . } } ^ { * } \\top \\pmb { x } = \\operatorname { i n f }$ while keeping others ${ \\pmb w } _ { i } ^ { \\top } { \\pmb x }$ , $i \\neq y$ , small enough. Because the head classes contain much more training examples, the network makes the weight norm $\\| \\pmb { w } \\|$ of the head classes become larger to near the optimal solution as much as possible, which results that their predicted probabilities mainly concentrate near 1.0 (see Fig. 3, the upper half part showing in light blue). Another fact we can get from Fig. 3 is that the distributions of predicted probability are severely related to the instance numbers. Unlike balanced recognition, we claim that applying different strategies for different classes is extremely necessary for the long-tailed problem. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we propose a label-aware smoothing to solve the over-confidence in cross-entropy and the different distributions of predicted probability issue. The mathematical computation of label-aware smoothing is: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/fed4b7254500dd06fecfb7e6a4488c1bd6ba32a39df179bee12c41dde7f0cf10.jpg",
+        "text": "$$\nl ( q , p ) = - \\sum _ { i = 1 } ^ { K } q _ { i } \\log p _ { i } , q _ { i } = \\left\\{ \\begin{array} { l l } { 1 - \\epsilon _ { y } = 1 - f ( N _ { y } ) , } & { i = y , } \\\\ { \\frac { \\epsilon _ { y } } { K - 1 } = \\frac { f ( N _ { y } ) } { K - 1 } , } & { \\mathrm { o t h e r w i s e } , } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\epsilon _ { y }$ is a small label smoothing factor for Class- $_ y$ and relates to its class number $N _ { y }$ . Now the optimal solution becomes: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5dbc3c8aef54a66b0436d26bf2acce8b03309765efe3a8cd1eef2a630d984d21.jpg",
+        "text": "$$\n{ \\pmb w } _ { i } ^ { * } ^ { \\top } { \\pmb x } = \\left\\{ \\begin{array} { l l } { \\log \\left( \\frac { ( K - 1 ) ( 1 - \\epsilon _ { y } ) } { \\epsilon _ { y } } \\right) + c , } & { i = y , } \\\\ { c , } & { \\mathrm { o t h e r w } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $c$ can be an arbitrary real number. Comparing with the infinite optimal solution in cross-entropy, the label-aware smoothing encourages a finite output, which can get more generalized results and remedy over-fitting. We suppose the labels of the long-tailed dataset are assigned in a descending manner concerning the number of instances, i.e., $N _ { 1 } \\geq N _ { 2 } \\geq . . . \\geq N _ { K }$ . Because the head classes contain more various and diverse examples, the predicted probabilities are more promising than them of tail classes. Thus, we suggest classes with larger instance numbers should be penalized larger label smoothing factors, that is, the related function $f ( N _ { y } )$ should be negatively correlated to $N _ { y }$ . We define three types of related function $f ( N _ { y } )$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/fb443d9dc3c047683891f75e4183473d2a96ace892f6ebb61bda7604c2bd3f5f.jpg",
+        "text": "$$\n\\epsilon _ { y } = f ( N _ { y } ) = \\left\\{ \\begin{array} { l l } { { \\mathrm { ( C o n c a v e ) } } } & { { \\epsilon _ { K } + \\left( \\epsilon _ { 1 } - \\epsilon _ { K } \\right) \\sin \\left[ \\frac { \\pi ( N _ { y } - N _ { K } ) } { 2 ( N _ { 1 } - N _ { K } ) } \\right] , \\qquad y = 1 , 2 , . . . , K , } } \\\\ { { } } & { { } } \\\\ { { \\mathrm { ( L i n e a r ) } } } & { { \\epsilon _ { K } + \\left( \\epsilon _ { 1 } - \\epsilon _ { K } \\right) \\frac { N _ { y } - N _ { K } } { N _ { 1 } - N _ { K } } , \\qquad y = 1 , 2 , . . . , K , } } \\\\ { { } } & { { } } \\\\ { { \\mathrm { ( C o n v e x ) } } } & { { \\epsilon _ { 1 } + \\left( \\epsilon _ { 1 } - \\epsilon _ { K } \\right) \\sin \\left[ \\frac { 3 \\pi } { 2 } + \\frac { \\pi ( N _ { y } - N _ { K } ) } { 2 ( N _ { 1 } - N _ { K } ) } \\right] , \\quad y = 1 , 2 , . . . , K , } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon _ { 1 }$ and $\\epsilon _ { K }$ are two hyperparameters. If we set $\\epsilon _ { 1 } \\geq \\epsilon _ { K }$ , then we can get $\\epsilon _ { 1 } \\geq \\epsilon _ { 2 } \\geq . . . \\geq \\epsilon _ { K }$ . It means that if the instance number $N _ { y }$ for Class- $y$ is larger, label-aware smoothing will allocate a larger smoothing factor and lower the fitting probability to relieve the over-confidence because the head and medium classes are more likely to be over-confident than the tail classes (see Fig. 3). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As the form of label-aware smoothing is more complicated than cross-entropy, we propose a more generalized classifier learning framework to fit it. Here we give a quick review about cRT and LWS: cRT tries to learn a new classifier weight, which contains $K M$ learnable parameters. LWS is restricted to learn the weight scaling vector $\\bar { \\mathbf { \\boldsymbol { s } } } \\in \\mathbb { R } ^ { K }$ , which contains only $K$ learnable parameters. By contrast, cRT has more learnable parameters. It means cRT has a more powerful representation ability. LWS tends to obtain better validation losses and performances on large-scale datasets (refer to the experiment part in Kang et al. (2020)). It means LWS has a better generalization property. To combine the advantages of both cRT and LWS, we redesign the classifier framework in Stage-2: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f136c6ba3826c8121d8101265d1a23ee85020522b605b893b1cd252494318458.jpg",
+        "text": "$$\nz = \\mathrm { d i a g } ( s ) \\left( a { \\cal W } + \\Delta { \\cal W } \\right) ^ { \\top } x .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Eqn. (3), we fix the original classifier weight $W$ in Stage-2. If we make the learnable scaling vector $\\pmb { s }$ fixed, set $\\mathbf { \\delta } _ { \\mathbf { { s } } } = \\mathbf { 1 } \\ \\mathbf { \\delta } _ { \\mathbf { { l } } }$ , $a = 0$ , and just learn the new classifier weight $\\Delta W \\in \\mathbb { R } ^ { M \\times K }$ , Eqn. (4) will degrade to cRT. Because LWS fixes the original classifier weights $W$ and only learns the scaling $\\pmb { s }$ , Eqn. (4) will degrade to LWS if we set $a = 1$ and $\\Delta W = \\mathbf { 0 }$ . In most cases, LWS generally achieves better results than cRT on large scale datasets. Thus, we let $\\pmb { s }$ learnable and set $a = 1$ . We also make $\\Delta \\mathbf { W }$ learnable to improve the representation ability but optimize $\\Delta \\mathbf { W }$ by a different learning rate. $\\Delta \\mathbf { W }$ can be viewed as doing a shift transformation on $W$ . This transformation can change the direction of the original weight vector $\\pmb { w }$ in $W$ , which is what LWS cannot do. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.3 SHIFT LEARNING ON BATCH NORMALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            178,
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+    {
+        "type": "text",
+        "text": "In the two-stage training framework, models are first trained with instance-balanced sampling in Stage-1 and then trained with class-balanced sampling in Stage-2. Since the framework involves two samplers, or two datasets: the instance-balanced dataset $\\mathcal { D } _ { \\mathrm { I } }$ and the class-balanced dataset $\\mathcal { D } _ { \\mathrm { C } }$ , we can regard this two-stage training framework as a derivative of transfer learning approaches. However, if we view the two-stage decoupling training framework from the transfer learning perspective, fixing the backbone part and just fine-tuning the classifier in Stage-2 will be clearly unreasonable, especially for the batch normalization (BN) layers. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Concretely, we suppose that the input of the network is $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ , the input feature of some BN layer is $g ( \\pmb { x } _ { i } )$ , and the mini-batch size is $m$ . The running mean and the running variance of Channel- $j$ for these two stages are: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f1fd625d24e763d26ffef803dce734957f46dac07e13876949e14b1a5ba13cb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\pmb { \\mu } _ { \\mathrm { I } } ^ { ( j ) } = \\displaystyle \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } g ( { \\pmb x } _ { i } ) ^ { ( j ) } , \\quad \\pmb { \\sigma } _ { \\mathrm { I } } ^ { 2 ^ { ( j ) } } = \\displaystyle \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\left[ g ( { \\pmb x } _ { i } ) ^ { ( j ) } - \\pmb { \\mu } _ { \\mathrm { I } } ^ { ( j ) } \\right] ^ { 2 } , \\quad \\pmb { x } _ { i } \\sim P _ { { \\mathcal { D } } _ { \\mathrm { I } } } ( { \\pmb x } , y ) , } \\\\ & { \\pmb { \\mu } _ { \\mathrm { C } } ^ { ( j ) } = \\displaystyle \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } g ( { \\pmb x } _ { i } ) ^ { ( j ) } , \\quad \\pmb { \\sigma } _ { \\mathrm { C } } ^ { 2 ^ { ( j ) } } = \\displaystyle \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\left[ g ( { \\pmb x } _ { i } ) ^ { ( j ) } - \\pmb { \\mu } _ { \\mathrm { C } } ^ { ( j ) } \\right] ^ { 2 } , \\quad \\pmb { x } _ { i } \\sim P _ { { \\mathcal { D } } _ { \\mathrm { C } } } ( { \\pmb x } , y ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/19b86e3a88f558bf61a12c4d7e37ede85b460b9f37eb25257a310659fcc06882.jpg",
+        "image_caption": [
+            "Figure 4: Reliability diagrams of ResNet-32 trained on LT CIFAR-100, $\\mathrm { { I F } = 1 0 0 }$ . From left to right: cRT with mixup, LWS with mixup, LWS with mixup and shifted BN, and MiSLAS. It is better to look together with Fig. 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Due to the different sampling strategies, the composition ratios of the head, medium, and tail classes are also totally different, which leads to $P _ { \\mathcal { D } _ { \\mathrm { I } } } ( \\pmb { x } , y ) \\neq P _ { \\mathcal { D } _ { \\mathrm { C } } } ( \\pmb { x } , y )$ . Calculated by Eqn. (5) and (6), there exist some biases in $\\pmb { \\mu }$ and $\\pmb { \\sigma }$ under two sampling strategies, i.e., $\\mu _ { \\mathrm { I } } \\neq \\mu _ { \\mathrm { C } }$ , and $\\sigma _ { \\mathrm { I } } ^ { 2 } \\neq \\sigma _ { \\mathrm { C } } ^ { 2 }$ Thus, it is clearly infeasible for the decoupling framework that BN shares mean and variance across datasets with two sampling strategies. Motivated by AdaBN (Li et al., 2018) and TransNorm (Wang et al., 2019), we unfreeze the update procedures of the running mean $\\pmb { \\mu }$ and running variance $\\sigma$ but fix the learnable linear transformation parameters $_ { \\pmb { \\alpha } }$ and $\\beta$ for a better normalization in Stage-2. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our experimental setup including the implementation details and evaluation protocol mainly follows Cao et al. (2019) for LT CIFAR-10 and LT CIFAR-100, and Kang et al. (2020) for ImageNet-LT, Places-LT, and iNuturalist 2018. Please see Appendix A for further details. ",
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+    {
+        "type": "text",
+        "text": "4.2 ABLATION STUDY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Improving calibration. Here we show the reliability diagrams with 15 bins of our methods in Fig. 4. Comparing with Fig. 1 in the introduction part, both the mixup and label-aware smoothing can not only largely enhance the network calibration (even lower ECEs than them on balanced datasets) but also greatly improve the performance for long-tailed recognition. The similar trends can also be found on LT CIFAR-10, ImageNet-LT, and Places-LT (please see the figures in Appendix B for detail), which proves the powerful effects of the proposed method on calibration. According to all experiment results, training networks on imbalanced datasets leads to more severe over-confidence. Since the conventional mixup and label-smoothing both contain the operation of softening the ground truth labels, which may suggest that training with hard labels is likely to be another contributing factor leading to network over-confidence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Further analysis of label-aware smoothing. In our label-aware smoothing, there are two hyperparameters in Eqn. (3), $\\epsilon _ { 1 }$ and $\\epsilon _ { K }$ , which control the penalties of classes. In recognition system, if the predicted probability of some Class- $y$ is larger than 0.5, the classifier will classify the input to Class- $y$ . Thus, to ensure reasonability, we limit $0 \\le \\epsilon _ { K } \\le \\epsilon _ { 1 } \\le 0 . 5$ . Here we conduct a comparing experiment for varying $\\epsilon _ { 1 }$ and $\\epsilon _ { K }$ both from 0.0 to 0.5 on LT CIFAR-100 with imbalanced factor 100. We plot the performance matrix upon $\\epsilon _ { 1 }$ and $\\epsilon _ { k }$ in Fig. 5 for all possible variants. From it, the classification accuracy can be further improved by $0 . 9 \\%$ comparing with the conventional cross-entropy $( \\epsilon _ { 1 } = 0$ , $\\epsilon _ { K } = 0$ , green square) when we pick $\\epsilon _ { 1 } = 0 . 4$ , $\\epsilon _ { K } = 0 . 1$ (orange square) for label-aware smoothing. A more surprising improvement (growing by $3 . 3 \\%$ ) can be found on LT CIFAR-10 (see Appendix D.1 for detail). We also find that the concave related function $f ( \\cdot )$ in Eqn. (3) achieves the best performance but the gain is quite limited (refer Appendix D.2 for detail). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To visualize the change in predicted probability distributions, we train two LWS models, one with cross-entropy and the other with label-aware smoothing on long-tailed CIFAR-100 with imbalanced factor 100. The cross-entropy-based distributions of the head, medium, and tail classes are showing in the upper half part of Fig. 3 in light blue. The label-aware smoothing-based distributions are showing in the bottom half part in deep blue. We observe that the over-confidence of head and medium classes relieve greatly, and the whole distribution of the tail classes slightly moves right (a larger mean) when using label-aware smoothing. This empirical visualization is consistent with our analysis mentioned in Sec. 3.2. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2: Ablation study for all proposed modules on long-tailed CIFAR-100, $\\scriptstyle { \\mathrm { I F } = 1 0 0 }$ . MU: applying mixup just in Stage-1. SL: shift learning on batch normalization. LAS: label-aware smoothing. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Module</td><td colspan=\"2\">LT CIFAR-100</td></tr><tr><td>MU</td><td>SL</td><td>LAS</td><td>100 50</td><td>10</td></tr><tr><td>区</td><td>区</td><td>区</td><td>41.2</td><td>46.0 58.5</td></tr><tr><td>回</td><td>区</td><td>区</td><td>44.2 50.6</td><td>62.2</td></tr><tr><td>回</td><td>回</td><td>区</td><td>45.3 51.4</td><td>62.8</td></tr><tr><td>回</td><td>回</td><td>回</td><td>47.0</td><td>52.3 63.0</td></tr></table>",
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+        "image_caption": [
+            "Figure 5: Ablation study of two hyperparameters $\\epsilon _ { 1 }$ and $\\epsilon _ { K }$ in label-aware smoothing. ",
+            "Figure 6: Visualization of the changes in the running mean $\\pmb { \\mu }$ and variance $\\sigma ^ { 2 }$ . The ResNet-32 based model is trained on LT CIFAR-100 with imbalanced factor 100. Left: $\\pmb { \\mu }$ and $\\sigma ^ { 2 }$ in the first BN of ResNet-32, which contains 16 channels. Right: $\\pmb { \\mu }$ and $\\sigma ^ { 2 }$ in the last BN of ResNet-32, which contains 64 channels. "
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+        "text": "Further analysis of shift learning. In this part, we conduct an empirical experiment to show the effectiveness and reasonability of shift learning on BN. We train the LWS model on long-tailed CIFAR-100 with imbalanced factor 100. After 10 epochs finetuning in Stage-2, the model trained with BN shifting achieves accuracy at $4 5 . 3 \\%$ , which is $1 . 1 \\%$ higher than it without BN shifting. We also draw a visualization of the change in BN. As shown in Fig. 6, we see that there indeed exist biases in $\\pmb { \\mu }$ and $\\sigma ^ { 2 }$ between the dataset using different sampling strategies. Due to the composition ratios of the head classes, medium classes and tail classes are different in terms of different sampling strategies, the statistic running mean $\\pmb { \\mu }$ and running variance $\\sigma ^ { 2 }$ are certainly different. We also find some interesting phenomenons need for future exploration: (i) The changes in the running variance $\\sigma ^ { 2 }$ are larger than the changes in the running mean $\\pmb { \\mu }$ . (ii) The changes of $\\pmb { \\mu }$ and $\\sigma ^ { 2 }$ in deep BN layers are quite smaller than them in shallow BN layers. ",
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+        "type": "text",
+        "text": "Overall, Table 2 shows the ablation investigation on the effects of mixup (adding mixup in Stage1, MU), shift learning on batch normalization (SL), and label-aware smoothing (LAS). From it, each proposed module can further improve the performances on long-tailed CIFAR-100 for all commonly-used imbalanced factors, which firmly demonstrates the effectiveness. ",
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+    {
+        "type": "text",
+        "text": "4.3 COMPARISON WITH THE STATE-OF-THE-ART ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this subsection, we compare the proposed method against previous one-stage methods, such as Range Loss (Zhang et al., 2017), LDAM Loss (Cao et al., 2019), FSLwF (Gidaris & Komodakis, 2018), and OLTR (Liu et al., 2019), and against previous two-stage methods, such as DRS-like, DRW-like (Cao et al., 2019), LFME (Xiang & Ding, 2020), cRT, and LWS (Kang et al., 2020). For fair comparisons, we also add mixup on the LWS and cRT models. Remix (Chou et al., 2020) is a recently proposed augmentation method for long-tail recognition. Because BBN (Zhou et al., 2020) has double samplers and is trained in a mixup-like manner, we directly compare our method with it. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 3: Top-1 accuracy $( \\% )$ for ResNet-32 models trained on long tailed CIFAR-10 and CIFAR-100. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Long-tailed CIFAR-10</td><td colspan=\"3\">Long-tailed CIFAR-100</td></tr><tr><td>100</td><td>50</td><td>10</td><td>100</td><td>50</td><td>10</td></tr><tr><td>CE</td><td>70.4</td><td>74.8</td><td>86.4</td><td>38.4</td><td>43.9</td><td>55.8</td></tr><tr><td>mixup</td><td>73.1</td><td>77.8</td><td>87.1</td><td>39.6</td><td>45.0</td><td>58.2</td></tr><tr><td>LDAM+DRW</td><td>77.1</td><td>81.1</td><td>88.4</td><td>42.1</td><td>46.7</td><td>58.8</td></tr><tr><td>BBN(include mixup)</td><td>79.9</td><td>82.2</td><td>88.4</td><td>42.6</td><td>47.1</td><td>59.2</td></tr><tr><td>Remix+DRW(30 epochs)</td><td>79.8</td><td>=</td><td>89.1</td><td>46.8</td><td>-</td><td>61.3</td></tr><tr><td>cRT+mixup</td><td>79.1</td><td>84.2</td><td>89.8</td><td>45.1</td><td>50.9</td><td>62.1</td></tr><tr><td>LWS+mixup</td><td>76.3</td><td>82.6</td><td>89.6</td><td>44.2</td><td>50.6</td><td>62.2</td></tr><tr><td>MiSLAS</td><td>82.1</td><td>85.8</td><td>89.9</td><td>47.0</td><td>52.3</td><td>63.0</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 4: Top-1 accuracy $( \\% )$ on ImageNet-LT (left), iNaturalist 2018 (center) and Place-LT (right). "
+        ],
+        "table_footnote": [
+            "(a) ImageNet-LT (b) iNaturalist 2018 (c) Place-LT "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>ResNet-50</td></tr><tr><td>CE</td><td>44.6</td></tr><tr><td>CE+DRW</td><td>48.5</td></tr><tr><td>Focal+DRW</td><td>47.9</td></tr><tr><td>LDAM+DRW</td><td>48.8</td></tr><tr><td>CRT+mixup LWS+mixup MiSLAS</td><td>51.7 52.0 52.7</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/94d4e75dbd5c82d35e650a4e045b3ca7282a4422135d6d8aaebe27e109e72141.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=4>Method</td><td rowspan=1 colspan=1>ResNet-50</td></tr><tr><td rowspan=1 colspan=4>CB-FocalLDAM+DRW</td><td rowspan=1 colspan=1>61.168.0</td></tr><tr><td rowspan=1 colspan=4>BBN(include mixup)</td><td rowspan=1 colspan=1>69.6</td></tr><tr><td rowspan=1 colspan=4>Remix+DRW</td><td rowspan=1 colspan=1>70.5</td></tr><tr><td rowspan=1 colspan=3>cRT+mixup</td><td rowspan=1 colspan=1>up</td><td rowspan=1 colspan=1>70.2</td></tr><tr><td rowspan=2 colspan=4>LWS+mixupMiSLAS</td><td rowspan=1 colspan=1>mixup</td></tr><tr><td rowspan=1 colspan=1>71.6</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>ResNet-152</td></tr><tr><td>Range Loss</td><td>35.1</td></tr><tr><td>FSLwF</td><td>34.9</td></tr><tr><td>OLTR OLTR+LFME</td><td>35.9 36.2</td></tr><tr><td></td><td>38.3</td></tr><tr><td>cRT+mixup LWS+mixup</td><td>39.7</td></tr><tr><td>MiSLAS</td><td>40.4</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Experimental results on CIFAR-LT. We conduct extensive experiments on long-tailed CIFAR-10 and CIFAR-100 with imbalanced factors of 10, 50, and 100, which is the same as the previous setting (Cao et al., 2019; Zhou et al., 2020). The experimental results are summarized in Table 3. Compared with previous methods ( $^ +$ mixup, one/two-stage), our MiSLAS outperforms all previous methods by a large margin. Moreover, this superiority of the proposed method holds for all imbalanced factors on both long-tailed CIFAR-10 and CIFAR-100. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Experimental results on ImageNet-LT, iNaturalist 2018, and Place-LT. We further verify the effectiveness of our method on three large-scale imbalanced datasets, i.e., ImageNet-LT, iNaturalist 2018, and Place-LT. Table 4 lists experimental results on ImageNet-LT (left), iNaturalist 2018 (center), and Places-LT (right). Notably, our MiSLAS still outperforms all competing approaches and sets new state-of-the-art records for all three large-scale long-tailed benchmarks. More detailed results about the split class accuracies and different backbones on these three datasets are listed in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we discover that models trained on long-tailed datasets are more miscalibrated and over-confident than them trained on balanced datasets. The two-stage models suffer the same issue as well. To relieve over-confidence, we propose two solutions: (i) We find that mixup can remedy over-confidence and have a positive effect on representation learning but a negative or negligible effect on classifier learning. (ii) To further improve classifier learning and calibration, we propose label-aware smoothing to handle the different degrees of over-confidence for different classes. We are the first to note the dataset bias or domain shift in two-stage resampling methods for long-tailed recognition. To solve the dataset bias producing by different re-sampling in the decoupling framework, we propose shift learning on the batch normalization layer and this novel model can greatly improve the performance. Extensive quantitative and qualitative experiments on multiple benchmark datasets show that our MiSLAS achieves superior performances over the state-of-the-art methods. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A EXPERIMENT SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "A.1 DATASETS EXPLANATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10-LT and CIFAR-100-LT. CIFAR-10 and CIFAR-100 both have 60,000 images 50,000 for training and 10,000 for validation with 10 categories and 100 categories, respectively. For a fair comparison, we use the long-tailed versions of CIFAR datasets with the same setting as those used in Cao et al. (2019). They control the degrees of data imbalance with an imbalanced factor $\\beta$ . $\\begin{array} { r } { \\beta = \\frac { N _ { \\mathrm { m a x } } } { N _ { \\mathrm { m i n } } } } \\end{array}$ , where $N _ { \\mathrm { m a x } }$ and $N _ { \\mathrm { m i n } }$ are the numbers of training samples for the most frequent class and the least frequent class. Following Cao et al. (2019) and Zhou et al. (2020), we conduct experiments with imbalanced factors 100, 50, and 10. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "ImageNet-LT and Places-LT. ImageNet-LT and Places-LT were proposed by Liu et al. (2019). ImageNet-LT is a long-tailed version of the large-scale object classification dataset ImageNet (Russakovsky et al., 2015) by sampling a subset following the Pareto distribution with power value $\\alpha = 6$ . It contains 115.8K images from 1,000 categories, with class cardinality ranging from 5 to 1,280. Places-LT is a long-tailed version of the large-scale scene classification dataset Places (Zhou et al., 2017). It consists of 184.5K images from 365 categories with class cardinality ranging from 5 to 4,980. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "iNaturalist 2018. iNaturalist 2018 (Van Horn et al., 2018) is one species classification dataset, which is on a large scale and suffers from extremely imbalanced label distributions. It is composed of 437.5K images from 8,142 categories. In addition to the extreme imbalance, the iNaturalist 2018 dataset also confronts the fine-grained problem. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 EVALUATION PROTOCOL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            462,
+            387,
+            476
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Following Liu et al. (2019) and Kang et al. (2020), we report the commonly used top-1 accuracy over all classes on the balanced test/validation datasets, denoted as All. In more detail, further report accuracy on three splits of the set of classes: Head-Many (more than 100 images), Med.-Medium (20 to 100 images) and Tail-Few (less than 20 images). ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.3 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            560,
+            406,
+            574
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For all experiments, we use the SGD optimizer with momentum 0.9 to optimize networks. ",
+        "bbox": [
+            173,
+            585,
+            764,
+            601
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For long-tailed CIFAR, we mainly follow Cao et al. (2019). We train all MiSLAS models with the ResNet-32 backbone on one GPU and use the multistep learning rate schedule, which decreases the learning rate by 0.1 at the $1 6 0 ^ { \\mathrm { t h } }$ epoch and the $1 8 0 ^ { \\mathrm { t h } }$ epochs in Stage-1. ",
+        "bbox": [
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+            607,
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+            650
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For ImageNet-LT, Place-LT, and iNaturalist 2018, We mainly follow Kang et al. (2020) and use the cosine learning rate schedule (Loshchilov & Hutter, 2017) to train all MiSLAS models with the ResNet-10/50/101/152 backbones on 4 GPUs. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            698
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/c6aecb6b715c1e153f6b21fae9e822f806304864c94313aa78a8ea929bf18ec4.jpg",
+        "table_caption": [
+            "Table 5: Detailed experiment settings on five benchmark datasets. LR: learning rate, BS: batch size, WD: weight decay, and LRS: learning rate schedule, $\\Delta \\mathbf { W }$ : learning rate ratio of $\\Delta \\mathbf { W }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"3\">Common</td><td colspan=\"2\">Stage-1</td><td colspan=\"5\">Stage-2</td></tr><tr><td>LR</td><td>BS</td><td>WD</td><td>Epochs</td><td>LRS</td><td>Epochs</td><td>LRS</td><td>@1</td><td>EK</td><td>△W</td></tr><tr><td>LT CIFAR-10</td><td>0.1</td><td>128</td><td>2e-4</td><td>200</td><td>multi.</td><td>10</td><td>cosine</td><td>0.3</td><td>0.0</td><td>0.5x</td></tr><tr><td>LT CIFAR-100</td><td>0.1</td><td>128</td><td>2e-4</td><td>200</td><td>multi.</td><td>10</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.2x</td></tr><tr><td>ImageNet-LT</td><td>0.1</td><td>256</td><td>5e-4</td><td>180</td><td>cosine</td><td>10</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.05x</td></tr><tr><td>Places-LT</td><td>0.1</td><td>256</td><td>5e-4</td><td>90</td><td>cosine</td><td>10</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.05x</td></tr><tr><td>iNaturalist&#x27;18</td><td>0.1</td><td>256</td><td>1e-4</td><td>200</td><td>cosine</td><td>30</td><td>cosine</td><td>0.4</td><td>0.1</td><td>0.05x</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B CALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/eacc92a146d5ac8a9a51356e182064f8bab077b1209b16054aeef90c0ea78cb2.jpg",
+        "image_caption": [
+            "Figure 7: Reliability diagrams on CIFAR10 with 15 bins. From left to right: plain ResNet-32 model trained on the original CIFAR-10 dataset, plain model, cRT, LWS, and MiSLAS trained on long-tailed CIFAR-10 with imbalanced factor 100. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/ec26f5f2647f296366c96bb51460bb73017ab9b16d3e9ad67012cc02bd8b43d6.jpg",
+        "image_caption": [
+            "Figure 8: Reliability diagrams on ImageNet with 15 bins. From left to right: plain ResNet-50 model trained on the original ImageNet dataset, plain model, cRT, LWS, and MiSLAS trained on ImageNet-LT. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/4b1d456880deb636b722c684adcb5b082238c3880ce0191066e72f548d599faf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            512,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 9: Reliability diagrams of ResNet-152 trained on Places-LT with 15 bins. From left to right: cRT, LWS, cRT with mixup, LWS with mixup, and MiSLAS. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C MORE DETAILED RESULTS ON IMAGENET-LT, PLACES-LT AND INATURALIST 2018 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            735,
+            136
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/d4319799f64fa455e68b4371cb6dadf4c5aa23cf52c0c9a335e9cfced30f9854.jpg",
+        "table_caption": [
+            "Table 6: Comprehensive accuracy results on ImageNet-LT with different backbone networks (ResNet50, ResNet-101 & ResNet-152) and training 180 epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Backbone</td><td>Method</td><td>Many</td><td>Medium</td><td>Few</td><td>All</td></tr><tr><td rowspan=\"4\">ResNet-50</td><td>cRT</td><td>62.5</td><td>47.4</td><td>29.5</td><td>50.3</td></tr><tr><td>LWS</td><td>61.8</td><td>48.6</td><td>33.5</td><td>51.2</td></tr><tr><td>cRT+mixup</td><td>63.9</td><td>49.1</td><td>30.2</td><td>51.7</td></tr><tr><td>LWS+mixup MiSLAS</td><td>62.9 61.7</td><td>49.8 51.3</td><td>31.6 35.8</td><td>52.0 52.7</td></tr><tr><td rowspan=\"4\">ResNet-101</td><td>cRT LWS</td><td>63.8 63.1</td><td>48.5</td><td>30.0</td><td>51.4</td></tr><tr><td>cRT+mixup</td><td>65.2</td><td>49.9</td><td>33.8</td><td>52.3</td></tr><tr><td>LWS+mixup</td><td>64.5</td><td>50.6</td><td>31.6</td><td>53.1</td></tr><tr><td>MiSLAS</td><td>64.3</td><td>51.2 52.1</td><td>34.1 35.8</td><td>53.5 54.1</td></tr><tr><td rowspan=\"4\">ResNet-152</td><td>cRT LWS</td><td>64.9</td><td>50.4</td><td>30.6</td><td>52.7</td></tr><tr><td></td><td>64.1</td><td>51.8</td><td>35.5</td><td>53.8</td></tr><tr><td>cRT+mixup</td><td>66.5</td><td>51.6</td><td>32.8</td><td>54.2</td></tr><tr><td>LWS+mixup MiSLAS</td><td>66.1 65.4</td><td>52.2 53.2</td><td>34.5 37.1</td><td>54.6 55.2</td></tr></table>",
+        "bbox": [
+            200,
+            194,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/655893187ef592749ff42f5b43cd962fd9d584be055660a0d40acadb8a0c8a5f.jpg",
+        "table_caption": [
+            "Table 7: Comprehensive accuracy results on iNaturalist 2018 with ResNet-50 and training 200 epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Backbone</td><td>Method</td><td>Many</td><td>Medium</td><td>Few</td><td>All</td></tr><tr><td rowspan=\"5\">ResNet-50</td><td>cRT</td><td>73.2</td><td>68.8</td><td>66.1</td><td>68.2</td></tr><tr><td>T-normalized</td><td>71.1</td><td>68.9</td><td>69.3</td><td>69.3</td></tr><tr><td>LWS</td><td>71.0</td><td>69.8</td><td>68.8</td><td>69.5</td></tr><tr><td>cRT+mixup</td><td>74.2</td><td>71.1</td><td>68.2</td><td>70.2</td></tr><tr><td>LWS+mixup</td><td>72.8</td><td>71.6</td><td>69.8</td><td>70.9</td></tr><tr><td></td><td>MiSLAS</td><td>73.2</td><td>72.4</td><td>70.4</td><td>71.6</td></tr></table>",
+        "bbox": [
+            204,
+            494,
+            789,
+            617
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/d3c0b12be2a7d80ee85125745e390ca0fab5a9df6efd630a08ea43ce23504140.jpg",
+        "table_caption": [
+            "Table 8: Detailed accuracy results on Places-LT, starting from an ImageNet pre-trained ResNet-152. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Many      Medium      Few       All</td></tr><tr><td rowspan=10 colspan=1>ResNet-152</td><td rowspan=1 colspan=1>Lifted Loss</td><td rowspan=1 colspan=1>41.1         35.4        24.0      35.2</td></tr><tr><td rowspan=1 colspan=1>Focal Loss</td><td rowspan=1 colspan=1>41.1         34.8        22.4      34.6</td></tr><tr><td rowspan=1 colspan=1>Range Loss</td><td rowspan=1 colspan=1>41.1         35.4        23.2      35.1</td></tr><tr><td rowspan=1 colspan=1>FSLwFOLTR</td><td rowspan=1 colspan=1>43.9         29.9        29.5      34.944.7         37.0        25.3      35.9</td></tr><tr><td rowspan=1 colspan=1>OLTR+LFME</td><td rowspan=1 colspan=1>39.3         39.6        24.2      36.2</td></tr><tr><td rowspan=2 colspan=1>cRTT-normalizedLWS</td><td rowspan=1 colspan=1>42.0         37.6        24.9      36.7</td></tr><tr><td rowspan=1 colspan=1>37.8         40.7        31.8      37.940.6         39.1        28.6      37.6</td></tr><tr><td rowspan=3 colspan=1>cRT+mixupLWS+mixupMiSLAS</td><td rowspan=1 colspan=1>44.1         38.5        27.1      38.1</td></tr><tr><td rowspan=1 colspan=1>41.7         41.3        33.1      39.7</td></tr><tr><td rowspan=1 colspan=1>39.6         43.3        36.1      40.4</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D ABLATION STUDY OF LABEL-AWARE SMOOTHING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            630,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/16ab8345b0b49db5c1f6d862e06d90e6e500f84f1e06457336244442b23313d5.jpg",
+        "image_caption": [
+            "D.1 MORE RESULTS ABOUT THE HYPERPARAMETERS $\\epsilon _ { 1 }$ AND $\\epsilon _ { K }$ ",
+            "Figure 10: Ablation study of two hyperparameters $\\epsilon _ { 1 }$ and $\\epsilon _ { K }$ in label-aware smoothing. Our labelaware smoothing (orange square) outperforms cross-entropy (green square) by a large margin on both long-tailed CIFAR-10 (left) and long-tailed CIFAR-100 (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.2 FORM OF THE RELATED FUNCTION $f ( \\cdot )$ ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Sec. 3.2 and Sec. 4.2, the form of the related function $f ( \\cdot )$ may play a significant role for the final model performance. We draw the visualization of Eqn. (3) at the left part of Fig. 11. For the LT CIFAR-100 dataset with balanced factor 100, $N _ { 1 } = 5 0 0$ and $N _ { 1 0 0 } = 5$ . Based on the ablation study results of $\\epsilon _ { 1 }$ and $\\epsilon _ { K }$ mentioned in Sec. 4.2 and above, we set $\\epsilon _ { 1 } = 0 . 4$ and $\\epsilon _ { 1 0 0 } = 0 . 1 $ here. After fintuning for 10 epochs in Stage-2, the accuracy of the concave model is the best. We also design a power-like related function, which can be written as: ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/373b8363d3a977a4ad1eaa32adf167d05ea82a082c743a680c614ef8b1aaef5f.jpg",
+        "text": "$$\n\\epsilon _ { y } = f ( N _ { y } ) = \\epsilon _ { K } + ( \\epsilon _ { 1 } - \\epsilon _ { K } ) \\bigg ( { \\frac { N _ { y } - N _ { K } } { N _ { 1 } - N _ { K } } } \\bigg ) ^ { p } , \\qquad y = 1 , 2 , . . . , K ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            564,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $p$ is a hyperparameter to control the shape of the related function. For example, we will get concave related function if we set $p < 1$ and we will get convex related function if we set $p > 1$ . The visualization of Eqn. (7) is shown at the right part of Fig. 11. However, comparing the accuracies of all variants, the influence of the related function form is quite limited for the final performance (just growing by $0 . 3 \\%$ ). Because the concave related function in Eqn. (3) achieves the best performance among all variants, we choose it as the default setting of the related function $f ( \\cdot )$ for other experiments. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/98f172c1a250bbc32349248542a822455a2e8cc42d80596499181b72ec7f5323.jpg",
+        "image_caption": [
+            "Figure 11: Function visualization and accuracy of Eqn. (3) (left) and Eqn. (7) (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    }
+]