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+# CONTRASTIVE SYN-TO-REAL GENERALIZATION
+
+Wuyang Chen1∗, Zhiding $\mathbf { Y } \mathbf { u } ^ { 2 \dagger }$ , Shalini De Mello2, Sifei Liu2, Jose M. Alvarez2,   
+Zhangyang $\mathbf { W a n g ^ { 1 } }$ , Anima Anandkumar2,3   
+1The University of Texas at Austin 2NVIDIA 3California Institute of Technology   
+{wuyang.chen,atlaswang}@utexas.edu   
+{zhidingy,shalinig,sifeil,josea,aanandkumar}@nvidia.com   
+https://github.com/NVlabs/CSG
+
+# ABSTRACT
+
+Training on synthetic data can be beneficial for label or data-scarce scenarios. However, synthetically trained models often suffer from poor generalization in real domains due to domain gaps. In this work, we make a key observation that the diversity of the learned feature embeddings plays an important role in the generalization performance. To this end, we propose contrastive synthetic-to-real generalization (CSG), a novel framework that leverages the pre-trained ImageNet knowledge to prevent overfitting to the synthetic domain, while promoting the diversity of feature embeddings as an inductive bias to improve generalization. In addition, we enhance the proposed CSG framework with attentional pooling (A-pool) to let the model focus on semantically important regions and further improve its generalization. We demonstrate the effectiveness of CSG on various synthetic training tasks, exhibiting state-of-the-art performance on zero-shot domain generalization.
+
+# 1 INTRODUCTION
+
+Deep neural networks have pushed the boundaries of many visual recognition tasks. However, their success often hinges on the availability of both training data and labels. Obtaining data and labels can be difficult or expensive in many applications such as semantic segmentation, correspondence, 3D reconstruction, pose estimation, and reinforcement learning. In these cases, learning with synthetic data can greatly benefit the applications since large amounts of data and labels are available at relatively low costs. For this reason, synthetic training has recently gained significant attention (Wu et al., 2015; Richter et al., 2016; Shrivastava et al., 2017; Savva et al., 2019).
+
+Despite many benefits, synthetically trained models often have poor generalization on the real domain due to large domain gaps between synthetic and real images. Limitations on simulation and rendering can lead to degraded synthesis quality, such as aliased boundaries, unrealistic textures, fake appearance, over-simplified lighting conditions, and unreasonable scene layouts. These issues result in domain gaps between synthetic and real images, preventing the synthetically trained models from capturing meaningful representations and limiting their generalization ability on real images.
+
+To mitigate these issues, domain generalization and adaptation techniques have been proposed (Li et al., 2017; Pan et al., 2018; Yue et al., 2019). Domain adaptation assumes the availability of target data (labeled, partially labeled, or unlabeled) during training. On the other hand, domain generalization considers zero-shot generalization without seeing the target data of real images, and is therefore more challenging. An illustration of the domain generalization protocol on the
+
+![](images/373d72541256dc85741ef7b3a8176d5d397430972ddfae8e36100d32abf14f94.jpg)  
+Figure 1: An illustration of the domain generalization protocol on the VisDA-17 dataset, where real target domain (test) images are assumed unavailable during model training.
+
+VisDA-17 dataset (Peng et al., 2017) is shown in Figure 1. Considering that ImageNet pre-trained representation is widely used as model initialization, recent efforts on domain generalization show that such knowledge can be used to prevent overfitting to the synthetic domain (Chen et al., 2018; 2020c). Specifically, they impose a distillation loss to regularize the distance between the synthetically trained and the ImageNet pre-trained representations, which improves synthetic-to-real generalization.
+
+The above approaches still face limitations due to the challenging nature of this problem. Taking a closer look, we observe the following pitfalls in training on synthetic data. First, obtaining photorealistic appearance features at the micro-level, such as texture and illumination, is challenging due to the limits of simulation complexity and rendering granularity. Without special treatment, CNNs tend to be biased towards textures (Geirhos et al., 2019) and suffer from badly learned representations on synthetic data. Second, the common lack of texture and shape variations on synthetic images often leads to collapsed and trivial representations without any diversity. This is unlike training with natural images where models get sufficiently trained by seeing enough variations. Such a lack of diversity in the representation makes the learned models vulnerable to natural variations in the real world.
+
+# Summary of contributions and results:
+
+• We observe that the diversity of learned feature embedding plays an important role in syntheticto-real generalization. We show an example of collapsed representations learned by a synthetic model, which is in sharp contrast to features learned from real data (Section 2).
+
+• Motivated by the above observation, we propose a contrastive synthetic-to-real generalization framework that simultaneously regularizes the synthetically trained representation while promoting the diversity of the learned representation to improve generalization (Section 3.1).
+
+• We further enhance the CSG framework with attentional pooling (A-pool) where feature representations are guided by model attention. This allows the model to localize its attention to semantically more important regions, and thus improves synthetic-to-real generalization (Section 3.4).
+
+• We benchmark CSG on various synthetic training tasks including image classification (VisDA-17) and semantic segmentation $\mathrm { ( G T A 5  }$ Cityscapes). We show that CSG considerably improves the generalization performance without seeing target data. Our best model reaches $6 4 . 0 5 \%$ accuracy on VisDA-17 compared to previous state-of-the-art (Chen et al., 2020c) with $6 1 . 1 \%$ (Section 4).
+
+# 2 A MOTIVATING EXAMPLE
+
+We give a motivating example to show the significant differences between the features learned on synthetic and real images. Specifically, we use a ResNet-101 backbone and extract the $l _ { 2 }$ normalized feature embedding after global average pooling (defined as $\bar { \mathbf { \nabla } } \bar { \mathbf { v } }$ ). We consider the following three models: 1) model pre-trained on ImageNet, 2) model trained on VisDA-17 validation set (real images), and 3) model trained on VisDA-17 training set (synthetic images) 1. Both 2) and 3) are initialized with ImageNet pre-training, and fine-tuned on the 12 classes defined in VisDA-17.
+
+![](images/08c82af44f55d8f5cb2345e010695c49f516df4ace1ab78a5e7d6f32dbfb40fc.jpg)  
+Figure 2: Feature diversity on VisDA-17 test images in $\mathbb { R } ^ { 2 }$ with Gaussian kernel density estimation (KDE). Darker areas have more concentrated features. $E _ { s }$ : hyperspherical energy of features, lower the more diverse.
+
+Visualization of feature diversity. We visualize the normalized representations on a 2-dim sphere. A Gaussian kernel with bandwidth estimated by Scott’s Rule (Scott, 2015) is applied to estimate the probability density function. Darker areas have more concentrated features, and if the feature space (the 2-dim sphere) is covered by dark areas, it has more diversely placed features. In Figure 2, we can see that the ImageNet pretrained model can widely span the representations on the 2-dim feature space. The model trained on VisDA-17 validation set can also generate diverse features, although slightly affected by the class imbalance. However, when the model is trained on the training set (synthetic images), the features largely collapse to a narrow subspace, i.e., the model fails to fully leverage the whole feature space. This is clear that training on synthetic images can easily introduce poor bias to the model and the collapsed representations will fail to generalize to the real domain.
+
+Quantitive measurement of feature diversity. Inspired by (Liu et al., 2018), we also quantitatively measure the diversity of the feature embeddings using the following hyperspherical potential energy:
+
+$$
+E _ { s } \left( \bar { v } _ { i } | _ { i = 1 } ^ { N } \right) = \sum _ { i = 1 } ^ { N } \sum _ { \substack { j = 1 , j \neq i } } ^ { N } e _ { s } \left( \| \bar { v } _ { i } - \bar { v } _ { j } \| \right) = \left\{ \begin{array} { l l } { \sum _ { i \neq j } \| \bar { v } _ { i } - \bar { v } _ { j } \| ^ { - s } , \quad s > 0 } \\ { \sum _ { i \neq j } \log \left( \| \bar { v } _ { i } - \bar { v } _ { j } \| ^ { - 1 } \right) , \quad s = 0 } \end{array} \right.
+$$
+
+$N$ is the number of examples. The lower the hyperspherical energy (HSE) is, the more diverse the feature vectors will be scattered in the unit sphere. $s$ is the power factor, and we choose $s = 0$ in this example. Three training strategies exhibit energies as 0.2541, 0.3355, 0.4408, respectively. This validates that models trained on real images can capture diverse features, whereas the synthetic training will lead the model to highly collapsed feature space.
+
+Remarks. A conclusion can be drawn from the above examples: though assisted with ImageNet initialization, fine-tuning on synthetic images tends to give collapsed features with poor diversity in sharp contrast to training with real images. This indicates that the diversity of learned representation could play an important role in synthetic-to-real generalization.
+
+# 3 CONTRASTIVE SYNTHETIC-TO-REAL GENERALIZATION
+
+We consider the synthetic-to-real domain generalization problem following the protocols of Chen et al. (2020c). More specifically, the objective is to achieve the best zero-shot generalization on the unseen target domain real images without having access to them during synthetic training.
+
+# 3.1 NOTATION AND FRAMEWORK
+
+Our design of the model considers the following two aspects with a “push and pull” strategy:
+
+Pull: Without access to real images, the ImageNet pre-trained model presents the only source of real domain knowledge that can implicitly guide our training. As a result, we hope to impose some form of similarity between the features obtained by the synthetic model and the ImageNet pre-trained one. This helps to overcome the domain gaps from the unrealistic appearance of synthetic images.
+
+Push: Section 2 shows that synthetic training tends to generate collapsed features whereas models trained on natural images give many diverse ones. We treat this as an inductive bias to improve synthetic training, by pushing the feature embeddings away from each other across different images.
+
+The above “push and pull” strategy can be exactly formulated with a contrastive loss. This motivates us to propose a contrastive synthetic-to-real generalization framework as partly inspired by recent popular contrastive learning methods (He et al., 2020). Figure 3(b) illustrates our CSG framework. Specifically, we denote the frozen Imagenet pre-trained model as $f _ { e , o }$ and the synthetically trained model $f _ { e }$ , where $f _ { e }$ is supervised by the task loss $\mathcal { L } _ { \boldsymbol { s y n } }$ for the defined downstream task. We denote the input synthetic image as $\pmb { x } ^ { a }$ and treat it as an anchor. We treat the embeddings of $\pmb { x } ^ { a }$ obtained by $f _ { e }$ and $f _ { e , o }$ as anchor and positive embeddings, denoting them as $z ^ { a }$ and $z ^ { + }$ , respectively. Following a typical contrastive approach, we define $K$ negative images $\{ \pmb { x } _ { 1 } ^ { - } , \cdots , \pmb { x } _ { K } ^ { - } \}$ for every anchor $\pmb { x } ^ { a }$ , and denote their corresponding embeddings as $\{ z _ { 1 } ^ { - } , \cdots , z _ { K } ^ { - } \}$ . Similar to the design in (Chen et al., 2020d), we define $h / \widetilde { h } : \mathbb { R } ^ { C }  \mathbb { R } ^ { c }$ as the nonlinear projection heads with a two MLP layers and a ReLU layer between them. The CSG framework regularizes $f _ { e }$ in a contrastive manner: pulling $z ^ { a }$ and $z ^ { + }$ to be closer while pushing $z ^ { a }$ and $\{ z _ { 1 } ^ { - } , \cdots , z _ { K } ^ { - } \}$ apart. This regularizes the model by preventing its representation from deviating too far from that of a pre-trained ImageNet model and yet encouraging it to learn task-specific information from the synthetic data.
+
+![](images/7d30e3b74a188c3c1b9761d315e826059ca6905f8129c8afcb3e7314e2649b9d.jpg)  
+Figure 3: (a) Previous work (Chen et al., 2018; 2020c) consider “learning without forgetting” which minimizes a distillation loss between a synthetic model and an ImageNet pre-trained one (either on features or model parameters) to avoid catastrophic forgetting. (b) The proposed CSG framework with a “push and pull” strategy.
+
+Even though having connections to recent self-supervised contrastive representation learning methods (Oord et al., 2018; Wu et al., 2018; Chen et al., 2020a; He et al., 2020; Chen et al., 2020b; Jiang et al., 2020), our work differs in the following aspects: 1) Self-supervised learning and the addressed task are ill-posed in different manners - the former lacks the constraints from semantic labels, whereas the latter lacks the support of data distribution. 2) As a result, the motivations of contrastive learning are different. Our work is also related to the contrastive distillation framework in (Tian et al., 2020a). Again, the two works differ in both task and motivation despite the converging techniques.
+
+# 3.2 AUGMENTATION
+
+Augmentation has been an important part of effective contrastive learning. By perturbing or providing different views of the representations, augmentation forces a model to focus more on the mid-level and high-level representations of object parts and structures which are visually more realistic and reliable. To this end, we follow existing popular approaches to create augmentation at different levels:
+
+Image augmentation. We consider image-level augmentation using RandAugment (Cubuk et al., 2020) where a single global control factor $M$ is used to control the augmentation magnitude. We denote the transform operators of image-level augmentation as $\tau ( \cdot )$ .
+
+Model augmentation. We adopt a mean-teacher (Tarvainen & Valpola, 2017) styled moving average of a model to create different views of feature embeddings. Given an anchor image $\pmb { x } ^ { a }$ and $K$ negative images $\{ \pmb { x } _ { 1 } ^ { - } , \cdots , \pmb { x } _ { K } ^ { - } \}$ , we compute the embeddings as follows:
+
+$$
+\begin{array} { r } { z ^ { a } = f _ { e } \circ g \circ h ( \mathcal { T } ( \pmb { x } ^ { a } ) ) , z ^ { + } = f _ { e , o } \circ g \circ \widetilde { h } ( \mathcal { T } ( \pmb { x } ^ { a } ) ) , z _ { k } ^ { - } = f _ { e , o } \circ g \circ \widetilde { h } ( \mathcal { T } ( \pmb { x } _ { k } ^ { - } ) ) , } \end{array}
+$$
+
+where $g : \mathbb { R } ^ { C \times h \times w }  \mathbb { R } ^ { C }$ is a pooling operator transforming a feature map into a vector. Following (He et al., 2020), we define $\widetilde { h } ( \cdot )$ as an exponential moving average of the $h ( \cdot )$ across different iterations. Such difference in $h ( \cdot )$ and $\widetilde { h } ( \cdot )$ leads to augmented views of embeddings.
+
+# 3.3 CONTRASTIVE LOSS
+
+We use InfoNCE loss (Wu et al., 2018) to formulate the “push and pull” strategy:
+
+$$
+{ \mathcal { L } } _ { \mathrm { N C E } } = - \log { \frac { \exp { ( z ^ { a } \cdot z ^ { + } / \tau ) } } { \exp { ( z ^ { a } \cdot z ^ { + } / \tau ) } + \sum _ { z ^ { - } } \exp { ( z ^ { a } \cdot z ^ { - } / \tau ) } } } ,
+$$
+
+where $\tau = 0 . 0 7$ is a temperature hyper-parameter in our work. Together, we minimize the combination of the synthetic task loss and $\mathcal { L } _ { \mathrm { N C E } }$ during our transfer learning process:
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathrm { T a s k } } + \lambda \mathcal { L } _ { \mathrm { N C E } }
+$$
+
+Specifically, $\mathcal { L } _ { \mathrm { T a s k } }$ is the synthetic training task objective. For example, $\mathcal { L } _ { \mathrm { T a s k } }$ is a cross-entropy loss of a vector over the 12 defined classes on VisDA-17, whereas it is a per-pixel dense cross-entropy loss on GTA5. $\lambda$ is a balancing factor controlling the strength of the Contrastive Learning.
+
+Multi-layer contrastive learning. We are curious that on which layer(s) should we apply contrastive learning to achieve best generalization. We therefore propose a multi-layer CSG framework with different groups (combinations) of layer, denoted as $\mathcal { G }$ :
+
+$$
+{ \mathcal { L } } _ { \mathrm { N C E } } = \sum _ { l \in { \mathcal { G } } } { \mathcal { L } } _ { \mathrm { N C E } } ^ { l } = \sum _ { l \in { \mathcal { G } } } - \log { \frac { \exp \left( z ^ { l , a } \cdot z ^ { l , + } / \tau \right) } { \exp \left( z ^ { l , a } \cdot z ^ { l , + } / \tau \right) + \sum _ { z ^ { l , - } } \exp \left( z ^ { l , a } \cdot z ^ { l , - } / \tau \right) } }
+$$
+
+We conduct an ablation in Section 4.1.2 to study the generalization performance with respect to different $\mathcal { G }$ on ResNet- $1 0 1$ . Note that the non-linear projection heads $\bar { h } ^ { l } ( \cdot ) / \widetilde { h } ^ { l } ( \cdot )$ are layer-specific.
+
+Cross-task dense contrastive learning. Semantic segmentation presents a new form of task with per-pixel dense prediction, and the task naturally requires pixel-wise dense supervision $\mathcal { L } _ { \mathrm { T a s k } }$ . Unlike image classification, an image in semantic segmentation could contain rich amounts of objects. We therefore make $\mathcal { L } _ { \mathrm { N C E } }$ spatially denser in semantic segmentation to make it more compatible with the dense task loss $\mathcal { L } _ { \mathrm { T a s k } }$ . Specifically, the NCE losses are applied on cropped feature map patches:
+
+$$
+\mathcal { L } _ { \mathrm { N C E } } = \sum _ { l \in \mathcal { G } } \sum _ { i = 1 } ^ { N _ { l } } \mathcal { L } _ { \mathrm { N C E } } ^ { l , i } = \sum _ { l \in \mathcal { G } } \sum _ { i = 1 } ^ { N _ { l } } - \frac { 1 } { N _ { l } } \log \frac { \exp { \left( z _ { i } ^ { l , a } \cdot z _ { i } ^ { l , + } / \tau \right) } } { \exp { \left( z _ { i } ^ { l , a } \cdot z _ { i } ^ { l , + } / \tau \right) } + \sum _ { z _ { i } ^ { l , - } } \exp { \left( z _ { i } ^ { l , a } \cdot z _ { i } ^ { l , - } / \tau \right) } }
+$$
+
+where we crop $\pmb { x } ^ { a }$ into local patches $\mathbf { \Delta } \mathbf { x } _ { i } ^ { a }$ with $z _ { i } ^ { a } = f _ { e } \circ g \circ h ( \mathcal { T } ( x _ { i } ^ { a } ) )$ . Similar for ${ \pmb x } ^ { - }$ . In practice, we crop $_ { \textbf { \em x } }$ into $N _ { l } = 8 \times 8 = 6 4$ local patches during segmentation training.
+
+# 3.4 A-POOL: ATTENTIONAL POOLING FOR IMPROVED REPRESENTATION
+
+![](images/af13523c936a4303981417621f7848f41b185c17893aa7599b1b7a5c57b595a3.jpg)  
+Figure 4: (a) For each input image, A-pool computes an attention matrix $\textbf { \em a }$ based on the inner product between the global average pooled feature vector $\bar { \bf { v } }$ and vector at each position $\mathbf { \delta } _ { v : , i , j }$ $( \bar { \pmb { v } } , \pmb { v } _ { : , i , j } \in \mathbb { R } ^ { C } )$ ). (b) Example of four generated reweighting matrices on different images. Note that the values are defined as the ratio of the attention over uniform weight. The attention is visualized with upsampling to match the input size $2 2 4 \times 2 2 4 )$ ).
+
+The purpose of the pooling function $g ( \cdot )$ and the non-linear projection head $h ( \cdot )$ is to project a high dimensional feature map $\textbf {  { v } }$ from $\mathbb { R } ^ { C \bar { \times } h \bar { \times } w }$ to a low-dimensional embedding in $\mathbb { R } ^ { c }$ . With the feature pooled by $g ( \cdot )$ being more informative, we could also let the contrastive learning focus on more semantically meaningful representations. Inspired by recent works showing CNN’s capability of localizing salient objects (Zhou et al., 2016; Zhang et al., 2018) with only image-level supervision, we propose an attentional pooling (A-pool) module to improve the quality of the pooled feature.
+
+As shown in Figure 4(a), given a feature map $\textbf {  { v } }$ we first calculate its global average pooled vector $\begin{array} { r } { \bar { v } = g ( v ) = \frac { \bar { \Gamma } } { h w } [ \sum _ { i , j } \pmb { v } _ { 1 , i , j } ^ { - } , \cdot \cdot , \sum _ { i , j } \pmb { v } _ { C , i , j } ] , i \in [ 1 , h ] , j \in [ 1 , w ] } \end{array}$ , we then define the attention score for each pixel at (i, j) as ai,j = P hv:,i,j ,v¯ii0,j0 hv:,i0,j0 ,v¯i $\begin{array} { r } { \begin{array} { r } { \pmb { a } _ { i , j } = \frac { \langle \pmb { v } _ { : , i , j } , \pmb { \bar { v } } \rangle } { \sum _ { i ^ { \prime } , j ^ { \prime } } \langle \pmb { v } _ { : , i ^ { \prime } , j ^ { \prime } } , \pmb { \bar { v } } \rangle } ( i ^ { \prime } \in [ 1 , h ] , j ^ { \prime } \in [ 1 , w ] ) } \end{array} } \end{array}$ and use this score as the weight term in global pooling. Specifically, we define A-pool operator as $\hat { \pmb v } = g _ { a } ( \pmb v ) =$ $\begin{array} { r } { [ \sum _ { i , j } { \pmb v } _ { 1 , i , j } \cdot { \pmb a } _ { i , j } , \cdot \cdot \cdot , \sum _ { i , j } { \pmb v } _ { C , i , j } \cdot { \pmb a } _ { i , j } ] } \end{array}$ . This attention-weighted pooling procedure can effectively shift the focus of the pooled feature vector to the semantically salient regions, leading to more meaningful contrastive learning. In Figure 4(b), we plot the attention as the ratio of new attention score $^ { a }$ over uniform weights (i.e., the uniform score used in global average pooling as $\scriptstyle { \frac { 1 } { h \times w } }$ . For example, a value 1.5 in Figure 4(b) indicates an attention re of $\textstyle \frac { 1 . 5 } { h \times w } .$ ). Note that if any spatially$f _ { e , o }$ calculated by $f _ { e }$ , since $f _ { e }$ is the one adapted to the source domain with better attention.
+
+# 4 EXPERIMENT
+
+We follow (Chen et al., 2020c) to evaluate on two popular benchmarks: VisDA-17 COCO (classification) and GTA5 Cityscapes (segmentation). Codes is available at https://github.com/ NVlabs/CSG.
+
+# 4.1 IMAGE CLASSIFICATION
+
+Dataset. The VisDA-17 dataset (Peng et al., 2017) provides three subsets (domains), each with the same 12 object categories. Among them, the training set (source domain) is collected from synthetic renderings of 3D models under different angles and lighting conditions, whereas the validation set (target domain) contains real images cropped from the Microsoft COCO dataset (Lin et al., 2014).
+
+Implementation. For VisDA-17, we choose ImageNet pretrained ResNet-101 (He et al., 2016) as the backbone. We fine-tune the model on the source domain with SGD optimizer of learning rate $1 \times 1 0 ^ { - 4 }$ , weight decay $5 \times 1 0 ^ { - 4 }$ , and momentum 0.9. Batch size is set to 32, and the model is trained for 30 epochs. $\lambda$ for $\mathcal { L } _ { \mathrm { N C E } }$ is set as 0.1.
+
+# 4.1.1 MAIN RESULTS
+
+We compare with different distillation strategies in Table 1, including feature $l _ { 2 }$ regularization (Chen et al., 2018), parameter $l _ { 2 }$ regularization, importance weighted parameter $l _ { 2 }$ regularization (Zenke et al., 2017), and KL divergence (Chen et al., 2020c). All these approaches try to retain the ImageNet domain knowledge during the synthetic training, without feature diversity being explicitly promoted. One could see, CSG significantly improves generalizaiton performance over these baselines.
+
+We also verify that CSG promotes diverse representations, and that the diversity is correlated with generalization performance. To this end, we quantitatively measure the hyperspherical energy defined in Equation 1 on the feature embeddings extracted by different methods. From Table 1, one can see that the baseline suffers from the highest energy, and under different power settings, CSG consistently achieves the lowest energies. Table 1 indicates that a method that achieves lower HSE can better generalize from synthetic to the real domain. This confirms our motivation that forcing the model to capture more diversely scattered features will achieve better generalization performance.
+
+Table 1: Generalization performance and hyperspherical energy of the features extracted by different models (lower is better). Dataset: VisDA-17 (Peng et al., 2017) validation set. Model: ResNet-101.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Power</td><td rowspan="2">Accuracy (%)</td></tr><tr><td>0</td><td>1</td><td>2</td></tr><tr><td>Oracle on ImageNet3</td><td>=</td><td>=</td><td>=</td><td>53.3</td></tr><tr><td>Baseline (vanilla synthetic training)</td><td>0.4245</td><td>1.2500</td><td>1.6028</td><td>49.3</td></tr><tr><td>Weight l2 distance (Kirkpatrick et al., 2017)</td><td>0.4014</td><td>1.2296</td><td>1.5302</td><td>56.4</td></tr><tr><td>Synaptic Intelligence (Zenke et al., 2017)</td><td>0.3958</td><td>1.2261</td><td>1.5216</td><td>57.6</td></tr><tr><td>Feature l2 distance (Chen et al.,2018)</td><td>0.3337</td><td>1.1910</td><td>1.4449</td><td>57.1</td></tr><tr><td>ASG (Chen et al.,2020c)</td><td>0.3251</td><td>1.1840</td><td>1.4229</td><td>61.1</td></tr><tr><td>CSG (Ours)</td><td>0.3188</td><td>1.1806</td><td>1.4177</td><td>64.05</td></tr></table>
+
+# 4.1.2 ABLATION STUDY
+
+We perform ablation studies (Table 2, 3, 4) on the VisDA-17 image classification benchmark (Peng et al., 2017).
+
+Augmentation. We study different magnitudes of RandAugment (Cubuk et al., 2020) in our scenario (Section 3.2), as summarized in Table 2. By tuning the global magnitude control factor $M$ , we observe that too strong augmentations deteriorate generalization (e.g. $M = 1 2 , 1 8 , 2 4 ,$ , while mild augmentation brings limited help $M = 3$ ). A moderate augmentation $M = 6$ ) can improve contrastive learning.
+
+Multi-layer Contrastive Learning. Since features from the high-level layers are directly responsible for the downstream classification or other vision tasks, we suspect the last layer in the feature extractor $f _ { e }$ would be the most important. We conduct an ablation study on generalization performance with different layer combinations for multilayer contrastive learning (Section 3.3). From Table 3, one can see that applying $\mathcal { L } _ { \mathrm { N C E } }$ on layer 3 and 4 are most effective. Therefore, in our work we set $\mathcal { G } = \{ 3 , 4 \}$
+
+![](images/306292eeaab547c16940b7f355b035efc74439639cc5b64c6ee28d12ff0af3f5.jpg)  
+Figure 5: An illustration of model attention by GradCAM (Selvaraju et al., 2017) on the VisDA-17 validation set.
+
+guided pooling (Section 3.4), A-pool can further improve the generalization performance, compared with the vanilla global average pooling (GAP).
+
+Table 2: Ablation with $M$ .   
+
+<table><tr><td>M</td><td>Accuracy</td></tr><tr><td>O (no aug.)</td><td>60.86</td></tr><tr><td>3</td><td>61.36</td></tr><tr><td>6</td><td>62.88</td></tr><tr><td>12</td><td>62.61</td></tr><tr><td>18</td><td>62.00</td></tr></table>
+
+Table 3: Ablation with $\mathcal { G }$ .   
+
+<table><tr><td>Layer Groups G</td><td>Accuracy (%)</td></tr><tr><td>4</td><td>62.88</td></tr><tr><td>3+4</td><td>63.77</td></tr><tr><td>2+3+4</td><td>62.66</td></tr><tr><td>1+2+3+4</td><td>62.30</td></tr></table>
+
+Table 4: Ablation w./w.o. A-pool. GAP: global average pooling.   
+
+<table><tr><td>Pooling</td><td>Accuracy (%)</td></tr><tr><td>GAP</td><td>63.77</td></tr><tr><td>A-pool</td><td>64.05</td></tr></table>
+
+# 4.1.3 CSG BENEFITS VISUAL ATTENTION
+
+We further show the Grad-CAM3 attention on VisDA-17 validation set (Figure 5). We can see that our CSG framework also contributes to better visual attention on unseen real images.
+
+# 4.2 SEMANTIC SEGMENTATION
+
+Dataset. GTA5 (Richter et al., 2016) is a vehicle-egocentric image dataset collected in a computer game with pixel-wise semantic labels. It contains 24,966 images with a resolution of $1 0 5 2 \times 1 9 1 4$ There are 19 classes that are compatible with the Cityscapes dataset (Cordts et al., 2016).
+
+Cityscapes (Cordts et al., 2016) contains urban street images taken on a vehicle from some European cities. There are 5,000 images with pixel-wise annotations. The images have a resolution of $1 0 2 4 \times 2 0 4 8$ and are labeled into 19 semantic categories.
+
+Implementation. We study DeepLabv2 (Chen et al., 2017) with both ResNet-50 and ResNet-101 backbone. The backbones are pre-trained on ImageNet. We also use SGD optimizer, with learning rate as $1 \times 1 0 ^ { - 3 }$ , weight decay as $5 \times 1 0 ^ { - 4 }$ , and momentum are 0.9. Batch size is set to six. We crop the images into patches of $5 1 2 \times 5 1 2$ and train the model with multi-scale augmentation $( 0 . 7 5 \sim 1 . 2 5 )$ and horizontal flipping. The model is trained for 50 epochs, and $\lambda$ for $\mathcal { L } _ { \mathrm { N C E } }$ is set as 75.
+
+# 4.2.1 MAIN RESULTS
+
+We also evaluate the generalization performance of our CSG on semantic segmentation. In particular, we treat the GTA5 training set as the synthetic source domain and train segmentation models on it. We then treat the Cityscapes validation sets as real target domains, where we directly evaluate the synthetically trained models. We can see that in Table 5, CSG achieves the best performance gain. IBN-Net Pan et al. (2018) improves domain generalization by carefully mix the instance and batch normalization in the backbone, while Yue et al. (2019) transfers the real image styles from ImageNet to synthetic images. However, Yue et al. (2019) requires ImageNet images during synthetic training, and also implicitly leverages ImageNet labels as auxiliary domains.
+
+Table 5: Comparison to previous domain generalization methods for segmentation $\mathrm { G T A } 5 $ Cityscapes).   
+
+<table><tr><td>Methods</td><td>Backbone</td><td>mIoU %</td><td>mIoU↑%</td></tr><tr><td>No Adapt</td><td rowspan="6">ResNet-50</td><td>22.17</td><td rowspan="2">7.47</td></tr><tr><td>IBN-Net (Pan et al.,2018)</td><td>29.64</td></tr><tr><td>No Adapt</td><td>32.45</td><td rowspan="2">4.97</td></tr><tr><td>Yue et al. (Yue et al., 2019)</td><td>37.42</td></tr><tr><td>No Adapt</td><td>25.88</td><td>3.77</td></tr><tr><td>ASG (Chen et al.,2020c) No Adapt</td><td></td><td>29.65</td></tr><tr><td rowspan="2">CSG (ours)</td><td rowspan="6">ResNet-101</td><td>25.88</td><td rowspan="2">9.39</td></tr><tr><td>35.27</td></tr><tr><td>No Adapt</td><td>33.56</td><td>8.97</td></tr><tr><td>Yue et al. (Yue et al.,2019)</td><td>42.53</td><td rowspan="2">3.16</td></tr><tr><td>No Adapt</td><td>29.63</td></tr><tr><td>ASG (Chen et al.,2020c)</td><td>32.79</td><td></td></tr><tr><td>No Adapt CSG (ours)</td><td></td><td>29.63 38.88</td><td>9.25</td></tr></table>
+
+# 4.2.2 FEATURE DIVERSITY ON SEGMENTATION WITH BALANCED TRAINING SET
+
+We further conduct visualization and quantitative measures of feature diversity on the segmentation task. Similar to section 2, we randomly sample a subset of the GTA5 training set to match the size of the Cityscapes training set. We again have similar observations: models trained on real images have relatively diverse features, and synthetic training leads to collapsed features. Here we get lower $E _ { s }$ than classification since we follow the setting in Eq. 6 to study dense-level features. This leads to a larger total number of features on segmentation than classification.
+
+![](images/ec0bda1f0bd50d717b806533b4dc8d661062dda200f5712726d9a8ebb9b64769.jpg)  
+Figure 6: Feature diversity on Cityscapes test images in $\mathbb { R } ^ { 2 }$ with Gaussian kernel density estimation (KDE). Darker areas have more concentrated features. $E _ { s }$ : hyperspherical energy of features, lower the more diverse.
+
+# 4.2.3 VISUAL RESULTS
+
+By visualizing the segmentation results (Figure 7), we can see that as our CSG framework achieves better mIoU on unseen real images from the Cityscapes validation set, the model produces segmentation with much higher visual quality. In contrast, the baseline model suffers from much more misclassification.
+
+# 5 RELATED WORK
+
+Domain generalization considers the problem of generalizing a model to the unseen target domain without leveraging any target domain images (Muandet et al., 2013; Gan et al., 2016). The core challenge is how to close the domain gap and align feature spaces from different domains, without even seeing the target domain’s data. Muandet et al. (2013) proposed to use MMD (Maximum Mean Discrepancy) to align the distributions from different source domains and train their network with adversarial learning. Li et al. (2017) built separate networks for each source domain and used shared parameters for testing. By using a meta-learning approach on split training sets, Li et al. (2018) further improved generalization performance. Instance Normalization and Batch Normalization are carefully integrated into the backbone network by Pan et al. (2018) to boost network generalization. Differently, Yue et al. (2019) proposed to transfer information from the real domain as image styles to synthetic images. Most recently, (Chen et al., 2020c) formulated domain generalization as a life-long learning problem (Li & Hoiem, 2017), and try to avoid the catastrophic forgetting about the ImageNet pre-trained weights and to retain real-domain knowledge during transfer learning.
+
+![](images/e04c4a4fcb913cc3990207bbe93d170fbdf3913e9673ea453972ad5bed625a19.jpg)  
+Figure 7: Generalization results on $\mathrm { G T A } 5 $ Cityscapes. Rows correspond to sample images in Cityscapes validation set. From left to right, columns correspond to original images, ground truth, predication results of baseline (DeepLabv2-ResNet50 Chen et al. (2017)), and prediction by model trained with our CSG framework.
+
+Contrastive learning. Noise contrastive estimation loss (Wu et al., 2018) recently becomes a predominant design choice for self-supervised contrastive representation learning (Hjelm et al., 2018; Oord et al., 2018; Henaff et al. ´ , 2019; Tian et al., 2019; He et al., 2020; Misra & Maaten, 2020; Chen et al., 2020a). Studies show that self-supervised models can serve as powerful initializations for downstream tasks, even outperforming supervised pre-training on several. Besides engineering improvements, key factors towards better contrastive learning include employing large numbers of negative examples and designing more semantically meaningful augmentations to create different views of images. This leads to both maximize the mutual information between two views of the same instance and pushing examples from different instances apart (Tian et al., 2020b). As also observed by Wang & Isola (2020), contrastive learning tends to align the features belonging to the same instance, while scattering the normalized learned features on a hypersphere. However, most work focus on the representation learning for a real-to-real transfer learning setting where the main focus is to improve the performance of the downstream tasks. While having connections to these methods, our work pursues a different task with different motivations despite the converging techniques.
+
+# 6 CONCLUSIONS
+
+Motivated by the observation that models trained on synthetic images tend to generate collapsed feature representation, we make a hypothesis that the diversity of feature representation plays an important role in generalization performance. Taking this as an inductive bias, we propose a contrastive synthetic-to-real generalization framework that simultaneously regularizes the synthetically trained representations while promoting the diversity of the features to improve generalization. Experiments on VisDA-17 validate our hypothesis, showing that the diversity of features correlates with generalization performance across different models. Together with the multi-scale contrastive learning and attention-guided pooling strategy, the proposed framework outperforms previous state-of-the-arts on VisDA-17 with sizable gains, while giving competitive performance and the largest relative improvements on GTA5 Cityscapes without bells and whistles.
+
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+        "type": "text",
+        "text": "CONTRASTIVE SYN-TO-REAL GENERALIZATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Wuyang Chen1∗, Zhiding $\\mathbf { Y } \\mathbf { u } ^ { 2 \\dagger }$ , Shalini De Mello2, Sifei Liu2, Jose M. Alvarez2,   \nZhangyang $\\mathbf { W a n g ^ { 1 } }$ , Anima Anandkumar2,3   \n1The University of Texas at Austin 2NVIDIA 3California Institute of Technology   \n{wuyang.chen,atlaswang}@utexas.edu   \n{zhidingy,shalinig,sifeil,josea,aanandkumar}@nvidia.com   \nhttps://github.com/NVlabs/CSG ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Training on synthetic data can be beneficial for label or data-scarce scenarios. However, synthetically trained models often suffer from poor generalization in real domains due to domain gaps. In this work, we make a key observation that the diversity of the learned feature embeddings plays an important role in the generalization performance. To this end, we propose contrastive synthetic-to-real generalization (CSG), a novel framework that leverages the pre-trained ImageNet knowledge to prevent overfitting to the synthetic domain, while promoting the diversity of feature embeddings as an inductive bias to improve generalization. In addition, we enhance the proposed CSG framework with attentional pooling (A-pool) to let the model focus on semantically important regions and further improve its generalization. We demonstrate the effectiveness of CSG on various synthetic training tasks, exhibiting state-of-the-art performance on zero-shot domain generalization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks have pushed the boundaries of many visual recognition tasks. However, their success often hinges on the availability of both training data and labels. Obtaining data and labels can be difficult or expensive in many applications such as semantic segmentation, correspondence, 3D reconstruction, pose estimation, and reinforcement learning. In these cases, learning with synthetic data can greatly benefit the applications since large amounts of data and labels are available at relatively low costs. For this reason, synthetic training has recently gained significant attention (Wu et al., 2015; Richter et al., 2016; Shrivastava et al., 2017; Savva et al., 2019). ",
+        "bbox": [
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+        "type": "text",
+        "text": "Despite many benefits, synthetically trained models often have poor generalization on the real domain due to large domain gaps between synthetic and real images. Limitations on simulation and rendering can lead to degraded synthesis quality, such as aliased boundaries, unrealistic textures, fake appearance, over-simplified lighting conditions, and unreasonable scene layouts. These issues result in domain gaps between synthetic and real images, preventing the synthetically trained models from capturing meaningful representations and limiting their generalization ability on real images. ",
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+        "type": "text",
+        "text": "To mitigate these issues, domain generalization and adaptation techniques have been proposed (Li et al., 2017; Pan et al., 2018; Yue et al., 2019). Domain adaptation assumes the availability of target data (labeled, partially labeled, or unlabeled) during training. On the other hand, domain generalization considers zero-shot generalization without seeing the target data of real images, and is therefore more challenging. An illustration of the domain generalization protocol on the ",
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+        "image_caption": [
+            "Figure 1: An illustration of the domain generalization protocol on the VisDA-17 dataset, where real target domain (test) images are assumed unavailable during model training. "
+        ],
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+        "bbox": [
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+        "text": "VisDA-17 dataset (Peng et al., 2017) is shown in Figure 1. Considering that ImageNet pre-trained representation is widely used as model initialization, recent efforts on domain generalization show that such knowledge can be used to prevent overfitting to the synthetic domain (Chen et al., 2018; 2020c). Specifically, they impose a distillation loss to regularize the distance between the synthetically trained and the ImageNet pre-trained representations, which improves synthetic-to-real generalization. ",
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+        "text": "The above approaches still face limitations due to the challenging nature of this problem. Taking a closer look, we observe the following pitfalls in training on synthetic data. First, obtaining photorealistic appearance features at the micro-level, such as texture and illumination, is challenging due to the limits of simulation complexity and rendering granularity. Without special treatment, CNNs tend to be biased towards textures (Geirhos et al., 2019) and suffer from badly learned representations on synthetic data. Second, the common lack of texture and shape variations on synthetic images often leads to collapsed and trivial representations without any diversity. This is unlike training with natural images where models get sufficiently trained by seeing enough variations. Such a lack of diversity in the representation makes the learned models vulnerable to natural variations in the real world. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Summary of contributions and results: ",
+        "text_level": 1,
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+        "text": "• We observe that the diversity of learned feature embedding plays an important role in syntheticto-real generalization. We show an example of collapsed representations learned by a synthetic model, which is in sharp contrast to features learned from real data (Section 2). ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "• Motivated by the above observation, we propose a contrastive synthetic-to-real generalization framework that simultaneously regularizes the synthetically trained representation while promoting the diversity of the learned representation to improve generalization (Section 3.1). ",
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+        "text": "• We further enhance the CSG framework with attentional pooling (A-pool) where feature representations are guided by model attention. This allows the model to localize its attention to semantically more important regions, and thus improves synthetic-to-real generalization (Section 3.4). ",
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+        "text": "• We benchmark CSG on various synthetic training tasks including image classification (VisDA-17) and semantic segmentation $\\mathrm { ( G T A 5  }$ Cityscapes). We show that CSG considerably improves the generalization performance without seeing target data. Our best model reaches $6 4 . 0 5 \\%$ accuracy on VisDA-17 compared to previous state-of-the-art (Chen et al., 2020c) with $6 1 . 1 \\%$ (Section 4). ",
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+    {
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+        "text": "2 A MOTIVATING EXAMPLE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We give a motivating example to show the significant differences between the features learned on synthetic and real images. Specifically, we use a ResNet-101 backbone and extract the $l _ { 2 }$ normalized feature embedding after global average pooling (defined as $\\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { v } }$ ). We consider the following three models: 1) model pre-trained on ImageNet, 2) model trained on VisDA-17 validation set (real images), and 3) model trained on VisDA-17 training set (synthetic images) 1. Both 2) and 3) are initialized with ImageNet pre-training, and fine-tuned on the 12 classes defined in VisDA-17. ",
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+        "image_caption": [
+            "Figure 2: Feature diversity on VisDA-17 test images in $\\mathbb { R } ^ { 2 }$ with Gaussian kernel density estimation (KDE). Darker areas have more concentrated features. $E _ { s }$ : hyperspherical energy of features, lower the more diverse. "
+        ],
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+    {
+        "type": "text",
+        "text": "Visualization of feature diversity. We visualize the normalized representations on a 2-dim sphere. A Gaussian kernel with bandwidth estimated by Scott’s Rule (Scott, 2015) is applied to estimate the probability density function. Darker areas have more concentrated features, and if the feature space (the 2-dim sphere) is covered by dark areas, it has more diversely placed features. In Figure 2, we can see that the ImageNet pretrained model can widely span the representations on the 2-dim feature space. The model trained on VisDA-17 validation set can also generate diverse features, although slightly affected by the class imbalance. However, when the model is trained on the training set (synthetic images), the features largely collapse to a narrow subspace, i.e., the model fails to fully leverage the whole feature space. This is clear that training on synthetic images can easily introduce poor bias to the model and the collapsed representations will fail to generalize to the real domain. ",
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+    {
+        "type": "text",
+        "text": "Quantitive measurement of feature diversity. Inspired by (Liu et al., 2018), we also quantitatively measure the diversity of the feature embeddings using the following hyperspherical potential energy: ",
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+    {
+        "type": "equation",
+        "img_path": "images/52eb2deab1cd74d9099772f5e30bc31681e0f7048bcb6ebf66998714f0016b00.jpg",
+        "text": "$$\nE _ { s } \\left( \\bar { v } _ { i } | _ { i = 1 } ^ { N } \\right) = \\sum _ { i = 1 } ^ { N } \\sum _ { \\substack { j = 1 , j \\neq i } } ^ { N } e _ { s } \\left( \\| \\bar { v } _ { i } - \\bar { v } _ { j } \\| \\right) = \\left\\{ \\begin{array} { l l } { \\sum _ { i \\neq j } \\| \\bar { v } _ { i } - \\bar { v } _ { j } \\| ^ { - s } , \\quad s > 0 } \\\\ { \\sum _ { i \\neq j } \\log \\left( \\| \\bar { v } _ { i } - \\bar { v } _ { j } \\| ^ { - 1 } \\right) , \\quad s = 0 } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$N$ is the number of examples. The lower the hyperspherical energy (HSE) is, the more diverse the feature vectors will be scattered in the unit sphere. $s$ is the power factor, and we choose $s = 0$ in this example. Three training strategies exhibit energies as 0.2541, 0.3355, 0.4408, respectively. This validates that models trained on real images can capture diverse features, whereas the synthetic training will lead the model to highly collapsed feature space. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remarks. A conclusion can be drawn from the above examples: though assisted with ImageNet initialization, fine-tuning on synthetic images tends to give collapsed features with poor diversity in sharp contrast to training with real images. This indicates that the diversity of learned representation could play an important role in synthetic-to-real generalization. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 CONTRASTIVE SYNTHETIC-TO-REAL GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider the synthetic-to-real domain generalization problem following the protocols of Chen et al. (2020c). More specifically, the objective is to achieve the best zero-shot generalization on the unseen target domain real images without having access to them during synthetic training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 NOTATION AND FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our design of the model considers the following two aspects with a “push and pull” strategy: ",
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+    {
+        "type": "text",
+        "text": "Pull: Without access to real images, the ImageNet pre-trained model presents the only source of real domain knowledge that can implicitly guide our training. As a result, we hope to impose some form of similarity between the features obtained by the synthetic model and the ImageNet pre-trained one. This helps to overcome the domain gaps from the unrealistic appearance of synthetic images. ",
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+    {
+        "type": "text",
+        "text": "Push: Section 2 shows that synthetic training tends to generate collapsed features whereas models trained on natural images give many diverse ones. We treat this as an inductive bias to improve synthetic training, by pushing the feature embeddings away from each other across different images. ",
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+    {
+        "type": "text",
+        "text": "The above “push and pull” strategy can be exactly formulated with a contrastive loss. This motivates us to propose a contrastive synthetic-to-real generalization framework as partly inspired by recent popular contrastive learning methods (He et al., 2020). Figure 3(b) illustrates our CSG framework. Specifically, we denote the frozen Imagenet pre-trained model as $f _ { e , o }$ and the synthetically trained model $f _ { e }$ , where $f _ { e }$ is supervised by the task loss $\\mathcal { L } _ { \\boldsymbol { s y n } }$ for the defined downstream task. We denote the input synthetic image as $\\pmb { x } ^ { a }$ and treat it as an anchor. We treat the embeddings of $\\pmb { x } ^ { a }$ obtained by $f _ { e }$ and $f _ { e , o }$ as anchor and positive embeddings, denoting them as $z ^ { a }$ and $z ^ { + }$ , respectively. Following a typical contrastive approach, we define $K$ negative images $\\{ \\pmb { x } _ { 1 } ^ { - } , \\cdots , \\pmb { x } _ { K } ^ { - } \\}$ for every anchor $\\pmb { x } ^ { a }$ , and denote their corresponding embeddings as $\\{ z _ { 1 } ^ { - } , \\cdots , z _ { K } ^ { - } \\}$ . Similar to the design in (Chen et al., 2020d), we define $h / \\widetilde { h } : \\mathbb { R } ^ { C }  \\mathbb { R } ^ { c }$ as the nonlinear projection heads with a two MLP layers and a ReLU layer between them. The CSG framework regularizes $f _ { e }$ in a contrastive manner: pulling $z ^ { a }$ and $z ^ { + }$ to be closer while pushing $z ^ { a }$ and $\\{ z _ { 1 } ^ { - } , \\cdots , z _ { K } ^ { - } \\}$ apart. This regularizes the model by preventing its representation from deviating too far from that of a pre-trained ImageNet model and yet encouraging it to learn task-specific information from the synthetic data. ",
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+    {
+        "type": "image",
+        "img_path": "images/7d30e3b74a188c3c1b9761d315e826059ca6905f8129c8afcb3e7314e2649b9d.jpg",
+        "image_caption": [
+            "Figure 3: (a) Previous work (Chen et al., 2018; 2020c) consider “learning without forgetting” which minimizes a distillation loss between a synthetic model and an ImageNet pre-trained one (either on features or model parameters) to avoid catastrophic forgetting. (b) The proposed CSG framework with a “push and pull” strategy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Even though having connections to recent self-supervised contrastive representation learning methods (Oord et al., 2018; Wu et al., 2018; Chen et al., 2020a; He et al., 2020; Chen et al., 2020b; Jiang et al., 2020), our work differs in the following aspects: 1) Self-supervised learning and the addressed task are ill-posed in different manners - the former lacks the constraints from semantic labels, whereas the latter lacks the support of data distribution. 2) As a result, the motivations of contrastive learning are different. Our work is also related to the contrastive distillation framework in (Tian et al., 2020a). Again, the two works differ in both task and motivation despite the converging techniques. ",
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+    {
+        "type": "text",
+        "text": "3.2 AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Augmentation has been an important part of effective contrastive learning. By perturbing or providing different views of the representations, augmentation forces a model to focus more on the mid-level and high-level representations of object parts and structures which are visually more realistic and reliable. To this end, we follow existing popular approaches to create augmentation at different levels: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Image augmentation. We consider image-level augmentation using RandAugment (Cubuk et al., 2020) where a single global control factor $M$ is used to control the augmentation magnitude. We denote the transform operators of image-level augmentation as $\\tau ( \\cdot )$ . ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Model augmentation. We adopt a mean-teacher (Tarvainen & Valpola, 2017) styled moving average of a model to create different views of feature embeddings. Given an anchor image $\\pmb { x } ^ { a }$ and $K$ negative images $\\{ \\pmb { x } _ { 1 } ^ { - } , \\cdots , \\pmb { x } _ { K } ^ { - } \\}$ , we compute the embeddings as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d8c4e68a1bedd8984cefaf85bc69abe860d1bc12901e828bc56d9d4c40d2c61b.jpg",
+        "text": "$$\n\\begin{array} { r } { z ^ { a } = f _ { e } \\circ g \\circ h ( \\mathcal { T } ( \\pmb { x } ^ { a } ) ) , z ^ { + } = f _ { e , o } \\circ g \\circ \\widetilde { h } ( \\mathcal { T } ( \\pmb { x } ^ { a } ) ) , z _ { k } ^ { - } = f _ { e , o } \\circ g \\circ \\widetilde { h } ( \\mathcal { T } ( \\pmb { x } _ { k } ^ { - } ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            647,
+            767,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g : \\mathbb { R } ^ { C \\times h \\times w }  \\mathbb { R } ^ { C }$ is a pooling operator transforming a feature map into a vector. Following (He et al., 2020), we define $\\widetilde { h } ( \\cdot )$ as an exponential moving average of the $h ( \\cdot )$ across different iterations. Such difference in $h ( \\cdot )$ and $\\widetilde { h } ( \\cdot )$ leads to augmented views of embeddings. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.3 CONTRASTIVE LOSS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use InfoNCE loss (Wu et al., 2018) to formulate the “push and pull” strategy: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0c2e1f8e0fb96e7980d7c359b5d3f1715c4ed360b7c883e4769ebaffe3c030d6.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { N C E } } = - \\log { \\frac { \\exp { ( z ^ { a } \\cdot z ^ { + } / \\tau ) } } { \\exp { ( z ^ { a } \\cdot z ^ { + } / \\tau ) } + \\sum _ { z ^ { - } } \\exp { ( z ^ { a } \\cdot z ^ { - } / \\tau ) } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau = 0 . 0 7$ is a temperature hyper-parameter in our work. Together, we minimize the combination of the synthetic task loss and $\\mathcal { L } _ { \\mathrm { N C E } }$ during our transfer learning process: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c0e041bc43c06c7233125e8d1dbcdd8fdeaed2671df813506c8bde9b7705c3e3.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { \\mathrm { T a s k } } + \\lambda \\mathcal { L } _ { \\mathrm { N C E } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            859,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Specifically, $\\mathcal { L } _ { \\mathrm { T a s k } }$ is the synthetic training task objective. For example, $\\mathcal { L } _ { \\mathrm { T a s k } }$ is a cross-entropy loss of a vector over the 12 defined classes on VisDA-17, whereas it is a per-pixel dense cross-entropy loss on GTA5. $\\lambda$ is a balancing factor controlling the strength of the Contrastive Learning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Multi-layer contrastive learning. We are curious that on which layer(s) should we apply contrastive learning to achieve best generalization. We therefore propose a multi-layer CSG framework with different groups (combinations) of layer, denoted as $\\mathcal { G }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/94b3ca2effa8c70ce1ff69f3154225dd6250dedf5c6f32df53c27bd51638f653.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { N C E } } = \\sum _ { l \\in { \\mathcal { G } } } { \\mathcal { L } } _ { \\mathrm { N C E } } ^ { l } = \\sum _ { l \\in { \\mathcal { G } } } - \\log { \\frac { \\exp \\left( z ^ { l , a } \\cdot z ^ { l , + } / \\tau \\right) } { \\exp \\left( z ^ { l , a } \\cdot z ^ { l , + } / \\tau \\right) + \\sum _ { z ^ { l , - } } \\exp \\left( z ^ { l , a } \\cdot z ^ { l , - } / \\tau \\right) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We conduct an ablation in Section 4.1.2 to study the generalization performance with respect to different $\\mathcal { G }$ on ResNet- $1 0 1$ . Note that the non-linear projection heads $\\bar { h } ^ { l } ( \\cdot ) / \\widetilde { h } ^ { l } ( \\cdot )$ are layer-specific. ",
+        "bbox": [
+            176,
+            193,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Cross-task dense contrastive learning. Semantic segmentation presents a new form of task with per-pixel dense prediction, and the task naturally requires pixel-wise dense supervision $\\mathcal { L } _ { \\mathrm { T a s k } }$ . Unlike image classification, an image in semantic segmentation could contain rich amounts of objects. We therefore make $\\mathcal { L } _ { \\mathrm { N C E } }$ spatially denser in semantic segmentation to make it more compatible with the dense task loss $\\mathcal { L } _ { \\mathrm { T a s k } }$ . Specifically, the NCE losses are applied on cropped feature map patches: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/63376829fb008b27e7d1f568227bdce8374523e7a4369a085709f3feb620bcd4.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { N C E } } = \\sum _ { l \\in \\mathcal { G } } \\sum _ { i = 1 } ^ { N _ { l } } \\mathcal { L } _ { \\mathrm { N C E } } ^ { l , i } = \\sum _ { l \\in \\mathcal { G } } \\sum _ { i = 1 } ^ { N _ { l } } - \\frac { 1 } { N _ { l } } \\log \\frac { \\exp { \\left( z _ { i } ^ { l , a } \\cdot z _ { i } ^ { l , + } / \\tau \\right) } } { \\exp { \\left( z _ { i } ^ { l , a } \\cdot z _ { i } ^ { l , + } / \\tau \\right) } + \\sum _ { z _ { i } ^ { l , - } } \\exp { \\left( z _ { i } ^ { l , a } \\cdot z _ { i } ^ { l , - } / \\tau \\right) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we crop $\\pmb { x } ^ { a }$ into local patches $\\mathbf { \\Delta } \\mathbf { x } _ { i } ^ { a }$ with $z _ { i } ^ { a } = f _ { e } \\circ g \\circ h ( \\mathcal { T } ( x _ { i } ^ { a } ) )$ . Similar for ${ \\pmb x } ^ { - }$ . In practice, we crop $_ { \\textbf { \\em x } }$ into $N _ { l } = 8 \\times 8 = 6 4$ local patches during segmentation training. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 A-POOL: ATTENTIONAL POOLING FOR IMPROVED REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/af13523c936a4303981417621f7848f41b185c17893aa7599b1b7a5c57b595a3.jpg",
+        "image_caption": [
+            "Figure 4: (a) For each input image, A-pool computes an attention matrix $\\textbf { \\em a }$ based on the inner product between the global average pooled feature vector $\\bar { \\bf { v } }$ and vector at each position $\\mathbf { \\delta } _ { v : , i , j }$ $( \\bar { \\pmb { v } } , \\pmb { v } _ { : , i , j } \\in \\mathbb { R } ^ { C } )$ ). (b) Example of four generated reweighting matrices on different images. Note that the values are defined as the ratio of the attention over uniform weight. The attention is visualized with upsampling to match the input size $2 2 4 \\times 2 2 4 )$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            422,
+            820,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The purpose of the pooling function $g ( \\cdot )$ and the non-linear projection head $h ( \\cdot )$ is to project a high dimensional feature map $\\textbf {  { v } }$ from $\\mathbb { R } ^ { C \\bar { \\times } h \\bar { \\times } w }$ to a low-dimensional embedding in $\\mathbb { R } ^ { c }$ . With the feature pooled by $g ( \\cdot )$ being more informative, we could also let the contrastive learning focus on more semantically meaningful representations. Inspired by recent works showing CNN’s capability of localizing salient objects (Zhou et al., 2016; Zhang et al., 2018) with only image-level supervision, we propose an attentional pooling (A-pool) module to improve the quality of the pooled feature. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 4(a), given a feature map $\\textbf {  { v } }$ we first calculate its global average pooled vector $\\begin{array} { r } { \\bar { v } = g ( v ) = \\frac { \\bar { \\Gamma } } { h w } [ \\sum _ { i , j } \\pmb { v } _ { 1 , i , j } ^ { - } , \\cdot \\cdot , \\sum _ { i , j } \\pmb { v } _ { C , i , j } ] , i \\in [ 1 , h ] , j \\in [ 1 , w ] } \\end{array}$ , we then define the attention score for each pixel at (i, j) as ai,j = P hv:,i,j ,v¯ii0,j0 hv:,i0,j0 ,v¯i $\\begin{array} { r } { \\begin{array} { r } { \\pmb { a } _ { i , j } = \\frac { \\langle \\pmb { v } _ { : , i , j } , \\pmb { \\bar { v } } \\rangle } { \\sum _ { i ^ { \\prime } , j ^ { \\prime } } \\langle \\pmb { v } _ { : , i ^ { \\prime } , j ^ { \\prime } } , \\pmb { \\bar { v } } \\rangle } ( i ^ { \\prime } \\in [ 1 , h ] , j ^ { \\prime } \\in [ 1 , w ] ) } \\end{array} } \\end{array}$ and use this score as the weight term in global pooling. Specifically, we define A-pool operator as $\\hat { \\pmb v } = g _ { a } ( \\pmb v ) =$ $\\begin{array} { r } { [ \\sum _ { i , j } { \\pmb v } _ { 1 , i , j } \\cdot { \\pmb a } _ { i , j } , \\cdot \\cdot \\cdot , \\sum _ { i , j } { \\pmb v } _ { C , i , j } \\cdot { \\pmb a } _ { i , j } ] } \\end{array}$ . This attention-weighted pooling procedure can effectively shift the focus of the pooled feature vector to the semantically salient regions, leading to more meaningful contrastive learning. In Figure 4(b), we plot the attention as the ratio of new attention score $^ { a }$ over uniform weights (i.e., the uniform score used in global average pooling as $\\scriptstyle { \\frac { 1 } { h \\times w } }$ . For example, a value 1.5 in Figure 4(b) indicates an attention re of $\\textstyle \\frac { 1 . 5 } { h \\times w } .$ ). Note that if any spatially$f _ { e , o }$ calculated by $f _ { e }$ , since $f _ { e }$ is the one adapted to the source domain with better attention. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We follow (Chen et al., 2020c) to evaluate on two popular benchmarks: VisDA-17 COCO (classification) and GTA5 Cityscapes (segmentation). Codes is available at https://github.com/ NVlabs/CSG. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Dataset. The VisDA-17 dataset (Peng et al., 2017) provides three subsets (domains), each with the same 12 object categories. Among them, the training set (source domain) is collected from synthetic renderings of 3D models under different angles and lighting conditions, whereas the validation set (target domain) contains real images cropped from the Microsoft COCO dataset (Lin et al., 2014). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation. For VisDA-17, we choose ImageNet pretrained ResNet-101 (He et al., 2016) as the backbone. We fine-tune the model on the source domain with SGD optimizer of learning rate $1 \\times 1 0 ^ { - 4 }$ , weight decay $5 \\times 1 0 ^ { - 4 }$ , and momentum 0.9. Batch size is set to 32, and the model is trained for 30 epochs. $\\lambda$ for $\\mathcal { L } _ { \\mathrm { N C E } }$ is set as 0.1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 MAIN RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare with different distillation strategies in Table 1, including feature $l _ { 2 }$ regularization (Chen et al., 2018), parameter $l _ { 2 }$ regularization, importance weighted parameter $l _ { 2 }$ regularization (Zenke et al., 2017), and KL divergence (Chen et al., 2020c). All these approaches try to retain the ImageNet domain knowledge during the synthetic training, without feature diversity being explicitly promoted. One could see, CSG significantly improves generalizaiton performance over these baselines. ",
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+    {
+        "type": "text",
+        "text": "We also verify that CSG promotes diverse representations, and that the diversity is correlated with generalization performance. To this end, we quantitatively measure the hyperspherical energy defined in Equation 1 on the feature embeddings extracted by different methods. From Table 1, one can see that the baseline suffers from the highest energy, and under different power settings, CSG consistently achieves the lowest energies. Table 1 indicates that a method that achieves lower HSE can better generalize from synthetic to the real domain. This confirms our motivation that forcing the model to capture more diversely scattered features will achieve better generalization performance. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/57f79b2714a680ed5124e45cfceca2bdffd9c162a53e553cf70cc8be52312263.jpg",
+        "table_caption": [
+            "Table 1: Generalization performance and hyperspherical energy of the features extracted by different models (lower is better). Dataset: VisDA-17 (Peng et al., 2017) validation set. Model: ResNet-101. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"3\">Power</td><td rowspan=\"2\">Accuracy (%)</td></tr><tr><td>0</td><td>1</td><td>2</td></tr><tr><td>Oracle on ImageNet3</td><td>=</td><td>=</td><td>=</td><td>53.3</td></tr><tr><td>Baseline (vanilla synthetic training)</td><td>0.4245</td><td>1.2500</td><td>1.6028</td><td>49.3</td></tr><tr><td>Weight l2 distance (Kirkpatrick et al., 2017)</td><td>0.4014</td><td>1.2296</td><td>1.5302</td><td>56.4</td></tr><tr><td>Synaptic Intelligence (Zenke et al., 2017)</td><td>0.3958</td><td>1.2261</td><td>1.5216</td><td>57.6</td></tr><tr><td>Feature l2 distance (Chen et al.,2018)</td><td>0.3337</td><td>1.1910</td><td>1.4449</td><td>57.1</td></tr><tr><td>ASG (Chen et al.,2020c)</td><td>0.3251</td><td>1.1840</td><td>1.4229</td><td>61.1</td></tr><tr><td>CSG (Ours)</td><td>0.3188</td><td>1.1806</td><td>1.4177</td><td>64.05</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1.2 ABLATION STUDY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We perform ablation studies (Table 2, 3, 4) on the VisDA-17 image classification benchmark (Peng et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "Augmentation. We study different magnitudes of RandAugment (Cubuk et al., 2020) in our scenario (Section 3.2), as summarized in Table 2. By tuning the global magnitude control factor $M$ , we observe that too strong augmentations deteriorate generalization (e.g. $M = 1 2 , 1 8 , 2 4 ,$ , while mild augmentation brings limited help $M = 3$ ). A moderate augmentation $M = 6$ ) can improve contrastive learning. ",
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+    {
+        "type": "text",
+        "text": "Multi-layer Contrastive Learning. Since features from the high-level layers are directly responsible for the downstream classification or other vision tasks, we suspect the last layer in the feature extractor $f _ { e }$ would be the most important. We conduct an ablation study on generalization performance with different layer combinations for multilayer contrastive learning (Section 3.3). From Table 3, one can see that applying $\\mathcal { L } _ { \\mathrm { N C E } }$ on layer 3 and 4 are most effective. Therefore, in our work we set $\\mathcal { G } = \\{ 3 , 4 \\}$ ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/306292eeaab547c16940b7f355b035efc74439639cc5b64c6ee28d12ff0af3f5.jpg",
+        "image_caption": [
+            "Figure 5: An illustration of model attention by GradCAM (Selvaraju et al., 2017) on the VisDA-17 validation set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "guided pooling (Section 3.4), A-pool can further improve the generalization performance, compared with the vanilla global average pooling (GAP). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8197bbcb3c6a77384d5b0c45144d37da9d590f3b133f1f1425a3b0a7f7313fd9.jpg",
+        "table_caption": [
+            "Table 2: Ablation with $M$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>M</td><td>Accuracy</td></tr><tr><td>O (no aug.)</td><td>60.86</td></tr><tr><td>3</td><td>61.36</td></tr><tr><td>6</td><td>62.88</td></tr><tr><td>12</td><td>62.61</td></tr><tr><td>18</td><td>62.00</td></tr></table>",
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+    },
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+        "type": "table",
+        "img_path": "images/5680f38dfd7d7725d74c905439e736b2dc81361bf76aedd548b26066dbd822df.jpg",
+        "table_caption": [
+            "Table 3: Ablation with $\\mathcal { G }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Layer Groups G</td><td>Accuracy (%)</td></tr><tr><td>4</td><td>62.88</td></tr><tr><td>3+4</td><td>63.77</td></tr><tr><td>2+3+4</td><td>62.66</td></tr><tr><td>1+2+3+4</td><td>62.30</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d243791b917a09c7d123c359fb9e5888bf4064fe00cfcc4a1f603e30d37bec66.jpg",
+        "table_caption": [
+            "Table 4: Ablation w./w.o. A-pool. GAP: global average pooling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Pooling</td><td>Accuracy (%)</td></tr><tr><td>GAP</td><td>63.77</td></tr><tr><td>A-pool</td><td>64.05</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.1.3 CSG BENEFITS VISUAL ATTENTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We further show the Grad-CAM3 attention on VisDA-17 validation set (Figure 5). We can see that our CSG framework also contributes to better visual attention on unseen real images. ",
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+    {
+        "type": "text",
+        "text": "4.2 SEMANTIC SEGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Dataset. GTA5 (Richter et al., 2016) is a vehicle-egocentric image dataset collected in a computer game with pixel-wise semantic labels. It contains 24,966 images with a resolution of $1 0 5 2 \\times 1 9 1 4$ There are 19 classes that are compatible with the Cityscapes dataset (Cordts et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "Cityscapes (Cordts et al., 2016) contains urban street images taken on a vehicle from some European cities. There are 5,000 images with pixel-wise annotations. The images have a resolution of $1 0 2 4 \\times 2 0 4 8$ and are labeled into 19 semantic categories. ",
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+    },
+    {
+        "type": "text",
+        "text": "Implementation. We study DeepLabv2 (Chen et al., 2017) with both ResNet-50 and ResNet-101 backbone. The backbones are pre-trained on ImageNet. We also use SGD optimizer, with learning rate as $1 \\times 1 0 ^ { - 3 }$ , weight decay as $5 \\times 1 0 ^ { - 4 }$ , and momentum are 0.9. Batch size is set to six. We crop the images into patches of $5 1 2 \\times 5 1 2$ and train the model with multi-scale augmentation $( 0 . 7 5 \\sim 1 . 2 5 )$ and horizontal flipping. The model is trained for 50 epochs, and $\\lambda$ for $\\mathcal { L } _ { \\mathrm { N C E } }$ is set as 75. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 MAIN RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also evaluate the generalization performance of our CSG on semantic segmentation. In particular, we treat the GTA5 training set as the synthetic source domain and train segmentation models on it. We then treat the Cityscapes validation sets as real target domains, where we directly evaluate the synthetically trained models. We can see that in Table 5, CSG achieves the best performance gain. IBN-Net Pan et al. (2018) improves domain generalization by carefully mix the instance and batch normalization in the backbone, while Yue et al. (2019) transfers the real image styles from ImageNet to synthetic images. However, Yue et al. (2019) requires ImageNet images during synthetic training, and also implicitly leverages ImageNet labels as auxiliary domains. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/013bde572ef0f9510f7d01c205256d56d155bd4436f4530365e402db51cc769b.jpg",
+        "table_caption": [
+            "Table 5: Comparison to previous domain generalization methods for segmentation $\\mathrm { G T A } 5 $ Cityscapes). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Backbone</td><td>mIoU %</td><td>mIoU↑%</td></tr><tr><td>No Adapt</td><td rowspan=\"6\">ResNet-50</td><td>22.17</td><td rowspan=\"2\">7.47</td></tr><tr><td>IBN-Net (Pan et al.,2018)</td><td>29.64</td></tr><tr><td>No Adapt</td><td>32.45</td><td rowspan=\"2\">4.97</td></tr><tr><td>Yue et al. (Yue et al., 2019)</td><td>37.42</td></tr><tr><td>No Adapt</td><td>25.88</td><td>3.77</td></tr><tr><td>ASG (Chen et al.,2020c) No Adapt</td><td></td><td>29.65</td></tr><tr><td rowspan=\"2\">CSG (ours)</td><td rowspan=\"6\">ResNet-101</td><td>25.88</td><td rowspan=\"2\">9.39</td></tr><tr><td>35.27</td></tr><tr><td>No Adapt</td><td>33.56</td><td>8.97</td></tr><tr><td>Yue et al. (Yue et al.,2019)</td><td>42.53</td><td rowspan=\"2\">3.16</td></tr><tr><td>No Adapt</td><td>29.63</td></tr><tr><td>ASG (Chen et al.,2020c)</td><td>32.79</td><td></td></tr><tr><td>No Adapt CSG (ours)</td><td></td><td>29.63 38.88</td><td>9.25</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 FEATURE DIVERSITY ON SEGMENTATION WITH BALANCED TRAINING SET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further conduct visualization and quantitative measures of feature diversity on the segmentation task. Similar to section 2, we randomly sample a subset of the GTA5 training set to match the size of the Cityscapes training set. We again have similar observations: models trained on real images have relatively diverse features, and synthetic training leads to collapsed features. Here we get lower $E _ { s }$ than classification since we follow the setting in Eq. 6 to study dense-level features. This leads to a larger total number of features on segmentation than classification. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ec0bda1f0bd50d717b806533b4dc8d661062dda200f5712726d9a8ebb9b64769.jpg",
+        "image_caption": [
+            "Figure 6: Feature diversity on Cityscapes test images in $\\mathbb { R } ^ { 2 }$ with Gaussian kernel density estimation (KDE). Darker areas have more concentrated features. $E _ { s }$ : hyperspherical energy of features, lower the more diverse. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.3 VISUAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "By visualizing the segmentation results (Figure 7), we can see that as our CSG framework achieves better mIoU on unseen real images from the Cityscapes validation set, the model produces segmentation with much higher visual quality. In contrast, the baseline model suffers from much more misclassification. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Domain generalization considers the problem of generalizing a model to the unseen target domain without leveraging any target domain images (Muandet et al., 2013; Gan et al., 2016). The core challenge is how to close the domain gap and align feature spaces from different domains, without even seeing the target domain’s data. Muandet et al. (2013) proposed to use MMD (Maximum Mean Discrepancy) to align the distributions from different source domains and train their network with adversarial learning. Li et al. (2017) built separate networks for each source domain and used shared parameters for testing. By using a meta-learning approach on split training sets, Li et al. (2018) further improved generalization performance. Instance Normalization and Batch Normalization are carefully integrated into the backbone network by Pan et al. (2018) to boost network generalization. Differently, Yue et al. (2019) proposed to transfer information from the real domain as image styles to synthetic images. Most recently, (Chen et al., 2020c) formulated domain generalization as a life-long learning problem (Li & Hoiem, 2017), and try to avoid the catastrophic forgetting about the ImageNet pre-trained weights and to retain real-domain knowledge during transfer learning. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e04c4a4fcb913cc3990207bbe93d170fbdf3913e9673ea453972ad5bed625a19.jpg",
+        "image_caption": [
+            "Figure 7: Generalization results on $\\mathrm { G T A } 5 $ Cityscapes. Rows correspond to sample images in Cityscapes validation set. From left to right, columns correspond to original images, ground truth, predication results of baseline (DeepLabv2-ResNet50 Chen et al. (2017)), and prediction by model trained with our CSG framework. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Contrastive learning. Noise contrastive estimation loss (Wu et al., 2018) recently becomes a predominant design choice for self-supervised contrastive representation learning (Hjelm et al., 2018; Oord et al., 2018; Henaff et al. ´ , 2019; Tian et al., 2019; He et al., 2020; Misra & Maaten, 2020; Chen et al., 2020a). Studies show that self-supervised models can serve as powerful initializations for downstream tasks, even outperforming supervised pre-training on several. Besides engineering improvements, key factors towards better contrastive learning include employing large numbers of negative examples and designing more semantically meaningful augmentations to create different views of images. This leads to both maximize the mutual information between two views of the same instance and pushing examples from different instances apart (Tian et al., 2020b). As also observed by Wang & Isola (2020), contrastive learning tends to align the features belonging to the same instance, while scattering the normalized learned features on a hypersphere. However, most work focus on the representation learning for a real-to-real transfer learning setting where the main focus is to improve the performance of the downstream tasks. While having connections to these methods, our work pursues a different task with different motivations despite the converging techniques. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the observation that models trained on synthetic images tend to generate collapsed feature representation, we make a hypothesis that the diversity of feature representation plays an important role in generalization performance. Taking this as an inductive bias, we propose a contrastive synthetic-to-real generalization framework that simultaneously regularizes the synthetically trained representations while promoting the diversity of the features to improve generalization. Experiments on VisDA-17 validate our hypothesis, showing that the diversity of features correlates with generalization performance across different models. Together with the multi-scale contrastive learning and attention-guided pooling strategy, the proposed framework outperforms previous state-of-the-arts on VisDA-17 with sizable gains, while giving competitive performance and the largest relative improvements on GTA5 Cityscapes without bells and whistles. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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parse/train/HJsjkMb0Z/HJsjkMb0Z.md

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+# $i$ -REVNET: DEEP INVERTIBLE NETWORKS
+
+Jorn-Henrik Jacobsen ¨ †‡, Arnold Smeulders †, Edouard Oyallon §   
+†University of Amsterdam   
+joern.jacobsen@bethgelab.org
+
+# ABSTRACT
+
+It is widely believed that the success of deep convolutional networks is based on progressively discarding uninformative variability about the input with respect to the problem at hand. This is supported empirically by the difficulty of recovering images from their hidden representations, in most commonly used network architectures. In this paper we show via a one-to-one mapping that this loss of information is not a necessary condition to learn representations that generalize well on complicated problems, such as ImageNet. Via a cascade of homeomorphic layers, we build the $i$ -RevNet, a network that can be fully inverted up to the final projection onto the classes, i.e. no information is discarded. Building an invertible architecture is difficult, for one, because the local inversion is ill-conditioned, we overcome this by providing an explicit inverse. An analysis of i-RevNets learned representations suggests an alternative explanation for the success of deep networks by a progressive contraction and linear separation with depth. To shed light on the nature of the model learned by the $i$ -RevNet we reconstruct linear interpolations between natural image representations.
+
+# 1 INTRODUCTION
+
+A CNN may be very effective in classifying images of all sorts (He et al., 2016; Krizhevsky et al., 2012), but the cascade of linear and nonlinear operators reveals little about the contribution of the internal representation to the classification. The learning process is characterized by a steady reduction of large amounts of uninformative variability in the images while simultaneously revealing the essence of the visual class. It is widely believed that this process is based on progressively discarding uninformative variability about the input with respect to the problem at hand (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016; Shwartz-Ziv & Tishby, 2017; Achille & Soatto, 2017). However, the extent to which information is discarded is lost somewhere in the intermediate nonlinear processing steps. In this paper, we aim to provide insight into the variability reduction process by proposing an invertible convolutional network, that does not discard any information about the input.
+
+The difficulty to recover images from their hidden representations is found in many commonly used network architectures (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016). This poses the question if a substantial loss of information is necessary for successful classification. We show information does not have to be discarded. By using homeomorphic layers, the invariance can be built only at the very last layer via a projection.
+
+In Shwartz-Ziv & Tishby (2017), minimal sufficient statistics are proposed as a candidate to explain the reduction of variability. Tishby & Zaslavsky (2015) introduces the information bottleneck principle which states that an optimal representation must reduce the mutual information between an input and its representation to reduce as much uninformative variability as possible. At the same time, the network should maximize the mutual information between the desired output and its representation to effectively preserve each class from collapsing onto other classes. The effect of the information bottleneck was demonstrated on small datasets in Shwartz-Ziv & Tishby (2017); Achille & Soatto (2017).
+
+However, in this work, we show it is not a necessary condition and we build a cascade of homeomorphic layers, which preserves the mutual information between input and hidden representation and shows that the loss of information can only occur at the final layer. This way we demonstrate that a loss of information can be avoided while maintaining discriminability, even for large-scale problems like ImageNet. One way to reduce variability is progressive contraction with respect to a meaningful $\ell ^ { 2 }$ metric in the intermediate representations.
+
+Several works (Oyallon, 2017; Zeiler & Fergus, 2014) observed a phenomenon of progressive separation and contraction in non-invertible networks on limited datasets. Those progressive improvements can be interpreted as the creation of progressively stronger invariants for classification. Ideally, the contraction should not be too brutal to avoid removing important information from the intermediate signal. This shows that a good trade-off between discriminability and invariance has to be progressively built. In this paper, we extend some findings of Zeiler & Fergus (2014); Oyallon (2017) to ImageNet (Russakovsky et al., 2015) and, most importantly, show that a loss of information is not necessary for observing a progressive contraction.
+
+The duality between invariance and separation of the classes is discussed in Mallat (2016). Here, intra-class variabilities are modeled as Lie groups that are processed by performing a parallel transport along those symmetries. Filters are adapted through learning to the specific bias of the dataset and avoid to contract along discriminative directions. However, using groups beyond the Euclidean case for image classification is hard. Mainly because groups associated with abstract variabilities are difficult to estimate due to their high-dimensional nature, as well as the appropriate degree of invariance required. An illustration of this framework on the Euclidean group is given by the scattering transform (Mallat, 2012), which builds invariance to small translations while being recoverable to a certain extent. In this work, we introduce a network that cannot discard any information except at the final classification stage, while we demonstrate numerically progressive contraction and separation of the signal classes.
+
+We introduce the $i$ -RevNet, an invertible deep network.1 $i$ -RevNets retain all information about the input signal in any of their intermediate representations up until the last layer. Our architecture builds upon the recently introduced RevNet (Gomez et al., 2017), where we replace the non-invertible components of the original RevNets by invertible ones. $i$ -RevNets achieve the same performance on Imagenet compared to similar non-invertible RevNet and ResNet architectures (Gomez et al., 2017; He et al., 2016).
+
+To shed light on the mechanism underlying the generalization-ability of the learned representation, we show that $i$ -RevNets progressively separate and contract signals with depth. Our results are evidence for an effective reduction of variability through a contraction with a recoverable input obtained from a series of one-to-one mappings.
+
+# 2 RELATED WORK
+
+Several recent works show that significant information about the input images is lost with depth in successful Imagenet classification CNNs (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016). To understand the loss of information, the references propose to invert the representations by means of learned or hand-engineered priors. The approximate inversions indicate increased geometric and photometric invariance with depth. Multiple other works report progressive properties of deep networks that may be linked to discarded information in the representations as well, such as linearization (Radford et al., 2015), linear separability (Zeiler & Fergus, 2014), contraction (Oyallon, 2017) and low-dimensional embeddings (Aubry & Russell, 2015). However, it is not clear from above observations if the loss of information is a necessity for the observed progressive phenomena. In this work, we show that progressive separation and contraction can be obtained while at the same time allowing an exact reconstruction of the signal.
+
+Multiple frameworks have been introduced that permit to learn invertible representations under certain conditions. Parseval networks (Cisse et al., 2017) have been introduced to increase the robustness of learned representations with respect to adversarial attacks. In this framework, the spectrum of convolutional operators is constrained to norm 1 during learning. The linear operator is thus injective.
+
+As a consequence, the input of Parseval networks can be recovered if but only if the built-in nonlinearities are invertible as well, which is typically not the case. Bruna et al. (2013) derive conditions under which pooling representations are, but our method directly overcomes this issue. The Scattering transform (Mallat, 2012) is an example of predefined deep representation, approximately invariant to translations, that can be reconstructed when the degree of invariance specified is small. Yet, it requires a gradient descent optimization and no guarantee of convergences are known. In summary, the references make clear that invertibility requires special care in designing the architecture or special care in designing the optimization procedure. In this paper, we introduce a network, that overcomes these issues and has an exact inverse by construction.
+
+Our main inspiration for this work is the recent reversible residual network (RevNet), introduced in Gomez et al. (2017). RevNets are in turn closely related to NICE and Real-NVP architectures (Dinh et al., 2016; 2014), which make use of constrained Jacobian determinants for generative modeling. All these architectures are similar to the lifting scheme (Sweldens, 1998) and Feistel cipher diagrams (Menezes et al., 1996), as we will show. RevNets illustrate how to build invertible ResNet-type blocks that avoid storing intermediate activations necessary for the backward pass. However, RevNets still employ multiple non-invertible operators like max-pooling and downsampling operators as part of the network. As such, RevNets are not invertible by construction. In this paper, we show how to build an invertible type of RevNet architecture that performs competitively with RevNets on Imagenet, which we call $i$ -RevNet for invertible RevNet.
+
+# 3 THE $i$ -REVNET
+
+This section introduces the general framework of the $i$ -RevNet architecture and explains how to explicitly build an inverse or a left-inverse to an $i$ -RevNet. Its practical implementation is discussed, and we demonstrate competitive numerical results.
+
+# 3.1 AN INVERTIBLE ARCHITECTURE
+
+![](images/5cd194f32b37730890221ba1d7413ff2ff91a6729965b3a9be0fa94b8d97be5b.jpg)  
+Figure 1: The main component of the $i$ -RevNet and its inverse. RevNet blocks are interleaved with convolutional bottlenecks ${ \mathcal { F } } _ { j }$ and reshuffling operations $S _ { j }$ to ensure invertibility of the architecture and computational efficiency. The input is processed through a splitting operator $\tilde { \cal S }$ , and output is merged through $\tilde { \mathcal { M } }$ . Observe that the inverse network is obtained with minimal adaptations.
+
+We describe $i$ -RevNets in their general setting. Their foundations are largely grounded in the recent RevNet architecture (Gomez et al., 2017). In an $i$ -RevNet, an initial input is split into two sublayers $( x _ { 0 } , \tilde { x } _ { 0 } )$ of equal size, thanks to a splitting operator $\tilde { S } x \triangleq ( x _ { 0 } , \tilde { x } _ { 0 } )$ , in this paper we choose to split the channel dimension as is done in RevNets. The operator $\tilde { \cal S }$ is linear, injective, reduces the spatial resolution of the coefficients and can potentially increase the layer size, as wider layers usually improve the classification performance (Zagoruyko $\&$ Komodakis, 2016). We can thus build a pseudo inverse ${ \tilde { S } } ^ { + }$ that will be used for the inversion. Recall that if $\tilde { \cal S }$ is invertible, then $\tilde { S } ^ { + } = \tilde { S } ^ { - 1 }$ .
+
+The number of coefficients of the next block is maintained, and at each depth $j$ , the representation $\Phi _ { j } x$ is again decoupled into two variables $\Phi _ { j } x \triangleq ( x _ { j } , \tilde { x } _ { j } )$ that play interlaced roles.
+
+The strategy implemented by an $i$ -RevNet consists in an alternation between additions, and nonlinear operators ${ \mathcal { F } } _ { j }$ , while progressively down-sampling the signal thanks to the operators $S _ { j }$ . Here, ${ \mathcal { F } } _ { j }$ consists of convolutions and non-linearity on $\tilde { x } _ { j }$ . The pair of the final layer is concatenated through a merging operator $\tilde { \mathcal { M } }$ . We will omit $\tilde { \mathcal { M } } , \tilde { \mathcal { M } } ^ { - 1 } , \tilde { \mathcal { S } } ^ { + }$ and $\tilde { \cal S }$ for the sake of simplicity, when not necessary. Figure 1 describes the blocks of an $i$ -RevNet. The design is similar to the Feistel cipher diagrams (Menezes et al., 1996) or a lifting scheme (Sweldens, 1998), which are invertible and efficient implementations of complex transforms like second generation wavelets.
+
+In this way, we avoid the non-invertible modules of a RevNet (e.g. max-pooling or strides) which are necessary to train them in a reasonable time and are designed to build invariance w.r.t. translation variability. Our method shows we can replace them by linear and invertible modules $S _ { j }$ , that can reduce the spatial resolution (we refer to it as a spatial down-sampling for the sake of simplicity) while maintaining the layer’s size by increasing the number of channels.
+
+We keep the computational cost manageable by tightly coupling downsampling and increase in width of the network. Reducing the spatial resolution can be undesirable, so $S _ { j }$ can potentially be the identity. We refer to such networks as $i$ -RevNets. This leads to the following equations:
+
+$$
+\left\{ \begin{array} { l l } { x _ { j + 1 } = S _ { j + 1 } \tilde { x } _ { j } } \\ { \tilde { x } _ { j + 1 } = x _ { j } + \mathcal { F } _ { j + 1 } \tilde { x } _ { j } } \end{array} \right. \iff \quad \left\{ \begin{array} { l l } { \tilde { x } _ { j } = S _ { j + 1 } ^ { - 1 } x _ { j + 1 } } \\ { x _ { j } = \tilde { x } _ { j + 1 } - \mathcal { F } _ { j + 1 } \tilde { x } _ { j } } \end{array} \right.
+$$
+
+Our downsampling layer can be written for $u$ the spatial variable and $\lambda$ the channel index:
+
+$$
+S _ { j } x ( u , \lambda ) = x ( \Psi ( u , \lambda ) )
+$$
+
+![](images/e760aaaf5fd2f19f2d2458df6118500adce3d86267e7acf371876f61f9b09078.jpg)  
+Figure 2: Illustration of the invertible down-sampling
+
+where $\Psi$ is some invertible mapping. In principle, any invertible downsampling operation like e.g. dilated convolutions (Yu & Koltun, 2015) can be considered here. We use the inverse of the operation described in Shi et al. (2016) as illustrated in Figure 2, since it preserves roughly the spatial ordering, and thus permits to avoid mixing different neighborhoods via the next convolution. $\tilde { \cal S }$ is similar, but also linearly increases the channel dimensionality, for example by concatenating 0.
+
+The final layer $\Phi x \triangleq \Phi _ { J } x = \left( x _ { J } , { \tilde { x } } _ { J } \right)$ is then averaged along the spatial dimension, followed by a ReLU non-linearity and finally a linear projection on the class probes, which are fed to a supervised training algorithm. From a given $i$ -RevNet, it is possible to define a left-inverse $\Phi ^ { + }$ , i.e. $\Phi ^ { + } \Phi x = x$ or even an inverse $\Phi ^ { - 1 }$ , i.e. $\Phi ^ { - 1 } \Phi x = \Phi ^ { - 1 } \Phi x = x$ if $\tilde { \cal S }$ is invertible. In these cases, the convolutional sections are as well some $i$ -RevNets. An $i$ -RevNet is the dual of its inverse, in the sense that it requires to replace $( S _ { j } , \mathcal { F } _ { j } )$ by $( S _ { j } ^ { - 1 } , - \mathcal { F } _ { j } )$ at each depth $j$ , and to apply ${ \tilde { S } } ^ { + }$ on the output. In consequence, its implementation is simple and specified by Equation (1). In Subsection 4.2, we discuss that the inverse of $\Phi$ does not suffer from significant round-off errors, while however being very sensitive to small variations of an input on a large subspace, as shown in Subsection 4.1.
+
+# 3.2 ARCHITECTURE, TRAINING AND PERFORMANCES
+
+In this subsection, we describe two models that we trained: an injective $i$ -RevNet (a) and a bijective $i$ -RevNet (b), with fewer parameters. The hyper-parameters were selected to be either close to the ResNet and RevNet baselines in terms of the number of layers (a) or parameters (b) while keeping performance competitive. For the same reasons as in Gomez et al. (2017), our scheme also allows avoiding storing any intermediate activations at training time, making memory consumption for very deep $i$ -RevNets not an issue in practice. We compare our implementation with a RevNet with 56 layers corresponding to $2 8 M$ parameters, as provided in the open source release of Gomez et al. (2017), and with a standard ResNet of 50 layers, with $2 6 M$ parameters (He et al., 2016).
+
+Each block ${ \mathcal { F } } _ { j }$ is a bottleneck block, which consists of a succession of 3 convolutional operators, each preceded by Batchnormalization (Ioffe & Szegedy, 2015) and ReLU non-linearity. The second layer has four times fewer channels than the other two, while their corresponding kernel sizes are respectively $1 \times 1 , 3 \times 3 , 1 \times 1$ .
+
+Table 1: Comparison of different architectures trained on ILSVRC-2012, in terms of classification accuracy and number of parameters   
+
+<table><tr><td>Architecture</td><td>Injective</td><td>Bijective</td><td>Top-1error</td><td>Parameters</td></tr><tr><td>ResNet</td><td></td><td></td><td>24.7</td><td>26M</td></tr><tr><td>RevNet</td><td>=</td><td></td><td>25.2</td><td>28M</td></tr><tr><td>i-RevNet (a)</td><td>yes</td><td>=</td><td>24.7</td><td>181M</td></tr><tr><td>i-RevNet (b)</td><td>yes</td><td>yes</td><td>26.7</td><td>29M</td></tr></table>
+
+The final representation is spatially averaged and projected onto the 1000 classes after a ReLU nonlinearity. We now discuss how we progressively decrease the spatial resolution, while increasing the number of channels per layer by use of the operators $S _ { j }$ .
+
+We first describe the model (a), that consists of 56 layers which have been optimized to match the performances of a RevNet or a ResNet with approximatively the same number of layers. In particular, we explain how we progressively decrease the spatial resolution, while increasing the number of channels per block by use of the operators $S _ { j }$ .
+
+The splitting operator $\tilde { \cal S }$ consists in a linear and injective embedding that downsamples by a factor $4 ^ { 2 }$ the spatial resolution by increasing the number of output channels from 48 to 96 by simply adding 0. The latter permits to increase the initial layer size, and consequently, the size of the next layers as performed in Gomez et al. (2017); it is thus not a bijective yet an injective $i$ -RevNet. At depth $j , { \mathcal { S } } _ { j }$ allows us to reduce the number of computations while maintaining good classification performance. It will correspond to a downsampling operator respectively at the depth $3 j = 1 5 , 2 7 , 4 $ 5 (3j as one block corresponds to three layers), similar to a normal RevNet. The spatial resolution of these layers is reduced by a factor $2 ^ { 2 }$ while increasing the number of channels by a factor of 4 respectively to 48, 192, 768 and 3072. Furthermore, it means that the corresponding spatial resolutions for an input of size $2 2 4 ^ { 2 }$ are respectively $1 1 2 ^ { 2 } , 5 6 ^ { 2 } , 2 8 ^ { 2 } , 1 4 ^ { 2 } , 7 ^ { 2 }$ . The total number of coefficients at each layer is then about $0 . 3 M$ . All the remaining blocks $S _ { j }$ are kept fix to the identity as explained in the section above.
+
+Architecture (b) is bijective, it consists of 300 layers (100 blocks), whose total numbers of parameters have been optimized to match those of a RevNet with 56 layers. Initially, the input is split via $\tilde { \cal S }$ , which corresponds to an invertible spatial downsampling of $2 ^ { \frac { 5 } { 2 } }$ that increases the number of channels from 3 to 12. It thus keeps the dimension constant and permits building a bijective $i$ -RevNet. Then, at depth $3 j = 3 , 2 1$ , 69, 285, the spatial resolution is reduced by $2 ^ { 2 }$ via $S _ { j }$ . Contrary to the architecture (a), the dimensionality of each layer is constantly equal to $3 \times 2 2 4 ^ { 2 }$ , until the final layer, with channel sizes of 24, 96, 384, 1536.
+
+For both networks, the training on Imagenet follows the same setup as Gomez et al. (2017). We train with SGD and momentum of 0.9. We regularized the model with a $\ell ^ { 2 }$ weight decay of $1 0 ^ { - 4 }$ and batch normalization. The dataset is processed for $6 0 0 \mathrm { k }$ iterations on a batch size of 256, distributed on 4GPUs. The initial learning rate is 0.1, dropped by a factor of ten every $1 6 0 \mathrm { k }$ iterations. The dataset was augmented according to Gomez et al. (2017). The images values are mapped to [0, 1] while following geometric transformations were applied: random scaling, random horizontal flipping, random cropping of size $2 2 4 ^ { 2 }$ , and finally color distortions. No other regularizations were incorporated into the classification pipeline. At test time, we rescale the image size to $2 5 6 ^ { 2 }$ and perform a center crop of size $2 2 4 ^ { 2 }$ .
+
+![](images/9b84f3e11c0bdeb5dd1a42d4cbb16eb51d1e10b7988637944ff79b527e7a6073.jpg)  
+Figure 3: Training loss of the $i$ -RevNet (b), compared to the ResNet, on ImageNet.
+
+We report the training loss (i.e. Cross entropy) curves in Figure 3 of our $i$ -RevNet (b) and the ResNet baseline, displayed is a moving average over 100 iterations. Observe that the decrease of both training-losses are very similar which indicates that the constraint of invertibility does not interfere negatively with the learning process. However, we observed one third longer wall-clock times for $i$ -RevNets compared to plain RevNets because the channel size becomes larger. The Table 1 reports the performances of our $i$ -RevNets, with comparable RevNet and ResNet. First, we compare the $i$ -RevNet (a) with the RevNet and ResNet. Indeed, those CNNs have the same number of layers, and the $i$ -RevNet (a) increases the channel width of the initial layer as done in Gomez et al. (2017). The drawback of this technique is that the kernel sizes will be larger for all subsequent layers.
+
+The $i$ -RevNet (a) has about 6 times more parameters than a RevNet and a ResNet but leads to a similar accuracy on the validation set of ImageNet. On the contrary, the $i$ -RevNet (b) is designed to have roughly the same number of parameters as the RevNet and ResNet, while being bijective. Its accuracy decreases by $1 . 5 \%$ absolute percent on ImageNet compared to the RevNet baseline, which is not surprising because the number of channels was not drastically increased in the earlier layers as done in the baselines (Gomez et al., 2017; Krizhevsky et al., 2012; He et al., 2016); we did not explore wide ranges of hyper-parameters, thus the gap between (a) and (b) can likely be reduced with additional engineering.
+
+# 4 ANALYSIS OF THE INVERSE
+
+We now analyze the representation $\Phi$ built by our bijective neural network $i$ -RevNet (b) and its inverse $\Phi ^ { - 1 }$ , as trained on ILSVRC-2012. We first explain why obtaining $\Phi ^ { - 1 }$ is challenging, even locally. We then discuss the reconstruction, while displaying in the image space linear interpolations between representations.
+
+# 4.1 AN ILL-CONDITIONED INVERSION
+
+In the previous section, we have described the $i$ -RevNet architecture, that permits defining a deep network with an explicit inverse. We explain now why this is normally difficult, by studying its local inversion. We study the local stability of a network $\Phi$ and its inverse $\Phi ^ { - 1 }$ w.r.t. to its input, which means that we will quantify locally the variations of the network and its inverse w.r.t. to small variations of an input. As $\Phi$ is differentiable (and its inverse as well), an equivalent way to perform this study is to analyze the singular values of the differential $\partial \Phi$ at some point, as for $( a , b )$ close the following holds:
+
+$$
+\Phi \boldsymbol { a } \approx \Phi \boldsymbol { b } + \partial \Phi _ { b } ( \boldsymbol { a } - \boldsymbol { b } ) .
+$$
+
+![](images/ce28b51d37c721b4a781c7d9d9b2bcd0ff508b71714b9075ab8d17c86d3d3d9a.jpg)  
+Figure 4: Normalized sorted singular values of $\partial \Phi _ { x }$ .
+
+Ideally, a well-conditioned operator has all its singular values constant equal to 1, for instance as achieved by the isometric operators of Cisse et al. (2017).
+
+In our numerical application to an image $x$ , $\partial \Phi _ { x }$ corresponds to a very large matrix (square of the number of coefficients of the image at least) whose computations are expensive. Figure 4 corresponds to the singular values of the differential (i.e. the square roots of the eigen values of $\partial \Phi ^ { * } \partial \Phi$ ), in decreasing order, for a given natural image from ImageNet. The example we plot is typical of the behavior of $\partial \Phi$ . Observe there is a fast decay: numerically, the first $\mathrm { 1 0 ^ { 3 } }$ and $1 0 ^ { \bar { 4 } }$ singular values are responsible respectively for $8 0 \%$ and $9 7 \%$ of the cumulated energy (i.e. sum of squared singular values). This indicates $\Phi$ linearizes the space locally in a considerably smaller space in comparison to the original input dimension. However, the dimensionality is still quite large (i.e. $> 1 0$ ) and thus we can not infer that $\Phi$ lays locally in a low-dimensional manifold. It also proves that inversing $\Phi$ is difficult and is an ill-conditioned problem. Thus obtaining implicitly this inverse would be a challenging task that we avoided, thanks to the formal reconstruction algorithm provided by Subsection 3.1.
+
+![](images/a6fe6b5bd4ff7c42fa5cbd145f44da0d3e48b046022b4d75069d5e7eefe3144c.jpg)  
+Figure 5: This graphic displays several reconstructed sequences $\{ x ^ { t } \} _ { t }$ . The left image corresponds to ${ \bar { \mathbf { \Gamma } } } _ { x } 0$ and the right image to $x ^ { 1 }$ .
+
+# 4.2 LINEAR INTERPOLATION AND RECONSTRUCTION
+
+Visualizing or understanding the important directions in the representation of inner layers of a CNN, and in particular, the final layer is complex because typically the cascade is either not invertible or unstable. One approach to reconstruct from an output layer consists in finding the input image that matches the activation through via gradient descent. However, this technique leads only to a partial or informal reconstruction (Mahendran & Vedaldi, 2015).
+
+Another method consists in embedding the representation in a lower dimensional space and comparing the common attributes of nearest neighbors (Szegedy et al., 2013). It is also possible to train a CNN to reconstruct the representation (Dosovitskiy & Brox, 2016). Yet these methods require a priori knowledge in order to find the appropriate embeddings or training sets. We now discuss the improvements achieved by the $i$ -RevNet.
+
+Our main claim is that while the local inversion is ill-conditioned, the inverse $\Phi ^ { - 1 }$ computations do not involve significant round-off errors. The forward pass of the network does not seem to suffer from significant instabilities, thus it seems coherent to assume that this will hold for $\Phi ^ { - 1 }$ as well. For example, adding constraints beyond vanishing moments in the case of a Lifting scheme is difficult (Sweldens, 1998; Mallat, 1999), and this is a weakness of this method. We validate our claim by computing the empirical relative error on several subsets $\mathcal { X }$ of data:
+
+$$
+\epsilon ( \mathcal { X } ) = \frac { 1 } { | \mathcal { X } | } \sum _ { \boldsymbol { x } \in \mathcal { X } } \frac { \| \boldsymbol { x } - \Phi ^ { - 1 } \Phi \boldsymbol { x } \| } { \| \boldsymbol { x } \| }
+$$
+
+We evaluate this measure on a subset $\mathcal { X } _ { 1 }$ of $| \mathcal { X } _ { 1 } | = 1 0 ^ { 4 }$ independent uniform noises and on the validation set $\mathcal { X } _ { 2 }$ of ImageNet. We report $\epsilon ( \mathcal { X } _ { 1 } ) = 5 \times 1 0 ^ { - 6 }$ and $\epsilon ( \mathcal { X } _ { 2 } ) = 3 \times 1 0 ^ { - 6 }$ respectively, which are close to the machine error and indicates that the inversion does not suffer from significant round-off errors.
+
+Given a pair of images $\{ x ^ { 0 } , x ^ { 1 } \}$ , we propose to study linear interpolations between the pair of representations $\{ \Phi x ^ { \bar { 0 } } , \Phi x ^ { \mathrm { { 1 } } } \}$ , in the feature domain. Those interpolations correspond to existing images as $\Phi ^ { - 1 }$ is an exact inverse. We reconstruct a convex path between two input points; it means that if:
+
+$$
+\phi ^ { t } = t \Phi x ^ { 0 } + ( 1 - t ) \Phi x ^ { 1 } ,
+$$
+
+then: $x ^ { t } = \Phi ^ { - 1 } \phi ^ { t }$ is a signal that corresponds to an image.
+
+![](images/293bac6aae019b36fb6785db214dbbee593d7a2ee2f0f04de109fb47c7cb0a9a.jpg)  
+Figure 6: Accuracy at depth $j$ for a linear SVM and a 1-nearest neighbor classifier applied to the spatially averaged $\Phi _ { j }$ .
+
+We discretized $[ 0 , 1 ]$ into $\{ t _ { 1 } , . . . , t _ { k } \}$ , adapt the step size manually and reconstruct the sequence of $\{ x ^ { t _ { 1 } } , . . . , x ^ { t _ { k } } \}$ . Results are displayed in the Figure 5. We selected images from the basel face dataset (Paysan et al., 2009), describable texture dataset (Cimpoi et al., 2014) and imagenet.
+
+We now interpret the results. First, observe that a linear interpolation in the feature space is not a linear interpolation in the image space and that intermediary images are noisy, even for small deformations, yet they mostly remain recognizable. However, some geometric transformations such as a 3D-rotation seem to have been linearized, as suggested in Aubry & Russell (2015). In the next section, we thus investigate how the linear separation progresses with depth.
+
+# 5 A CONTRACTION
+
+In this section, we study again the bijective $i$ -RevNet. We first show that a localized or linear classifier progressively improves with depth. Then, we describe the linear subspace spanned by $\Phi$ , namely the feature space, showing that the classification can be performed on a much smaller subspace, which can be built via a PCA.
+
+# 5.1 PROGRESSIVE LINEAR SEPARATION AND CONTRACTION
+
+We show that both a ResNet and an $i$ -RevNet build a progressively more linearly separable and contracted representation as measured in Oyallon (2017). Observe this property holds for the $i$ - RevNet despite the fact that it can not discard any information.
+
+We investigate these properties in each block, with the following experimental protocol. To reduce the computational burden we used a subset of 100 randomly selected imagenet classes, that consist of $N = 1 2 0 k$ images, and keep the same subset during all our following experiments. At each depth $j$ , we extract the features $\{ \Phi _ { j } x ^ { n } \} _ { n \leq N }$ of the training set, we average them along the spatial variable and standardize them in order to avoid any ill-conditioning effects. We used both a nearest neighbor classifier and a linear SVM. The former is a localized classifier that indicates that the $\ell ^ { 2 }$ metric is progressively more important for classification, while a linear SVM measures the linear separation of the different classes. The parameters of the linear SVM are cross-validated on a small subset of the training set, prior to training on the 100 classes. We evaluate both classifiers for each model on the validation set of ImageNet and report the Top-1 accuracy in Figure 6.
+
+We observe that both classifiers progressively improve similarly with depth for each model, the linear SVM performing slightly better than the nearest neighbor classifier because it is the more robust and discriminative classifier of the two. In the case of the $i$ -RevNet, the classification performed by the CNN leads to $7 7 \%$ , and the linear SVM performs slightly better because we did not fine-tune the model to 100 classes. Observe that there is a more intense jump of performance on the 3 last layers, which seems to indicate that the former layers have prepared the representation to be more contracted and linearly separated for the final layers.
+
+The results suggest a low-dimensional embedding of the data, but this is difficult to validate as estimating local dimensionality in high dimensions is an open problem. However, in the next section, we try to compute the dimension of the discriminative part of the representation built by an $i$ -RevNet.
+
+# 5.2 DIMENSIONALITY ANALYSIS OF THE FEATURE SPACE
+
+In this section, we investigate if we can refine the dimensionality of informative variabilities in the final layer of an $i$ -RevNet. Indeed, the cascade of convolutional operators has been trained on the training set to separate the 1000 different classes while being a homeomorphism on its feature space. Thus, the dimensionality of the feature space is potentially large.
+
+As shown in the previous subsection, the final layer is progressively prepared to be projected on the final probes corresponding to the classes. This indicates that the non-informative variabilities for classification can be removed via a linear projection on the final layer $\Phi$ , which lie in a space of dimension 1000, at most. However, this projection has been built via supervision, which can still retain directions that have been contracted and thus will not be selected by an algorithm such as PCA. We show in fact a PCA retains the necessary information for classification in a small subspace.
+
+To do so, we build the linear projectors $\pi _ { d }$ on the subspace of the $d$ first principal components, and we propose to measure the classification power of the projected representation with a supervised classifier, e.g. nearest neighbor or a linear SVM, on the previous 100 class task. Again, the feature representation $\{ \Phi x ^ { n } \} _ { n \leq N }$ are spatially averaged to remove the translation variability, and standardized on the training set. We apply both classifiers, and we report the classification accuracy of $\{ \pi _ { d } \Phi x ^ { n } \} _ { n \leq N }$ w.r.t. to $d$ on the Figure 7. A linear projection removes some information that can not be recovered by a linear classifier, therefore we observe that the classification accuracy only decreases significantly for $d \leq 2 0 0$ . This shows that the signal indeed lies in a subspace much lower dimensional than the original feature dimensions
+
+![](images/93bdd4b0bf41956f8e5f34e143db6c00cc121e2956561dec013eee6385bf9daf.jpg)  
+Figure 7: Accuracy of a linear SVM and nearest neighbor against the number of principal components retained.
+
+that can be extracted simply with a PCA that only considers directions of largest variances, illustrating a successful contraction of the representation.
+
+# 6 CONCLUSION
+
+Invertible representations and their relationship to loss of information are on the agenda of deep learning for some time. Understanding how transformations in feature space are related to the corresponding input is an important step towards interpretable deep networks, invertible deep networks may play an important role in such analysis since, for example, one could potentially back-track a property from the feature space to the input space. To the best of our knowledge, this work provides the first empirical evidence that learning invertible representations that do not discard any information about their input on large-scale supervised problems is possible.
+
+To achieve this we introduce the $i$ -RevNet class of CNN which is fully invertible and permits to exactly recover the input from its last convolutional layer. $i$ -RevNets achieve the same classification accuracy in the classification of complex datasets as illustrated on ILSVRC-2012, when compared to the RevNet (Gomez et al., 2017) and ResNet (He et al., 2016) architectures with a similar number of layers. Furthermore, the inverse network is obtained for free when training an $i$ -RevNet, requiring only minimal adaption to recover inputs from the hidden representations.
+
+The absence of loss of information is surprising, given the wide believe, that discarding information is essential for learning representations that generalize well to unseen data. We show that this is not the case and propose to explain the generalization property with empirical evidence of progressive separation and contraction with depth, on ImageNet.
+
+# ACKNOWLEDGEMENTS
+
+Jorn-Henrik Jacobsen was partially funded by the STW perspective program ImaGene. Edouard ¨ Oyallon was partially funded by the ERC grant InvariantClass 320959, via a grant for PhD Students of the Conseil regional dIle-de-France (RDM-IdF), and a postdoctoral grant from the from DPEI ´ of Inria (AAR 2017POD057) for the collaboration with CWI. We thank Berkay Kicanaoglu for the Basel Face data, Mathieu Andreux, Eugene Belilovsky, Amal Rannen, Patrick Putzky and Kyriacos Shiarlis for feedback on drafts of the paper.
+
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+        "text": "$i$ -REVNET: DEEP INVERTIBLE NETWORKS ",
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+        "type": "text",
+        "text": "Jorn-Henrik Jacobsen ¨ †‡, Arnold Smeulders †, Edouard Oyallon §   \n†University of Amsterdam   \njoern.jacobsen@bethgelab.org ",
+        "bbox": [
+            184,
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            224,
+            544,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
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+        "text": "It is widely believed that the success of deep convolutional networks is based on progressively discarding uninformative variability about the input with respect to the problem at hand. This is supported empirically by the difficulty of recovering images from their hidden representations, in most commonly used network architectures. In this paper we show via a one-to-one mapping that this loss of information is not a necessary condition to learn representations that generalize well on complicated problems, such as ImageNet. Via a cascade of homeomorphic layers, we build the $i$ -RevNet, a network that can be fully inverted up to the final projection onto the classes, i.e. no information is discarded. Building an invertible architecture is difficult, for one, because the local inversion is ill-conditioned, we overcome this by providing an explicit inverse. An analysis of i-RevNets learned representations suggests an alternative explanation for the success of deep networks by a progressive contraction and linear separation with depth. To shed light on the nature of the model learned by the $i$ -RevNet we reconstruct linear interpolations between natural image representations. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "A CNN may be very effective in classifying images of all sorts (He et al., 2016; Krizhevsky et al., 2012), but the cascade of linear and nonlinear operators reveals little about the contribution of the internal representation to the classification. The learning process is characterized by a steady reduction of large amounts of uninformative variability in the images while simultaneously revealing the essence of the visual class. It is widely believed that this process is based on progressively discarding uninformative variability about the input with respect to the problem at hand (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016; Shwartz-Ziv & Tishby, 2017; Achille & Soatto, 2017). However, the extent to which information is discarded is lost somewhere in the intermediate nonlinear processing steps. In this paper, we aim to provide insight into the variability reduction process by proposing an invertible convolutional network, that does not discard any information about the input. ",
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+    {
+        "type": "text",
+        "text": "The difficulty to recover images from their hidden representations is found in many commonly used network architectures (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016). This poses the question if a substantial loss of information is necessary for successful classification. We show information does not have to be discarded. By using homeomorphic layers, the invariance can be built only at the very last layer via a projection. ",
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+    {
+        "type": "text",
+        "text": "In Shwartz-Ziv & Tishby (2017), minimal sufficient statistics are proposed as a candidate to explain the reduction of variability. Tishby & Zaslavsky (2015) introduces the information bottleneck principle which states that an optimal representation must reduce the mutual information between an input and its representation to reduce as much uninformative variability as possible. At the same time, the network should maximize the mutual information between the desired output and its representation to effectively preserve each class from collapsing onto other classes. The effect of the information bottleneck was demonstrated on small datasets in Shwartz-Ziv & Tishby (2017); Achille & Soatto (2017). ",
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+        "text": "However, in this work, we show it is not a necessary condition and we build a cascade of homeomorphic layers, which preserves the mutual information between input and hidden representation and shows that the loss of information can only occur at the final layer. This way we demonstrate that a loss of information can be avoided while maintaining discriminability, even for large-scale problems like ImageNet. One way to reduce variability is progressive contraction with respect to a meaningful $\\ell ^ { 2 }$ metric in the intermediate representations. ",
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+    {
+        "type": "text",
+        "text": "Several works (Oyallon, 2017; Zeiler & Fergus, 2014) observed a phenomenon of progressive separation and contraction in non-invertible networks on limited datasets. Those progressive improvements can be interpreted as the creation of progressively stronger invariants for classification. Ideally, the contraction should not be too brutal to avoid removing important information from the intermediate signal. This shows that a good trade-off between discriminability and invariance has to be progressively built. In this paper, we extend some findings of Zeiler & Fergus (2014); Oyallon (2017) to ImageNet (Russakovsky et al., 2015) and, most importantly, show that a loss of information is not necessary for observing a progressive contraction. ",
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+    {
+        "type": "text",
+        "text": "The duality between invariance and separation of the classes is discussed in Mallat (2016). Here, intra-class variabilities are modeled as Lie groups that are processed by performing a parallel transport along those symmetries. Filters are adapted through learning to the specific bias of the dataset and avoid to contract along discriminative directions. However, using groups beyond the Euclidean case for image classification is hard. Mainly because groups associated with abstract variabilities are difficult to estimate due to their high-dimensional nature, as well as the appropriate degree of invariance required. An illustration of this framework on the Euclidean group is given by the scattering transform (Mallat, 2012), which builds invariance to small translations while being recoverable to a certain extent. In this work, we introduce a network that cannot discard any information except at the final classification stage, while we demonstrate numerically progressive contraction and separation of the signal classes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We introduce the $i$ -RevNet, an invertible deep network.1 $i$ -RevNets retain all information about the input signal in any of their intermediate representations up until the last layer. Our architecture builds upon the recently introduced RevNet (Gomez et al., 2017), where we replace the non-invertible components of the original RevNets by invertible ones. $i$ -RevNets achieve the same performance on Imagenet compared to similar non-invertible RevNet and ResNet architectures (Gomez et al., 2017; He et al., 2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To shed light on the mechanism underlying the generalization-ability of the learned representation, we show that $i$ -RevNets progressively separate and contract signals with depth. Our results are evidence for an effective reduction of variability through a contraction with a recoverable input obtained from a series of one-to-one mappings. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Several recent works show that significant information about the input images is lost with depth in successful Imagenet classification CNNs (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016). To understand the loss of information, the references propose to invert the representations by means of learned or hand-engineered priors. The approximate inversions indicate increased geometric and photometric invariance with depth. Multiple other works report progressive properties of deep networks that may be linked to discarded information in the representations as well, such as linearization (Radford et al., 2015), linear separability (Zeiler & Fergus, 2014), contraction (Oyallon, 2017) and low-dimensional embeddings (Aubry & Russell, 2015). However, it is not clear from above observations if the loss of information is a necessity for the observed progressive phenomena. In this work, we show that progressive separation and contraction can be obtained while at the same time allowing an exact reconstruction of the signal. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Multiple frameworks have been introduced that permit to learn invertible representations under certain conditions. Parseval networks (Cisse et al., 2017) have been introduced to increase the robustness of learned representations with respect to adversarial attacks. In this framework, the spectrum of convolutional operators is constrained to norm 1 during learning. The linear operator is thus injective. ",
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+    {
+        "type": "text",
+        "text": "As a consequence, the input of Parseval networks can be recovered if but only if the built-in nonlinearities are invertible as well, which is typically not the case. Bruna et al. (2013) derive conditions under which pooling representations are, but our method directly overcomes this issue. The Scattering transform (Mallat, 2012) is an example of predefined deep representation, approximately invariant to translations, that can be reconstructed when the degree of invariance specified is small. Yet, it requires a gradient descent optimization and no guarantee of convergences are known. In summary, the references make clear that invertibility requires special care in designing the architecture or special care in designing the optimization procedure. In this paper, we introduce a network, that overcomes these issues and has an exact inverse by construction. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our main inspiration for this work is the recent reversible residual network (RevNet), introduced in Gomez et al. (2017). RevNets are in turn closely related to NICE and Real-NVP architectures (Dinh et al., 2016; 2014), which make use of constrained Jacobian determinants for generative modeling. All these architectures are similar to the lifting scheme (Sweldens, 1998) and Feistel cipher diagrams (Menezes et al., 1996), as we will show. RevNets illustrate how to build invertible ResNet-type blocks that avoid storing intermediate activations necessary for the backward pass. However, RevNets still employ multiple non-invertible operators like max-pooling and downsampling operators as part of the network. As such, RevNets are not invertible by construction. In this paper, we show how to build an invertible type of RevNet architecture that performs competitively with RevNets on Imagenet, which we call $i$ -RevNet for invertible RevNet. ",
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+    {
+        "type": "text",
+        "text": "3 THE $i$ -REVNET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This section introduces the general framework of the $i$ -RevNet architecture and explains how to explicitly build an inverse or a left-inverse to an $i$ -RevNet. Its practical implementation is discussed, and we demonstrate competitive numerical results. ",
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+        "type": "text",
+        "text": "3.1 AN INVERTIBLE ARCHITECTURE ",
+        "text_level": 1,
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+        "img_path": "images/5cd194f32b37730890221ba1d7413ff2ff91a6729965b3a9be0fa94b8d97be5b.jpg",
+        "image_caption": [
+            "Figure 1: The main component of the $i$ -RevNet and its inverse. RevNet blocks are interleaved with convolutional bottlenecks ${ \\mathcal { F } } _ { j }$ and reshuffling operations $S _ { j }$ to ensure invertibility of the architecture and computational efficiency. The input is processed through a splitting operator $\\tilde { \\cal S }$ , and output is merged through $\\tilde { \\mathcal { M } }$ . Observe that the inverse network is obtained with minimal adaptations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We describe $i$ -RevNets in their general setting. Their foundations are largely grounded in the recent RevNet architecture (Gomez et al., 2017). In an $i$ -RevNet, an initial input is split into two sublayers $( x _ { 0 } , \\tilde { x } _ { 0 } )$ of equal size, thanks to a splitting operator $\\tilde { S } x \\triangleq ( x _ { 0 } , \\tilde { x } _ { 0 } )$ , in this paper we choose to split the channel dimension as is done in RevNets. The operator $\\tilde { \\cal S }$ is linear, injective, reduces the spatial resolution of the coefficients and can potentially increase the layer size, as wider layers usually improve the classification performance (Zagoruyko $\\&$ Komodakis, 2016). We can thus build a pseudo inverse ${ \\tilde { S } } ^ { + }$ that will be used for the inversion. Recall that if $\\tilde { \\cal S }$ is invertible, then $\\tilde { S } ^ { + } = \\tilde { S } ^ { - 1 }$ . ",
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+    {
+        "type": "text",
+        "text": "The number of coefficients of the next block is maintained, and at each depth $j$ , the representation $\\Phi _ { j } x$ is again decoupled into two variables $\\Phi _ { j } x \\triangleq ( x _ { j } , \\tilde { x } _ { j } )$ that play interlaced roles. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The strategy implemented by an $i$ -RevNet consists in an alternation between additions, and nonlinear operators ${ \\mathcal { F } } _ { j }$ , while progressively down-sampling the signal thanks to the operators $S _ { j }$ . Here, ${ \\mathcal { F } } _ { j }$ consists of convolutions and non-linearity on $\\tilde { x } _ { j }$ . The pair of the final layer is concatenated through a merging operator $\\tilde { \\mathcal { M } }$ . We will omit $\\tilde { \\mathcal { M } } , \\tilde { \\mathcal { M } } ^ { - 1 } , \\tilde { \\mathcal { S } } ^ { + }$ and $\\tilde { \\cal S }$ for the sake of simplicity, when not necessary. Figure 1 describes the blocks of an $i$ -RevNet. The design is similar to the Feistel cipher diagrams (Menezes et al., 1996) or a lifting scheme (Sweldens, 1998), which are invertible and efficient implementations of complex transforms like second generation wavelets. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In this way, we avoid the non-invertible modules of a RevNet (e.g. max-pooling or strides) which are necessary to train them in a reasonable time and are designed to build invariance w.r.t. translation variability. Our method shows we can replace them by linear and invertible modules $S _ { j }$ , that can reduce the spatial resolution (we refer to it as a spatial down-sampling for the sake of simplicity) while maintaining the layer’s size by increasing the number of channels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We keep the computational cost manageable by tightly coupling downsampling and increase in width of the network. Reducing the spatial resolution can be undesirable, so $S _ { j }$ can potentially be the identity. We refer to such networks as $i$ -RevNets. This leads to the following equations: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/da0995a53b3995df67a5ec1405289067fd48910cb6ad5f7bd3f7226e7dcd5d2a.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { x _ { j + 1 } = S _ { j + 1 } \\tilde { x } _ { j } } \\\\ { \\tilde { x } _ { j + 1 } = x _ { j } + \\mathcal { F } _ { j + 1 } \\tilde { x } _ { j } } \\end{array} \\right. \\iff \\quad \\left\\{ \\begin{array} { l l } { \\tilde { x } _ { j } = S _ { j + 1 } ^ { - 1 } x _ { j + 1 } } \\\\ { x _ { j } = \\tilde { x } _ { j + 1 } - \\mathcal { F } _ { j + 1 } \\tilde { x } _ { j } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Our downsampling layer can be written for $u$ the spatial variable and $\\lambda$ the channel index: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/82df32856045c35be0eba3c2ee24262b514350416991477d0f1a642428812415.jpg",
+        "text": "$$\nS _ { j } x ( u , \\lambda ) = x ( \\Psi ( u , \\lambda ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/e760aaaf5fd2f19f2d2458df6118500adce3d86267e7acf371876f61f9b09078.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the invertible down-sampling "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\Psi$ is some invertible mapping. In principle, any invertible downsampling operation like e.g. dilated convolutions (Yu & Koltun, 2015) can be considered here. We use the inverse of the operation described in Shi et al. (2016) as illustrated in Figure 2, since it preserves roughly the spatial ordering, and thus permits to avoid mixing different neighborhoods via the next convolution. $\\tilde { \\cal S }$ is similar, but also linearly increases the channel dimensionality, for example by concatenating 0. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The final layer $\\Phi x \\triangleq \\Phi _ { J } x = \\left( x _ { J } , { \\tilde { x } } _ { J } \\right)$ is then averaged along the spatial dimension, followed by a ReLU non-linearity and finally a linear projection on the class probes, which are fed to a supervised training algorithm. From a given $i$ -RevNet, it is possible to define a left-inverse $\\Phi ^ { + }$ , i.e. $\\Phi ^ { + } \\Phi x = x$ or even an inverse $\\Phi ^ { - 1 }$ , i.e. $\\Phi ^ { - 1 } \\Phi x = \\Phi ^ { - 1 } \\Phi x = x$ if $\\tilde { \\cal S }$ is invertible. In these cases, the convolutional sections are as well some $i$ -RevNets. An $i$ -RevNet is the dual of its inverse, in the sense that it requires to replace $( S _ { j } , \\mathcal { F } _ { j } )$ by $( S _ { j } ^ { - 1 } , - \\mathcal { F } _ { j } )$ at each depth $j$ , and to apply ${ \\tilde { S } } ^ { + }$ on the output. In consequence, its implementation is simple and specified by Equation (1). In Subsection 4.2, we discuss that the inverse of $\\Phi$ does not suffer from significant round-off errors, while however being very sensitive to small variations of an input on a large subspace, as shown in Subsection 4.1. ",
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+    {
+        "type": "text",
+        "text": "3.2 ARCHITECTURE, TRAINING AND PERFORMANCES ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this subsection, we describe two models that we trained: an injective $i$ -RevNet (a) and a bijective $i$ -RevNet (b), with fewer parameters. The hyper-parameters were selected to be either close to the ResNet and RevNet baselines in terms of the number of layers (a) or parameters (b) while keeping performance competitive. For the same reasons as in Gomez et al. (2017), our scheme also allows avoiding storing any intermediate activations at training time, making memory consumption for very deep $i$ -RevNets not an issue in practice. We compare our implementation with a RevNet with 56 layers corresponding to $2 8 M$ parameters, as provided in the open source release of Gomez et al. (2017), and with a standard ResNet of 50 layers, with $2 6 M$ parameters (He et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Each block ${ \\mathcal { F } } _ { j }$ is a bottleneck block, which consists of a succession of 3 convolutional operators, each preceded by Batchnormalization (Ioffe & Szegedy, 2015) and ReLU non-linearity. The second layer has four times fewer channels than the other two, while their corresponding kernel sizes are respectively $1 \\times 1 , 3 \\times 3 , 1 \\times 1$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/251ee7588dce774e1181820f0000e4bc7c393edb175e76ba0ff406a4c8e79c70.jpg",
+        "table_caption": [
+            "Table 1: Comparison of different architectures trained on ILSVRC-2012, in terms of classification accuracy and number of parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>Injective</td><td>Bijective</td><td>Top-1error</td><td>Parameters</td></tr><tr><td>ResNet</td><td></td><td></td><td>24.7</td><td>26M</td></tr><tr><td>RevNet</td><td>=</td><td></td><td>25.2</td><td>28M</td></tr><tr><td>i-RevNet (a)</td><td>yes</td><td>=</td><td>24.7</td><td>181M</td></tr><tr><td>i-RevNet (b)</td><td>yes</td><td>yes</td><td>26.7</td><td>29M</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The final representation is spatially averaged and projected onto the 1000 classes after a ReLU nonlinearity. We now discuss how we progressively decrease the spatial resolution, while increasing the number of channels per layer by use of the operators $S _ { j }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first describe the model (a), that consists of 56 layers which have been optimized to match the performances of a RevNet or a ResNet with approximatively the same number of layers. In particular, we explain how we progressively decrease the spatial resolution, while increasing the number of channels per block by use of the operators $S _ { j }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The splitting operator $\\tilde { \\cal S }$ consists in a linear and injective embedding that downsamples by a factor $4 ^ { 2 }$ the spatial resolution by increasing the number of output channels from 48 to 96 by simply adding 0. The latter permits to increase the initial layer size, and consequently, the size of the next layers as performed in Gomez et al. (2017); it is thus not a bijective yet an injective $i$ -RevNet. At depth $j , { \\mathcal { S } } _ { j }$ allows us to reduce the number of computations while maintaining good classification performance. It will correspond to a downsampling operator respectively at the depth $3 j = 1 5 , 2 7 , 4 $ 5 (3j as one block corresponds to three layers), similar to a normal RevNet. The spatial resolution of these layers is reduced by a factor $2 ^ { 2 }$ while increasing the number of channels by a factor of 4 respectively to 48, 192, 768 and 3072. Furthermore, it means that the corresponding spatial resolutions for an input of size $2 2 4 ^ { 2 }$ are respectively $1 1 2 ^ { 2 } , 5 6 ^ { 2 } , 2 8 ^ { 2 } , 1 4 ^ { 2 } , 7 ^ { 2 }$ . The total number of coefficients at each layer is then about $0 . 3 M$ . All the remaining blocks $S _ { j }$ are kept fix to the identity as explained in the section above. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Architecture (b) is bijective, it consists of 300 layers (100 blocks), whose total numbers of parameters have been optimized to match those of a RevNet with 56 layers. Initially, the input is split via $\\tilde { \\cal S }$ , which corresponds to an invertible spatial downsampling of $2 ^ { \\frac { 5 } { 2 } }$ that increases the number of channels from 3 to 12. It thus keeps the dimension constant and permits building a bijective $i$ -RevNet. Then, at depth $3 j = 3 , 2 1$ , 69, 285, the spatial resolution is reduced by $2 ^ { 2 }$ via $S _ { j }$ . Contrary to the architecture (a), the dimensionality of each layer is constantly equal to $3 \\times 2 2 4 ^ { 2 }$ , until the final layer, with channel sizes of 24, 96, 384, 1536. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For both networks, the training on Imagenet follows the same setup as Gomez et al. (2017). We train with SGD and momentum of 0.9. We regularized the model with a $\\ell ^ { 2 }$ weight decay of $1 0 ^ { - 4 }$ and batch normalization. The dataset is processed for $6 0 0 \\mathrm { k }$ iterations on a batch size of 256, distributed on 4GPUs. The initial learning rate is 0.1, dropped by a factor of ten every $1 6 0 \\mathrm { k }$ iterations. The dataset was augmented according to Gomez et al. (2017). The images values are mapped to [0, 1] while following geometric transformations were applied: random scaling, random horizontal flipping, random cropping of size $2 2 4 ^ { 2 }$ , and finally color distortions. No other regularizations were incorporated into the classification pipeline. At test time, we rescale the image size to $2 5 6 ^ { 2 }$ and perform a center crop of size $2 2 4 ^ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b84f3e11c0bdeb5dd1a42d4cbb16eb51d1e10b7988637944ff79b527e7a6073.jpg",
+        "image_caption": [
+            "Figure 3: Training loss of the $i$ -RevNet (b), compared to the ResNet, on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We report the training loss (i.e. Cross entropy) curves in Figure 3 of our $i$ -RevNet (b) and the ResNet baseline, displayed is a moving average over 100 iterations. Observe that the decrease of both training-losses are very similar which indicates that the constraint of invertibility does not interfere negatively with the learning process. However, we observed one third longer wall-clock times for $i$ -RevNets compared to plain RevNets because the channel size becomes larger. The Table 1 reports the performances of our $i$ -RevNets, with comparable RevNet and ResNet. First, we compare the $i$ -RevNet (a) with the RevNet and ResNet. Indeed, those CNNs have the same number of layers, and the $i$ -RevNet (a) increases the channel width of the initial layer as done in Gomez et al. (2017). The drawback of this technique is that the kernel sizes will be larger for all subsequent layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The $i$ -RevNet (a) has about 6 times more parameters than a RevNet and a ResNet but leads to a similar accuracy on the validation set of ImageNet. On the contrary, the $i$ -RevNet (b) is designed to have roughly the same number of parameters as the RevNet and ResNet, while being bijective. Its accuracy decreases by $1 . 5 \\%$ absolute percent on ImageNet compared to the RevNet baseline, which is not surprising because the number of channels was not drastically increased in the earlier layers as done in the baselines (Gomez et al., 2017; Krizhevsky et al., 2012; He et al., 2016); we did not explore wide ranges of hyper-parameters, thus the gap between (a) and (b) can likely be reduced with additional engineering. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "4 ANALYSIS OF THE INVERSE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now analyze the representation $\\Phi$ built by our bijective neural network $i$ -RevNet (b) and its inverse $\\Phi ^ { - 1 }$ , as trained on ILSVRC-2012. We first explain why obtaining $\\Phi ^ { - 1 }$ is challenging, even locally. We then discuss the reconstruction, while displaying in the image space linear interpolations between representations. ",
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+    {
+        "type": "text",
+        "text": "4.1 AN ILL-CONDITIONED INVERSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In the previous section, we have described the $i$ -RevNet architecture, that permits defining a deep network with an explicit inverse. We explain now why this is normally difficult, by studying its local inversion. We study the local stability of a network $\\Phi$ and its inverse $\\Phi ^ { - 1 }$ w.r.t. to its input, which means that we will quantify locally the variations of the network and its inverse w.r.t. to small variations of an input. As $\\Phi$ is differentiable (and its inverse as well), an equivalent way to perform this study is to analyze the singular values of the differential $\\partial \\Phi$ at some point, as for $( a , b )$ close the following holds: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15d1e4ab319c8f85cfdb3eb8af46b5fd022df59920c092b7db8b9ba844949389.jpg",
+        "text": "$$\n\\Phi \\boldsymbol { a } \\approx \\Phi \\boldsymbol { b } + \\partial \\Phi _ { b } ( \\boldsymbol { a } - \\boldsymbol { b } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            698,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce28b51d37c721b4a781c7d9d9b2bcd0ff508b71714b9075ab8d17c86d3d3d9a.jpg",
+        "image_caption": [
+            "Figure 4: Normalized sorted singular values of $\\partial \\Phi _ { x }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            501,
+            501,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Ideally, a well-conditioned operator has all its singular values constant equal to 1, for instance as achieved by the isometric operators of Cisse et al. (2017). ",
+        "bbox": [
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+            722,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our numerical application to an image $x$ , $\\partial \\Phi _ { x }$ corresponds to a very large matrix (square of the number of coefficients of the image at least) whose computations are expensive. Figure 4 corresponds to the singular values of the differential (i.e. the square roots of the eigen values of $\\partial \\Phi ^ { * } \\partial \\Phi$ ), in decreasing order, for a given natural image from ImageNet. The example we plot is typical of the behavior of $\\partial \\Phi$ . Observe there is a fast decay: numerically, the first $\\mathrm { 1 0 ^ { 3 } }$ and $1 0 ^ { \\bar { 4 } }$ singular values are responsible respectively for $8 0 \\%$ and $9 7 \\%$ of the cumulated energy (i.e. sum of squared singular values). This indicates $\\Phi$ linearizes the space locally in a considerably smaller space in comparison to the original input dimension. However, the dimensionality is still quite large (i.e. $> 1 0$ ) and thus we can not infer that $\\Phi$ lays locally in a low-dimensional manifold. It also proves that inversing $\\Phi$ is difficult and is an ill-conditioned problem. Thus obtaining implicitly this inverse would be a challenging task that we avoided, thanks to the formal reconstruction algorithm provided by Subsection 3.1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/a6fe6b5bd4ff7c42fa5cbd145f44da0d3e48b046022b4d75069d5e7eefe3144c.jpg",
+        "image_caption": [
+            "Figure 5: This graphic displays several reconstructed sequences $\\{ x ^ { t } \\} _ { t }$ . The left image corresponds to ${ \\bar { \\mathbf { \\Gamma } } } _ { x } 0$ and the right image to $x ^ { 1 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 LINEAR INTERPOLATION AND RECONSTRUCTION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            433,
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+            445
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Visualizing or understanding the important directions in the representation of inner layers of a CNN, and in particular, the final layer is complex because typically the cascade is either not invertible or unstable. One approach to reconstruct from an output layer consists in finding the input image that matches the activation through via gradient descent. However, this technique leads only to a partial or informal reconstruction (Mahendran & Vedaldi, 2015). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Another method consists in embedding the representation in a lower dimensional space and comparing the common attributes of nearest neighbors (Szegedy et al., 2013). It is also possible to train a CNN to reconstruct the representation (Dosovitskiy & Brox, 2016). Yet these methods require a priori knowledge in order to find the appropriate embeddings or training sets. We now discuss the improvements achieved by the $i$ -RevNet. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our main claim is that while the local inversion is ill-conditioned, the inverse $\\Phi ^ { - 1 }$ computations do not involve significant round-off errors. The forward pass of the network does not seem to suffer from significant instabilities, thus it seems coherent to assume that this will hold for $\\Phi ^ { - 1 }$ as well. For example, adding constraints beyond vanishing moments in the case of a Lifting scheme is difficult (Sweldens, 1998; Mallat, 1999), and this is a weakness of this method. We validate our claim by computing the empirical relative error on several subsets $\\mathcal { X }$ of data: ",
+        "bbox": [
+            173,
+            611,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/faf617bbc82767bf83b06069a10c7f59321cf2682e02e0004f4de34f1200046d.jpg",
+        "text": "$$\n\\epsilon ( \\mathcal { X } ) = \\frac { 1 } { | \\mathcal { X } | } \\sum _ { \\boldsymbol { x } \\in \\mathcal { X } } \\frac { \\| \\boldsymbol { x } - \\Phi ^ { - 1 } \\Phi \\boldsymbol { x } \\| } { \\| \\boldsymbol { x } \\| }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            713,
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+            753
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate this measure on a subset $\\mathcal { X } _ { 1 }$ of $| \\mathcal { X } _ { 1 } | = 1 0 ^ { 4 }$ independent uniform noises and on the validation set $\\mathcal { X } _ { 2 }$ of ImageNet. We report $\\epsilon ( \\mathcal { X } _ { 1 } ) = 5 \\times 1 0 ^ { - 6 }$ and $\\epsilon ( \\mathcal { X } _ { 2 } ) = 3 \\times 1 0 ^ { - 6 }$ respectively, which are close to the machine error and indicates that the inversion does not suffer from significant round-off errors. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Given a pair of images $\\{ x ^ { 0 } , x ^ { 1 } \\}$ , we propose to study linear interpolations between the pair of representations $\\{ \\Phi x ^ { \\bar { 0 } } , \\Phi x ^ { \\mathrm { { 1 } } } \\}$ , in the feature domain. Those interpolations correspond to existing images as $\\Phi ^ { - 1 }$ is an exact inverse. We reconstruct a convex path between two input points; it means that if: ",
+        "bbox": [
+            173,
+            830,
+            825,
+            886
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0fc9ca852c1841fc195fdab2bbbe01c79b13bcab5247de6e4056b08676f6395.jpg",
+        "text": "$$\n\\phi ^ { t } = t \\Phi x ^ { 0 } + ( 1 - t ) \\Phi x ^ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            886,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "then: $x ^ { t } = \\Phi ^ { - 1 } \\phi ^ { t }$ is a signal that corresponds to an image. ",
+        "bbox": [
+            171,
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/293bac6aae019b36fb6785db214dbbee593d7a2ee2f0f04de109fb47c7cb0a9a.jpg",
+        "image_caption": [
+            "Figure 6: Accuracy at depth $j$ for a linear SVM and a 1-nearest neighbor classifier applied to the spatially averaged $\\Phi _ { j }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            78,
+            803,
+            275
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We discretized $[ 0 , 1 ]$ into $\\{ t _ { 1 } , . . . , t _ { k } \\}$ , adapt the step size manually and reconstruct the sequence of $\\{ x ^ { t _ { 1 } } , . . . , x ^ { t _ { k } } \\}$ . Results are displayed in the Figure 5. We selected images from the basel face dataset (Paysan et al., 2009), describable texture dataset (Cimpoi et al., 2014) and imagenet. ",
+        "bbox": [
+            174,
+            347,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We now interpret the results. First, observe that a linear interpolation in the feature space is not a linear interpolation in the image space and that intermediary images are noisy, even for small deformations, yet they mostly remain recognizable. However, some geometric transformations such as a 3D-rotation seem to have been linearized, as suggested in Aubry & Russell (2015). In the next section, we thus investigate how the linear separation progresses with depth. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 A CONTRACTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            491,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we study again the bijective $i$ -RevNet. We first show that a localized or linear classifier progressively improves with depth. Then, we describe the linear subspace spanned by $\\Phi$ , namely the feature space, showing that the classification can be performed on a much smaller subspace, which can be built via a PCA. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 PROGRESSIVE LINEAR SEPARATION AND CONTRACTION ",
+        "text_level": 1,
+        "bbox": [
+            183,
+            604,
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+            617
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We show that both a ResNet and an $i$ -RevNet build a progressively more linearly separable and contracted representation as measured in Oyallon (2017). Observe this property holds for the $i$ - RevNet despite the fact that it can not discard any information. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We investigate these properties in each block, with the following experimental protocol. To reduce the computational burden we used a subset of 100 randomly selected imagenet classes, that consist of $N = 1 2 0 k$ images, and keep the same subset during all our following experiments. At each depth $j$ , we extract the features $\\{ \\Phi _ { j } x ^ { n } \\} _ { n \\leq N }$ of the training set, we average them along the spatial variable and standardize them in order to avoid any ill-conditioning effects. We used both a nearest neighbor classifier and a linear SVM. The former is a localized classifier that indicates that the $\\ell ^ { 2 }$ metric is progressively more important for classification, while a linear SVM measures the linear separation of the different classes. The parameters of the linear SVM are cross-validated on a small subset of the training set, prior to training on the 100 classes. We evaluate both classifiers for each model on the validation set of ImageNet and report the Top-1 accuracy in Figure 6. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We observe that both classifiers progressively improve similarly with depth for each model, the linear SVM performing slightly better than the nearest neighbor classifier because it is the more robust and discriminative classifier of the two. In the case of the $i$ -RevNet, the classification performed by the CNN leads to $7 7 \\%$ , and the linear SVM performs slightly better because we did not fine-tune the model to 100 classes. Observe that there is a more intense jump of performance on the 3 last layers, which seems to indicate that the former layers have prepared the representation to be more contracted and linearly separated for the final layers. ",
+        "bbox": [
+            174,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results suggest a low-dimensional embedding of the data, but this is difficult to validate as estimating local dimensionality in high dimensions is an open problem. However, in the next section, we try to compute the dimension of the discriminative part of the representation built by an $i$ -RevNet. ",
+        "bbox": [
+            176,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 DIMENSIONALITY ANALYSIS OF THE FEATURE SPACE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            164,
+            581,
+            178
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we investigate if we can refine the dimensionality of informative variabilities in the final layer of an $i$ -RevNet. Indeed, the cascade of convolutional operators has been trained on the training set to separate the 1000 different classes while being a homeomorphism on its feature space. Thus, the dimensionality of the feature space is potentially large. ",
+        "bbox": [
+            174,
+            189,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As shown in the previous subsection, the final layer is progressively prepared to be projected on the final probes corresponding to the classes. This indicates that the non-informative variabilities for classification can be removed via a linear projection on the final layer $\\Phi$ , which lie in a space of dimension 1000, at most. However, this projection has been built via supervision, which can still retain directions that have been contracted and thus will not be selected by an algorithm such as PCA. We show in fact a PCA retains the necessary information for classification in a small subspace. ",
+        "bbox": [
+            174,
+            252,
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+            335
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To do so, we build the linear projectors $\\pi _ { d }$ on the subspace of the $d$ first principal components, and we propose to measure the classification power of the projected representation with a supervised classifier, e.g. nearest neighbor or a linear SVM, on the previous 100 class task. Again, the feature representation $\\{ \\Phi x ^ { n } \\} _ { n \\leq N }$ are spatially averaged to remove the translation variability, and standardized on the training set. We apply both classifiers, and we report the classification accuracy of $\\{ \\pi _ { d } \\Phi x ^ { n } \\} _ { n \\leq N }$ w.r.t. to $d$ on the Figure 7. A linear projection removes some information that can not be recovered by a linear classifier, therefore we observe that the classification accuracy only decreases significantly for $d \\leq 2 0 0$ . This shows that the signal indeed lies in a subspace much lower dimensional than the original feature dimensions ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/93bdd4b0bf41956f8e5f34e143db6c00cc121e2956561dec013eee6385bf9daf.jpg",
+        "image_caption": [
+            "Figure 7: Accuracy of a linear SVM and nearest neighbor against the number of principal components retained. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            503,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "that can be extracted simply with a PCA that only considers directions of largest variances, illustrating a successful contraction of the representation. ",
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            641,
+            318,
+            656
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Invertible representations and their relationship to loss of information are on the agenda of deep learning for some time. Understanding how transformations in feature space are related to the corresponding input is an important step towards interpretable deep networks, invertible deep networks may play an important role in such analysis since, for example, one could potentially back-track a property from the feature space to the input space. To the best of our knowledge, this work provides the first empirical evidence that learning invertible representations that do not discard any information about their input on large-scale supervised problems is possible. ",
+        "bbox": [
+            174,
+            672,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To achieve this we introduce the $i$ -RevNet class of CNN which is fully invertible and permits to exactly recover the input from its last convolutional layer. $i$ -RevNets achieve the same classification accuracy in the classification of complex datasets as illustrated on ILSVRC-2012, when compared to the RevNet (Gomez et al., 2017) and ResNet (He et al., 2016) architectures with a similar number of layers. Furthermore, the inverse network is obtained for free when training an $i$ -RevNet, requiring only minimal adaption to recover inputs from the hidden representations. ",
+        "bbox": [
+            174,
+            776,
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+            861
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The absence of loss of information is surprising, given the wide believe, that discarding information is essential for learning representations that generalize well to unseen data. We show that this is not the case and propose to explain the generalization property with empirical evidence of progressive separation and contraction with depth, on ImageNet. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            367,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jorn-Henrik Jacobsen was partially funded by the STW perspective program ImaGene. Edouard ¨ Oyallon was partially funded by the ERC grant InvariantClass 320959, via a grant for PhD Students of the Conseil regional dIle-de-France (RDM-IdF), and a postdoctoral grant from the from DPEI ´ of Inria (AAR 2017POD057) for the collaboration with CWI. We thank Berkay Kicanaoglu for the Basel Face data, Mathieu Andreux, Eugene Belilovsky, Amal Rannen, Patrick Putzky and Kyriacos Shiarlis for feedback on drafts of the paper. ",
+        "bbox": [
+            174,
+            133,
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+            217
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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parse/train/VD_ozqvBy4W/VD_ozqvBy4W.md

@@ -0,0 +1,375 @@
+# COCON: A SELF-SUPERVISED APPROACH FOR CONTROLLED TEXT GENERATION
+
+Alvin Chan1∗, Yew-Soon $\mathbf { O n g ^ { 1 } }$ , Bill $\mathbf { P u n g ^ { 1 } }$ , Aston Zhang2, Jie $\mathbf { F u ^ { 3 } }$ 1Nanyang Technological University, 2Amazon AI, 3Mila, Polytechnique Montreal
+
+# ABSTRACT
+
+Pretrained Transformer-based language models (LMs) display remarkable natural language generation capabilities. With their immense potential, controlling text generation of such LMs is getting attention. While there are studies that seek to control high-level attributes (such as sentiment and topic) of generated text, there is still a lack of more precise control over its content at the word- and phrase-level. Here, we propose Content-Conditioner (CoCon) to control an LM’s output text with a content input, at a fine-grained level. In our self-supervised approach, the CoCon block learns to help the LM complete a partially-observed text sequence by conditioning with content inputs that are withheld from the LM. Through experiments, we show that CoCon can naturally incorporate target content into generated texts and control high-level text attributes in a zero-shot manner.1
+
+# 1 INTRODUCTION
+
+Transformer-based (Vaswani et al., 2017; Tay et al., 2020) pretrained language models (LMs) have led a wave of new advances in natural language processing tasks as a means to extract contextualized word embeddings (Devlin et al., 2018; Dai et al., 2019b; Yang et al., 2019) and as text generators (Radford et al., 2019; Brown et al., 2020). These LMs are trained on huge amounts of text corpora to predict next tokens through a log-likelihood objective. Given its remarkably fluent text generation, there is growing interest in controlling output texts of such LMs (Keskar et al., 2019; Dathathri et al., 2019). Approaches like training a modified LM from scratch to incorporate target text attributes (Keskar et al., 2019) can be expensive while finetuning pretrained LMs for specific attributes (Ziegler et al., 2019) limits the scope of text control. Without changing the architecture or weights of pretrained LMs, one promising approach (PPLM) (Dathathri et al., 2019) controls generated text through attribute models. Though effective in controlling high-level text attributes such as topic and sentiment, the same target attribute may generate text samples with vastly different content at the word- and phrase-levels, leaving a gap for more fine-grained control over the content of LM-generated texts.
+
+We conceptualize Content-Conditioner (CoCon) as an approach to narrow this gap by guiding pretrained LMs’ text outputs through the incorporation of content input. This content input can take the form of a text sequence whose content we would like to condition on for text generation. Essentially, CoCon comprises two parts: 1) a pretrained LM and 2) a interleave CoCon layer. By employing a pretrained LM, CoCon incorporates the representations of a content input into the encoded text representations through the CoCon layer before passing the content-conditioned representations into $\mathrm { L M } _ { \beta }$ for generation. To train the CoCon block, we propose a self-supervised learning approach where training data consist of text samples generated by the pretrained LM itself $( \ S \ 3 . 1 )$ . By splitting each text sequence into two segments $\bar { \mathbf { \Gamma } } ( [ \mathbf { x } ^ { a } ; \mathbf { x } ^ { b } ] )$ , CoCon learns through a self reconstruction objective to help the LM reconstruct missing latter segments $( \mathbf { x } ^ { b } )$ by taking $\mathbf { x } ^ { b }$ itself as the content input. We use content masking for CoCon and also propose other loss functions such as cycle reconstruction to condition content from divergent sources while producing high-quality texts. Since the CoCon block’s size is a small fraction of the LM and no finetuning is conducted on the LM’s weights, the training cost is significantly lower than training an LM from scratch. We show that CoCon’s fine-grained content control can be extended to also influence higher-level text attributes such as topic and sentiment in a zero-shot manner, and compare it with strong controlled generation baselines. Furthermore, CoCon is versatile in assimilating multiple content inputs, and its strength of content-conditioning can be flexibly adjusted through a content bias term during inference. In this paper, we demonstrate the CoCon approach with the GPT-2 345M model (Radford et al., 2019) as the pretrained LM. Given CoCon’s modular nature, it can be used with other Transformer-based LMs or even other controlled generation methods. All in all, the core contributions of this paper are:
+
+• We propose CoCon for content-conditioned language generation.   
+• We introduce a self-supervised learning approach where CoCon learns to complete text sequences when given information about future tokens.   
+• Through ablation studies and comparisons with strong baselines like PPLM and CTRL (Keskar et al., 2019), we investigate how CoCon controls high-level attributes such as topic and sentiment while generating texts that have high content similarity to conditioning text.
+
+# 2 RELATED WORK
+
+There is a line of work that aims to generate output text of desired attributes with neural networks. Some of the earliest efforts involve conditional generative models (Kikuchi et al., 2016; Ficler & Goldberg, 2017) where the networks are trained on text data labeled with the target attributes. These models can be trained via reinforcement learning (Ziegler et al., 2019) or the generative adversarial network (Yu et al., 2017) framework. Unlike CoCon, the requirement of predetermined attributes in those methods limits the possible types of generated texts. CTRL (Keskar et al., 2019) is a recent approach that generated controlled fluent texts through the use of control codes which are meta-data prepended to the text during generation. Though it produces high-quality text with its GPT-2-like architecture, its control codes are also predetermined during the training. Closest to our work is Plug and Play Language Model (PPLM) (Dathathri et al., 2019) which seeks to control text on already pretrained LM without finetuning through relatively small ‘pluggable’ attribute models. While PPLM’s flexible design also enables controlled generation without retraining or finetuning the LM like in CoCon, our approach aims to control the generation at a content level, beyond high-level text attributes. Another core difference lies in the training where CoCon’s self-supervised learning absolves the need for labeled data, such as the ones employed to train PPLM’s attribute discriminator models. Weighted decoding (Ghazvininejad et al., 2017; Holtzman et al., 2018) seeks to control the output text token by upweighting the probabilities of targeted words during the decoding step but has been shown to produce incoherent text (See et al., 2019). Conditioning language generation has been used in question generation to enhance faithfulness by attending to textual context such as predicates, subject types or object types (Elsahar et al., 2018) rather than the content input used here in CoCon. Small adapter layers (Bapna et al., 2019) have been previously proposed for multilingual translation to also save on model size and training resources but differ from CoCon’s self-supervised training as they rely on annotated sentence pairs of different languages for training.
+
+Text style transfer is a related area that controls texts’ attributes by translating text from one style to another (Dai et al., 2019a). A few of such studies employ auto-encoders to separate texts’ style and non-style latent representation (Shen et al., 2017; Hu et al., 2017; Yang et al., 2018). This disentanglement enables style changes to the text at the latent space while retaining most of its content. Another work identifies attribute markers (Li et al., 2018) which are $n$ -grams correlated with a particular style in a text corpus and edit texts’ style by substituting them. Essentially, style transfer alters existing texts rather than generating texts and requires predefined attributes.
+
+# 3 CONTENT CONDITIONER (COCON)
+
+In the following sections, we discuss the motivation for CoCon, its model architecture and how we train the CoCon block.
+
+Motivation In text generation with language models, given the prompt text $x _ { : t - 1 } \quad =$ $\{ x _ { 1 } , \dots , x _ { t - 1 } \}$ , the following text $\{ x _ { t } , \ldots , x _ { l } \}$ is generated in an auto-regressive manner (Man
+
+ning et al., 1999; Bengio et al., 2003):
+
+$$
+p ( x _ { t } , \ldots , x _ { l } | x _ { 1 } , \ldots , x _ { t - 1 } ) = \prod _ { i = t } ^ { l } p ( x _ { i } | x _ { 1 } , \ldots , x _ { i - 1 } ) .
+$$
+
+Previous studies on controlled text generation in LM showed that $p ( \mathbf { x } )$ can be conditioned on target attributes (Dathathri et al., 2019) or control codes (Keskar et al., 2019) to control the text’s sentiment or topic, i.e.,
+
+$$
+p ( x _ { t } , \ldots , x _ { l } | x _ { 1 } , \ldots , x _ { t - 1 } ) = \prod _ { i = 1 } ^ { l } p ( x _ { i } | \mathbf { a } , \{ x _ { 1 } , \ldots , x _ { i - 1 } \} ) ,
+$$
+
+where a is the target attribute. While these methods show that the generation is fluent and can be aligned with the target attribute well, the output texts $\{ x _ { t } , \ldots , x _ { l } \}$ are controlled at a global attribute (e.g., sentiment/topic) level rather than at a more local content (e.g., words/phrases) level. Since there is a vast number of possible $\{ x _ { t } , \ldots , x _ { l } \}$ candidates which would align well with both the prompt text and target attribute, this results in generated text samples that contain very different content during the stochastic token sampling process. This motivates an approach to condition on an content input $\mathbf { c }$ for more fine-grained control over text generation:
+
+$$
+p ( x _ { t } , \ldots , x _ { l } | x _ { 1 } , \ldots , x _ { t - 1 } ) = \prod _ { i = 1 } ^ { l } p ( x _ { i } | \mathbf { c } , \{ x _ { 1 } , \ldots , x _ { i - 1 } \} ) ,
+$$
+
+where c can be a text sequence whose content we would like to condition on during text generation. Next, we propose the model architecture of Content-Conditioner (CoCon) as an approach for this control.
+
+Model Architecture Our proposed Content-Conditioner (Figure 1) controls the content of the generated text while maintaining fluency by incorporating a pretrained Transformer-based language model (LM), GPT-2 (Radford et al., 2019) in our experiments. Such LMs have shown remarkable natural text generation in the auto-regressive manner (Eq. 1) where the next token $x _ { t }$ is sampled based on the logits $\mathbf { o } _ { t } = \mathrm { L M } ( x _ { : t - 1 } )$ . These LMs are essentially stacks of Transformer blocks, each consisting of layer normalization (Ba et al., 2016), multi-head self-attention (Vaswani et al., 2017) and position-wise feed forward operations.
+
+An LM’s generation can be broken down into two separate parts: layers before the CoCon block $( \mathrm { L M } _ { \alpha } )$ ) and layers after $( \mathrm { L M } _ { \beta } )$ . The $\mathrm { L M } _ { \alpha }$ acts as a feature extractor that takes in the input sequence’s embeddings and outputs its intermediate representation at a breakpoint, i.e., $\mathbf { h } _ { : t - 1 } = \mathrm { L M } _ { \alpha } ( x _ { : t - 1 } )$ . Subsequently, $\mathrm { L M } _ { \beta }$ takes in this representation and outputs the logits for the next token, i.e., $\mathbf { o } _ { t } =$ $\mathrm { L M } _ { \beta } ( \mathbf { h } _ { : t - 1 } )$ , yielding
+
+$$
+\begin{array} { r } { \mathbf { o } _ { t } = \mathrm { L M } ( x _ { : t - 1 } ) = \mathrm { L M } _ { \beta } ( \mathrm { L M } _ { \alpha } ( x _ { : t - 1 } ) ) = \mathrm { L M } _ { \beta } ( \mathbf { h } _ { : t - 1 } ) . } \end{array}
+$$
+
+From Eq. 4, we can see that the representation $\mathbf { \eta } ^ { ( \mathbf { h } ) }$ is a medium to control next token logits $\mathbf { \tau } ( \mathbf { o } )$ and hence the text generation process. Indeed, we transform $\mathbf { h }$ by conditioning it with the content input (c) through a CoCon block such that
+
+$$
+\mathbf { h } _ { : t - 1 } ^ { \prime } = \mathrm { C o C o n } ( \mathbf { h } _ { : l _ { c } } ^ { ( \mathbf { c } ) } , \ \mathbf { h } _ { : t - 1 } ) ,
+$$
+
+where $\mathbf { h } _ { : l _ { c } } ^ { ( \mathbf { c } ) } = \mathrm { L M } _ { \alpha } ( \mathbf { c } )$ is the content representations and $l _ { c }$ is the length of the content text secquence. We parameterize the CoCon block as a single Transformer block with an attention and position-wise feed-forward operation. Similar to a typical LM attention layer, the query $( \mathbf { Q } )$ , key $( \mathbf { K } )$ , value $( \mathbf { V } )$ matrices are computed through linear transformations on the representations $\mathbf { h } _ { : t - 1 }$ , where $\mathbf { Q } , \mathbf { K } , \mathbf { V } \ \in \ \mathbb { R } ^ { ( t - 1 ) \times d }$ and $d$ is the representations’ dimension. To attend to the content representations $( \mathbf { h } _ { : l _ { c } } ^ { ( \mathbf { c } ) } )$ , the content keys and values $( \mathbf { K } ^ { ( \mathbf { c } ) } , \mathbf { V } ^ { ( \mathbf { c } ) } \in \mathbb { R } ^ { l _ { c } \times d } )$ are also computed, and concatenated to the original attention matrices before computing the CoCon attention output:
+
+$$
+\mathbf { K } ^ { \prime } = [ \mathbf { K } ^ { ( \mathbf { c } ) } ; \mathbf { K } ] , \mathbf { V } ^ { \prime } = [ \mathbf { V } ^ { ( \mathbf { c } ) } ; \mathbf { V } ] , \mathbf { A } = \mathrm { S o f t m a x } ( \mathbf { Q } \mathbf { K } ^ { \prime \top } ) \mathbf { V } ^ { \prime } = \mathrm { S o f t m a x } ( \mathbf { W } ) \mathbf { V } ^ { \prime } ,
+$$
+
+where $\mathbf { A } = \{ \mathbf { a } _ { 1 } , \dots , \mathbf { a } _ { t - 1 } \}$ and $\mathbf { W } \in \mathbb { R } ^ { ( t - 1 ) \times ( l _ { c } + t - 1 ) }$ represents the attention weights. The final CoCon outputs are computed with a position-wise feed-forward layer. By concatenating to the representations prior to $t - 1$ and passing them to $\mathrm { L M } _ { \beta }$ , the next logits, and consequently word token $\tilde { \mathbf { x } } _ { t }$ , is now conditioned on c:
+
+$$
+\begin{array} { r } { \mathbf { h } _ { i } ^ { \prime } = \mathrm { F F } ( \mathbf { a } _ { i } ) , \tilde { \mathbf { o } } _ { t } = \mathrm { L M } _ { \beta } ( [ \mathbf { h } _ { : t - 2 } ; \mathbf { h } _ { t - 1 } ^ { \prime } ] ) , p _ { \theta , \psi } ( \tilde { x } _ { t } | \mathbf { c } , x _ { : t - 1 } ) = \mathrm { S o f t m a x } ( \tilde { \mathbf { o } } _ { t } ) , } \end{array}
+$$
+
+where $\theta$ and $\psi$ are the paramterization of the CoCon block and LM respectively. Similar to a GPT-2 Transformer block, our CoCon block includes layer normalization before its multi-headed attention and feed-forward layers. Figure 1 summarizes the CoCon architecture which enables auto-regressive text generation by using $\tilde { x } _ { i }$ as the token input $( x _ { i } )$ to generate $\tilde { x } _ { i + 1 }$ where $i \geq t$ .
+
+![](images/eceae1ebe0061e7644b25b9b83ee4e4bfaf2b184677f0c6b41d038f5bc39c394.jpg)  
+Figure 1: Model architecture of proposed Content-Conditioner (CoCon).
+
+Multiple Content Inputs CoCon’s flexible design enables multiple content inputs for a single generation. In the case where we have $N$ content inputs $( \mathbf { c } ^ { 1 } , \ldots , \mathbf { c } ^ { \tilde { N } } )$ , the output text can be conditioned by these contents through their attention keys and values, similar to Eq. 6:
+
+$$
+{ \bf K } ^ { \prime } = [ { \bf K } ^ { ( \mathbf { c } ^ { 1 } ) } \ldots { \bf K } ^ { ( \mathbf { c } ^ { N } ) } ; { \bf K } ] , \quad { \bf V } ^ { \prime } = [ { \bf V } ^ { ( \mathbf { c } ^ { 1 } ) } \ldots { \bf V } ^ { ( \mathbf { c } ^ { N } ) } ; { \bf V } ] , \quad { \bf A } = \mathrm { S o f t m a x } ( { \bf Q } { \bf K } ^ { \prime \top } ) { \bf V } ^ { \prime } .
+$$
+
+Strength of Content Conditioning Within CoCon’s attention mechanism, we can vary the extent of content conditioning on the output text by biasing the attention weights in W (Eq. 6) that correspond to the content input (c). More specifically, the influence of c on the output text can be altered through the attention’s softmax weighting on the content values $( \mathbf { V } ^ { ( \mathbf { c } ) } )$ . During generation, a positive bias term $( \tau _ { \mathrm { c o n t e n t } } )$ can optionally be added to the content attention weights $\bar { \mathbf { W } } _ { : , : l _ { c } } \in \mathbb { R } ^ { ( t - \bar { 1 } ) \times l _ { c } }$ to increase influence of $\mathbf { V } ^ { ( \mathbf { c } ) }$ , boosting content conditioning, while a negative term can conversely reduce the content-conditioning effect. We discuss examples of varying $\tau _ { \mathrm { c o n t e n t } }$ in $\ S 4 . 4$ .
+
+# 3.1 SELF-SUPERVISED LEARNING
+
+We train CoCon with a self-supervised learning approach that is inspired by the diversity of content in natural language. Given a text sequence $\mathbf { x } = \{ x _ { 1 } , \ldots , x _ { t - 1 } , x _ { t } , \ldots , x _ { l } \}$ of length $l$ , we can break it into two contiguous segments: $\mathbf { x } ^ { a } = \{ x _ { 1 } , \ldots , x _ { t - 1 } \}$ and $\mathbf { x } _ { } ^ { b } = \{ x _ { t } , \ldots , x _ { l } \}$ where $\mathbf { x } = [ \mathbf { x } ^ { a } ; \mathbf { x } ^ { b } ]$ . In the real world, there may be numerous substitutes of $\mathbf { x } ^ { b }$ that could follow from $\mathbf { x } ^ { a }$ fluently. Coupled with the randomness in text sampling, this means that, without information about $\mathbf { x } ^ { b }$ , the probability of reconstructing the full $\mathbf { x }$ from $\mathbf { x } ^ { a }$ alone with an LM can be low.
+
+Self Reconstruction Loss Based on this intuition, our approach trains the CoCon block to help the LM reconstruct the original $\mathbf { x }$ by also conditioning with $\mathbf { x } ^ { b }$ as the content input, i.e., $\mathbf { c } = \mathbf { x } ^ { \hat { b } }$ (Figure 2b). More concretely, we first compute the intermediate representations of the input text $\mathbf { x }$ and c:
+
+$$
+\mathbf { h } _ { : l } = \mathrm { L M } _ { \alpha } ( \mathbf { x } ) = \mathrm { L M } _ { \alpha } ( x _ { : l } ) , \mathbf { h } _ { : l _ { c } } ^ { ( \mathbf { c } ) } = \mathrm { L M } _ { \alpha } ( \mathbf { c } ) = \mathrm { L M } _ { \alpha } ( x _ { t : l } ) ,
+$$
+
+where $l _ { c } = l - t + 1$ is the length of $\mathbf { c }$ . The content-conditioned representation can be computed by the CoCon block where $\mathbf { h } _ { : l _ { c } } ^ { ( \mathbf { c } ) }$ is the content representation:
+
+$$
+\mathbf { h } _ { i } ^ { \prime } = \mathrm { C o C o n } ( \mathbf { h } _ { : l _ { c } } ^ { ( \mathbf { c } ) } , \ \mathbf { h } _ { : i } ) , \forall i \geq t - 1 .
+$$
+
+Similar to Eq. 7, the CoCon transformed representations are concatenated to the original representation before $t - 1$ and passed into $\mathrm { L M } _ { \beta }$ to produce the LM logits:
+
+$$
+\begin{array} { r } { \widetilde { \mathbf { o } } _ { i + 1 } = \mathrm { L M } _ { \beta } \big ( [ \mathbf { h } _ { : t - 2 } ; \mathbf { h } _ { t - 1 : i } ^ { \prime } ] \big ) , p _ { \theta , \psi } \big ( \widetilde { x } _ { i + 1 } | \mathbf { c } , x _ { : i } \big ) = \mathrm { S o f t m a x } \big ( \widetilde { \mathbf { o } } _ { i + 1 } \big ) , \forall i \geq t - 1 . } \end{array}
+$$
+
+Through an LM training objective, we arrive at the self-reconstruction loss term which trains CoCon to predict tokens of $\mathbf { x } ^ { b }$ by conditioning on $\mathbf { x } ^ { b }$ itself as the content input (c):
+
+$$
+\mathcal { L } _ { \mathrm { s e l f } } = - \sum _ { i = t } ^ { l } \log p _ { \theta , \psi } \left( x _ { i } | ( \mathbf { c } = \mathbf { x } ^ { b } ) , \{ x _ { 1 } , \ldots , x _ { i - 1 } \} \right) .
+$$
+
+To avoid trivializing the prediction of the next token $x _ { i + 1 }$ during training, we apply a self-token c-mask at CoCon’s attention layer such that $\mathbf { h } _ { i } ^ { \prime }$ does not attend to the token $x _ { i + 1 }$ in c that it is trying to predict. This approach can be conducted in a self-supervised manner with any pretrained LM where the training samples $\mathbf { x }$ are generated text outputs stochastically sampled from the LM itself.
+
+Null Content Loss To encourage CoCon’s outputs to follow the prompt text $\mathbf { x } ^ { a }$ fluently without relying on $\mathbf { x } ^ { b }$ , we also train CoCon with a loss term similar to Eq. 12 but replaces the content input with a null token $( \emptyset )$ :
+
+$$
+\mathcal { L } _ { \mathrm { n u l l } } = - \sum _ { i = t } ^ { l } \log p _ { \theta , \psi } \left( x _ { i } | ( \mathbf { c } = \mathcal { O } ) , \{ x _ { 1 } , \ldots , x _ { i - 1 } \} \right) .
+$$
+
+![](images/a1d1c083c0fbc93e7dfb18d1d08fbff6be614afd45cc265d6373d252545ec057.jpg)  
+Figure 2: Illustrative examples of (b) self reconstruction and (c) cycle reconstruction training.
+
+Cycle Reconstruction Loss The self reconstruction loss relies on CoCon content input (c) and initial prompt text $\mathbf { \tau } ( \mathbf { p } )$ originating from one single text sample. To encourage generalization on cases where c and $\mathbf { p }$ are from divergent text sources, we employ a cycle reconstruction training that utilizes two different training samples (e.g., x, $\mathbf { x } ^ { \prime }$ in Figure 2a) and two CoCon forward steps (Figure 2c). We can express the output of a CoCon’s auto-regressive generation as
+
+$$
+\mathbf { y } = f _ { \theta , \psi } ( \mathbf { c } , \mathbf { p } ) ,
+$$
+
+where $\left[ \mathbf { p } ; \mathbf { y } \right]$ would be a fluent text sequence and $\mathbf { y }$ is conditioned on the content of c. The first step (Figure 2c(i)) computes the CoCon output with the content input (c) sourced from $\mathbf { x }$ and prompt text $\mathbf { \tau } ( \mathbf { p } )$ sourced from $\mathbf { x } ^ { \prime }$ :
+
+$$
+\mathbf { y } _ { \mathbf { x } , \mathbf { x } ^ { \prime } } = f _ { \boldsymbol { \theta } , \boldsymbol { \psi } } ( ( \mathbf { c } = \mathbf { x } ^ { b } ) , ( \mathbf { p } = \mathbf { x } ^ { \prime } { } ^ { a } ) ) ,
+$$
+
+where $\mathbf { x } = [ \mathbf { x } ^ { a } ; \mathbf { x } ^ { b } ]$ and $\mathbf { x } ^ { \prime } = [ \mathbf { x } ^ { \prime } { } ^ { a } ; \mathbf { x } ^ { \prime } { } ^ { b } ]$ . Since CoCon utilizes a pretrained LM for generation, $\mathbf { y } _ { \mathbf { x } , \mathbf { x } ^ { \prime } }$ would be a text sequence that fluently follows the prompt, ${ \bf { x } } ^ { \prime } \bar { \bf { \Lambda } }$ , while seeking to incorporate $\mathbf { x } ^ { b }$ ’s content. The second CoCon forward step (Figure $2 \mathrm { c } ( \mathrm { i i } ) ,$ ) takes $\mathbf { y } _ { \mathbf { x } , \mathbf { x } ^ { \prime } }$ as content input and $\mathbf { x } ^ { a }$ as prompt text:
+
+$$
+\begin{array} { r } { \mathbf { y } _ { \mathrm { c y c l e } } = f _ { \boldsymbol { \theta } , \psi } \big ( ( \mathbf { c } = \mathbf { y } _ { \mathbf { x } , \mathbf { x } ^ { \prime } } ) , ( \mathbf { p } = \mathbf { x } ^ { a } ) \big ) , } \end{array}
+$$
+
+Since $\mathbf { x } = [ \mathbf { x } ^ { a } ; \mathbf { x } ^ { b } ]$ , $\mathbf { x } ^ { b }$ is a valid continuation from the prompt $\mathbf { x } ^ { a }$ and recall that $\mathbf { y } _ { \mathbf { x } , \mathbf { x } ^ { \prime } }$ was contentconditioned on $\mathbf { x } ^ { b }$ in the first CoCon step (Eq. 15). This posits $\mathbf { x } ^ { b }$ as a training label for $\mathbf { y } _ { \mathrm { c y c l e } }$ which gives us the cycle reconstruction loss term:
+
+$$
+\mathcal { L } _ { \mathrm { c y c l e } } = - \sum _ { i = t } ^ { l } \log p _ { \theta , \psi } \left( \mathbf { y } _ { \mathrm { c y c l e } } = \mathbf { x } ^ { b } | ( \mathbf { c } = \mathbf { y } _ { \mathbf { x } , \mathbf { x } ^ { \prime } } ) , ( \mathbf { p } = \mathbf { x } ^ { a } ) \right) .
+$$
+
+Adversarial Loss Adversarial training objectives have shown to help in generating realistic text outputs (Yang et al., 2018). Here, we also employ an adversarial training loss (Goodfellow et al., 2014) to encourage the output texts’ representations $( \operatorname { L M } _ { \alpha } ( \mathbf { y } ) )$ to match those of the training samples $( \mathrm { L M } _ { \alpha } ( \mathbf { x } ) )$ by minimizing the loss:
+
+$$
+\mathcal { L } _ { \mathrm { a d v } } = \mathbb { E } _ { \mathbf { x } } [ \log f _ { \mathrm { d i s c } } ( \mathrm { L M } _ { \alpha } ( \mathbf { x } ) ) ] + \mathbb { E } _ { \mathbf { y } } [ \log ( 1 - f _ { \mathrm { d i s c } } ( \mathrm { L M } _ { \alpha } ( \mathbf { y } ) ) ] ,
+$$
+
+where $f _ { \mathrm { d i s c } }$ is a discriminator network that classifies whether the representations are of CoCongenerated texts. Through continuous approximation of discrete sampling of $y$ where token logits instead of one-hot vectors are fed as input into $\mathrm { L M } _ { \alpha }$ , CoCon and $f _ { \mathrm { d i s c } }$ can be trained with backpropagation in an end-to-end manner. Parameterizing the $f _ { \mathrm { d i s c } }$ with $\phi$ , the discriminator is trained to maximize ${ \mathcal { L } } _ { \mathrm { a d v } }$ rather than minimize it:
+
+$$
+\phi ^ { * } = \arg \operatorname* { m a x } _ { \phi } \mathcal { L } _ { \mathrm { a d v } }
+$$
+
+Full Training The full learning objective trains the CoCon to minimize the four loss terms through stochastic gradient descent:
+
+$$
+\theta ^ { * } = \underset { \theta } { \arg \operatorname* { m i n } } ( \lambda _ { \mathrm { s e l f } } \mathcal { L } _ { \mathrm { s e l f } } + \lambda _ { \mathrm { n u l l } } \mathcal { L } _ { \mathrm { n u l l } } + \lambda _ { \mathrm { c y c l e } } \mathcal { L } _ { \mathrm { c y c l e } } + \lambda _ { \mathrm { a d v } } \mathcal { L } _ { \mathrm { a d v } } ) ,
+$$
+
+where the $\lambda$ values control how much the loss terms dominate the training. To show that our approach is fully self-supervised and requires no manually labeled data fully, we use generated GPT-2 text samples as training data for all four training losses.
+
+# 4 EXPERIMENTS
+
+We conduct a range of experiments on CoCon to study its control over generated texts and the quality of these texts. Table 1 shows CoCon samples with content, topic and sentiment control.
+
+Table 1: CoCon samples with multiple content inputs, given same prompt text (underlined), exhibiting control over generations. More samples are in the Appendix (Table 18 and 19).
+
+<table><tr><td>Content Input (c1):officials predict there could be 5,8oo submerged + Target Topic: SCIENCE, Content Input (c²): Scientist + Target Sentiment: Positive, Content Input (c3): is perfect</td></tr><tr><td>The movie makers speculate there&#x27;s a perfect match. Expectations there could be up to 5O0 kilograms of clay could be thrown onto the surface of the ocean. The BBC reported that it could have taken up to a year and a half to add clay to the ocean floor, though experts believe it could be done within several days..</td></tr></table>
+
+CoCon Setup In all our experiments, the GPT-2 medium 345M model (Radford et al., 2019) is used as the pretrained LM for CoCon. The CoCon’s $\mathrm { L M } _ { \alpha }$ comprises the first 7 GPT-2 Transformer blocks while the remaining 17 blocks make up $\mathrm { L M } _ { \beta }$ in our experiments. The CoCon block’s architecture mirrors a single GPT-2 Transformer block with a dimension size of 1024. The training samples $\mathbf { \tau } ( \mathbf { x } )$ are 30-BPE long segments sampled from GPT-2 output texts2. Subsequently, the $\mathbf { x } ^ { a }$ and $\mathbf { x } ^ { b }$ segments are split from x at a breakpoint between the 8th to 12th BPE position, uniformly sampled during training. More details about the setup are deferred to $\ S \mathbf { A }$ of the Appendix.
+
+# 4.1 CONTENT SIMILARITY
+
+We perform evaluation of CoCon’s content control over generated text with automatic metrics such as BLEU (Papineni et al., 2002), NIST (Doddington, 2002) and METEOR (Lavie & Agarwal, 2007). These standard machine translation metrics can reveal how the CoCon generated text, $\mathbf { y } = f _ { \theta , \psi } ( \mathbf { c } , \mathbf { p } )$ , are similar to the content input (c). Similar to Dathathri et al. (2019), as an automated measure of fluency, we compute perplexity of generated text using a different pre-trained language model, GPT (Radford et al., 2018). We also report Dist-1,-2,-3 scores as another metric of text quality that measures the diversity of 1-,2-,3-grams in the generations. Apart from a GPT-2 plain baseline without content conditioning, we also compare with three CoCon variants that omit either the $\mathcal { L } _ { \mathrm { c y c l e } }$ , ${ \mathcal { L } } _ { \mathrm { n u l l } }$ or ${ \mathcal { L } } _ { \mathrm { a d v } }$ for an ablation study. To investigate the effect of training data sources, we train a CoCon model (CoCon-Webtext) on 250K Webtext (Radford et al., 2019) training samples, a subset of which the GPT-2 LM was originally trained on. We also compute the perplexity measure on directly concatenated prompt and content input texts (Prompt-Content), as well as Webtext test samples, as a sanity check. More setup details are in $\ S \operatorname { A . 1 }$ of the Appendix.
+
+Results Based on the content similarity results (Table 2), all the CoCon variants can incorporate the content of $\mathbf { c }$ in the generated text better than an unconditioned plain GPT-2 LM. While the CoCon ablated variants appear to be better at incorporating c’s content, it comes at a high cost of text quality for the case of omitted $\mathcal { L } _ { \mathrm { c y c l e } }$ and ${ \mathcal { L } } _ { \mathrm { n u l l } }$ . If $\mathcal { L } _ { \mathrm { c y c l e } }$ were removed, CoCon would train only on prompt text $\mathbf { p }$ and content input c segments that were sampled from the same parent $\mathbf { x }$ , which explains why the quality of its outputs drops during test time when prompt text p and content input c are from different sources. We can see this degenerate case from generated samples (Table 9) where $\mathcal { L } _ { \mathrm { c y c l e } }$ is vital to smoothly integrate content inputs that are far from the prompt text. Despite slightly improved text diversity, we observe that ${ \mathcal { L } } _ { \mathrm { a d v } }$ marginally reduces CoCon’s perplexity which we speculate is due to it being a non-LM type loss term, causing a trade-off in performance on the LM-aligned perplexity metric. In our human evaluation (Table 8 of Appendix), we observe that humans also perceive CoCon without ${ \mathcal { L } } _ { \mathrm { a d v } }$ as more fluent, indicating that the addition of ${ \mathcal { L } } _ { \mathrm { a d v } }$ may have made it more challenging for the CoCon model to converge in its training. Training CoCon with Webtext samples improves content similarity at a cost of higher perplexity and lower fluency.
+
+Table 2: Content similarity and quality of generated content-conditioned samples. BLEU, NIST and METEOR values are reported in scale of $\bar { ( \times 1 0 ^ { - 2 } ) }$ ).
+
+<table><tr><td>Model</td><td>BLEU-4 (↑better)</td><td>NIST-4 (↑better)</td><td>METEOR (↑better)</td><td>Perplexity (↓better)</td><td>Dist-1 (↑better)</td><td>Dist-2 (↑better)</td><td>Dist-3 (↑better)</td></tr><tr><td>GPT-2</td><td>0.22</td><td>7.09</td><td>6.14</td><td>105.7</td><td>0.057</td><td>0.49</td><td>0.82</td></tr><tr><td>CoCon</td><td>2.76</td><td>22.9</td><td>21.5</td><td>70.8</td><td>0.048</td><td>0.39</td><td>0.70</td></tr><tr><td>L w/o Lcycle</td><td>3.30</td><td>25.1</td><td>23.9</td><td>150.8</td><td>0.050</td><td>0.42</td><td>0.74</td></tr><tr><td>L w/o Lnull</td><td>4.44</td><td>28.3</td><td>26.8</td><td>73.2</td><td>0.046</td><td>0.37</td><td>0.68</td></tr><tr><td>L w/o Ladv</td><td>4.47</td><td>28.2</td><td>27.2</td><td>68.7</td><td>0.047</td><td>0.38</td><td>0.69</td></tr><tr><td>CoCon-Webtext</td><td>2.90</td><td>24.6</td><td>23.0</td><td>112.5</td><td>0.054</td><td>0.44</td><td>0.74</td></tr><tr><td>Prompt-Content Webtext</td><td>1 1</td><td>1 1</td><td>1 1</td><td>442.2 185.8</td><td>1 1</td><td>1 1</td><td>1 1</td></tr></table>
+
+# 4.2 TOPIC RELEVANCE
+
+Setup We evaluate CoCon’s ability to control the topic of the generated text by using topic words as single-token content inputs and compare with two strong LM-based controlled generation baselines (PPLM (Dathathri et al., 2019) and CTRL (Keskar et al., 2019)), using their Huggingface versions (Wolf et al., 2019). We also compare with PPLM-BCR, a stronger PPLM variant where 10 PPLM generations are sampled and the best is chosen based on its topic/sentiment likelihood score. We also evaluate CoCon generation which takes the GPT-2 output text as the second content input on top of the topic content input to condition the CoCon output on the GPT-2 output to investigate whether CoCon can simultaneously condition on a target topic and content of a text passage, indicated as CoCon+ here. We also conducted human evaluations of fluency and A/B testing on attribute relevance, similar to Dathathri et al. (2019). More setup details are presented in the Appendix $\ S \ A . 2$ .
+
+Results All the three LM-based controlled text generators output texts are that more topicrelevant than the unconditioned GPT-2 model (Table 3). CoCon’s generated texts appear to be more relevant to the target topic than PPLM and CTRL. Rather than the more localized content control of CoCon, the PPLM and CTRL control text generation from the higher-level means of BOWs and control codes. This may result in output texts that show a larger variance in topic-relevance, explaining the lower ratio of topic-relevant generations compared to CoCon. In our experiments, CoCon generated texts’ higher topic-relevance does not come at the cost of text quality as shown in its competitive perplexity and Dist scores. Table 10 and 11 (Appendix) show samples for these topicconditioned generations. ${ \mathrm { C o C o n } } { \mathrm { : } } { \mathrm { s } }$ topic accuracy is lower than CoCon but still higher than GPT-2 text indicating that adding another content input (GPT-2 output text) can reduce the conditioning strength of the target topic content input. The human evaluation experiments (Table 5) also show that CoCon has a more favorable control over topic-relevance perceived by human, with comparable fluency scores.
+
+Table 3: Evaluation of topic-controlled generations. Topic accuracy report ratio of samples that were classified as their target topic.   
+
+<table><tr><td>Model</td><td>Topic % (↑ better)</td><td>Perplexity (↓better)</td><td>Dist-1 (↑ better)</td><td>Dist-2 (↑better)</td><td>Dist-3 (↑better)</td></tr><tr><td>GPT-2 PPLM</td><td rowspan="6">22.5</td><td rowspan="2">84.7</td><td rowspan="2">0.23</td><td rowspan="2">0.74</td><td rowspan="2">0.91</td></tr><tr><td>PPLM-BCR</td></tr><tr><td>42.5</td><td>32.4 37.5</td><td>0.15</td><td>0.54</td><td>0.78</td></tr><tr><td>61.3 86.7</td><td>60.5</td><td>0.23 0.14</td><td>0.64 0.56</td><td>0.86</td></tr><tr><td>CTRL CoCon</td><td>52.4</td><td>0.17</td><td>0.60</td><td>0.77 0.86</td></tr><tr><td></td><td>90.4 46.2</td><td>83.6</td><td>0.21</td><td></td><td></td></tr><tr><td>CoCon+</td><td></td><td></td><td></td><td>0.67</td><td>0.87</td></tr></table>
+
+# 4.3 SENTIMENT CONTROL
+
+Setup We also evaluate CoCon’s sentiment control with PPLM and CTRL, in a setup similar to $\mathrm { ~ \normalfont ~ \ S ~ } 4 . 2$ . Sentiment attribute markers (Li et al., 2018) ‘is perfect’ and ‘is horrible’ are used as content inputs to generated CoCon outputs for the POSITIVE and NEGATIVE sentiment respectively. Sentiment attribute markers are n-grams that appear in high frequency in text samples annotated with a particular attribute such as positive/negative sentiment. Similar to Dathathri et al. (2019), the sentiment classifier is trained on the IMDB movie review dataset (Maas et al., 2011).
+
+Results Similar to the findings in $\ S 4 . 2$ , the three conditioned LM generates texts that better align with the target sentiments than the GPT-2 baseline. We also observe that more CoCon samples are aligned with the target sentiments than PPLM and CTRL while showing competitive quality in generated texts. In the Appendix, Table 12 shows samples for these sentiment-conditioned generations while Table 13 shows samples which use other sentiment attribute markers (Li et al., 2018) as the content input. Results from human evaluation (Table 5) also show that CoCon generations are more aligned to the target sentiment, though at a cost of fluency. Similar to $\ S 4 . 2$ , we also observe a similar tradeoff in $\mathrm { C o C o n + }$ ’s sentiment alignment when presented with another content input (GPT-2 output text).
+
+Table 4: Evaluation of sentiment-controlled generations. Sentiment accuracy report ratio of samples that were classified as their target sentiment.   
+
+<table><tr><td>Model</td><td>Sentiment % (↑ better)</td><td>Perplexity (↓better)</td><td>Dist-1 (↑better)</td><td>Dist-2 (个better)</td><td>Dist-3 (↑better)</td></tr><tr><td>GPT-2</td><td>50.0</td><td>101.2</td><td>0.38</td><td>0.82</td><td>0.92</td></tr><tr><td>PPLM</td><td>68.9</td><td>35.5</td><td>0.24</td><td>0.63</td><td>0.82</td></tr><tr><td>PPLM-BCR</td><td>96.7</td><td>34.1</td><td>0.30</td><td>0.65</td><td>0.79</td></tr><tr><td>CTRL</td><td>81.1</td><td>44.1</td><td>0.21</td><td>0.62</td><td>0.80</td></tr><tr><td>CoCon</td><td>98.9</td><td>50.3</td><td>0.20</td><td>0.61</td><td>0.80</td></tr><tr><td>CoCon+</td><td>85.6</td><td>111.0</td><td>0.32</td><td>0.73</td><td>0.87</td></tr></table>
+
+Table 5: Human evaluation of topic/sentiment-controlled generations on relevance with target topic or sentiment and their fluency scores $\uparrow$ better for all metrics).   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Topic</td><td colspan="2">Sentiment</td></tr><tr><td>Acc.%</td><td>Fluency</td><td>Acc.%</td><td>Fluency</td></tr><tr><td>GPT-2</td><td>22.0</td><td>4.01</td><td>36.7</td><td>3.84</td></tr><tr><td>CoCon</td><td>85.0</td><td>3.86</td><td>76.7</td><td>3.30</td></tr><tr><td>PPLM-BCR</td><td>46.0</td><td>3.98</td><td>50.0</td><td>3.48</td></tr><tr><td>CoCon</td><td>75.0</td><td>3.86</td><td>66.7</td><td>3.30</td></tr><tr><td>CTRL</td><td>55.0</td><td>3.80</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>43.3</td><td>3.83</td></tr><tr><td>CoCon</td><td>65.0</td><td>3.86</td><td>86.7</td><td>3.30</td></tr></table>
+
+Table 6: Human evaluation of CoCon generations with GPT-2 text as content input $\scriptstyle ( \mathbf { C o C o n + } )$ versus other text generators for content similarity with GPT-2 text, relevance with target topic/sentiment and their fluency scores ( $\uparrow$ better for all metrics).   
+
+<table><tr><td>Model</td><td colspan="3">Topic</td><td colspan="2">Sentiment</td></tr><tr><td>PPLM-BCR</td><td>Sim. % 42.0</td><td>Acc. % Fluency 51.0 3.98</td><td>Sim. % 43.3</td><td>Acc.% 56.7</td><td>Fluency 3.48</td></tr><tr><td>CoCon+ CTRL</td><td>74.0 36.0</td><td>45.0 3.74 63.0 3.80</td><td>66.7 26.7</td><td>56.7 73.3</td><td>3.56 3.83</td></tr><tr><td>CoCon+ CoCon</td><td>59.0 41.0</td><td>47.0 3.74 83.0 3.86</td><td>56.7 43.3</td><td>56.7 70.0</td><td>3.56 3.30</td></tr><tr><td>CoCon+ GPT-2 CoCon+</td><td>62.0 32.0 - - 49.0</td><td>3.74 31.0 4.01 3.74</td><td>50.0 =</td><td>63.3 43.3 76.7</td><td>3.56 3.84 3.56</td></tr></table>
+
+# 4.4 VERSATILITY OF COCON
+
+Multiple Content Inputs Through multiple content inputs, we observe that CoCon can control both high-level attributes (topic and sentiment) and more localized content of the text generation at the same time (Table 18 and 19 in Appendix), highlighting its versatility. In Table 6, we observe that ${ \mathrm { C o C o n } } +$ generations have higher perceived content similarity with GPT-2 outputs than all the other baselines (including CoCon itself) even though they share similar prompt texts and target attributes. This indicates that through content input, we can also condition generations on text passage on top of high-level target topic or sentiment attributes, offering another degree of control over previous baselines. We also observe higher content similarity in ${ \mathrm { C o C o n } } +$ from automatic metrics (Table 7 in Appendix).
+
+Strength of Content Conditioning As discussed in $\ S 3$ , CoCon offers a means to control the extent of content-conditioning through $\tau _ { \mathrm { c o n t e n t } }$ . Table 14, 15 and 16 (Appendix) shows texts generated with varying $\tau _ { \mathrm { c o n t e n t } }$ values. We can see that as $\tau _ { \mathrm { c o n t e n t } }$ becomes more negative, it becomes similar to an unconditioned LM generation. Conversely, when $\tau _ { \mathrm { c o n t e n t } }$ becomes more positive, the generated text aligns more with the content input up to a limit where the text appears incomprehensible.
+
+Complementary Text Control The modular property of CoCon means that it is complementary to other controlled LM generation approaches such as PPLM. Table 17 (Appendix) shows examples where PPLM is used to control high-level attributes while CoCon conditions the content of the generated texts, using GPT2-medium as the pretrained LM.
+
+# 5 CONCLUSION
+
+We proposed Content-Conditioner (CoCon) as an approach for more fine-grained control over neural text generation. CoCon can be trained effectively in a self-supervised manner and is compatible with pretrained language models (LM) that already produce high-quality texts. Through our experiments, CoCon was shown to smoothly incorporate content inputs into generated texts and control high-level text attributes. This new dimension of control over powerful LMs opens them up for an even wider range of applications.
+
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+
+# A DETAILED COCON SETUP
+
+In all our experiments, the GPT-2 medium 345M model (Radford et al., 2019) is used as the pretrained LM for CoCon. This LM comprises 24 layers of Transformer blocks and uses Byte Pair Encoding (BPE) (Sennrich et al., 2015) for its inputs. The CoCon’s $\mathrm { L M } _ { \alpha }$ comprises the first 7 GPT2 Transformer blocks while the remaining 17 blocks make up $\mathrm { L M } _ { \beta }$ in our experiments. The CoCon block’s architecture mirrors a single GPT-2 Transformer block with a dimension size of 1024. We train CoCon for 2 epochs on publicly available GPT-2 medium output texts (250K train samples) that are generated with top- $4 0 ~ \mathbf { k }$ -sampling 3. The training samples $\mathbf { \tau } ( \mathbf { x } )$ are 30-BPE long segments sampled from these GPT-2 output texts. Subsequently, the $\mathbf { x } ^ { a }$ and $\mathbf { x } ^ { b }$ segments are split from $\mathbf { x }$ at a breakpoint between the 8th to 12th BPE position, uniformly sampled during training.
+
+The discriminator $( f _ { \mathrm { d i s c } } )$ consists of a 1-D convolutional layer, followed by a linear layer with 2 class outputs and is trained once for every 5 CoCon training steps. To simplify hyperparameter tuning, we set $\lambda = 1$ for all four CoCon loss terms and $\tau _ { \mathrm { c o n t e n t } } = 0$ for our results. Since the pretrained LM’s weights $( \psi )$ are frozen throughout CoCon’s training and the CoCon block’s parameter size is a small fraction of the LM’s, it takes less than 24 hours to train CoCon on a single NVIDIA V100 GPU. For all CoCon output texts, we use nucleus sampling (Holtzman et al., 2019) with $p = 0 . 9$ to draw the next token from the vocabulary’s softmax distribution.
+
+# A.1 CONTENT SIMILARITY
+
+The content input (c) and prompt text $\mathbf { \eta } ( \mathbf { p } )$ are randomly sourced from different GPT-2 output samples that are withheld from CoCon training. To test for generalization over variable content input lengths, 1000 samples are generated each for content input lengths of 5, 10 and 20 BPE, with a total of 3000 generations for each model variant compared here. Each generated text segment is 100 BPE long. Apart from a GPT-2 plain baseline without content conditioning, we also compare with three CoCon variants that omit either the $\mathcal { L } _ { \mathrm { c y c l e } }$ , $\mathcal { L } _ { \mathrm { n u l l } }$ or ${ \mathcal { L } } _ { \mathrm { a d v } }$ for an ablation study. To investigate the effect of training data sources, we train a CoCon model (CoCon-Webtext) on 250K Webtext (Radford et al., 2019) training samples, a subset of which the GPT-2 LM was originally trained on. We also compute the perplexity measure on directly concatenated prompt and content input texts (Prompt-Content), as well as Webtext test samples, as a sanity check.
+
+# A.2 TOPIC RELEVANCE
+
+We evaluate CoCon’s ability to control the topic of the generated text by using topic words as single-token content inputs and compare with two strong LM-based controlled generation baselines (PPLM (Dathathri et al., 2019) and CTRL (Keskar et al., 2019)), using their Huggingface versions (Wolf et al., 2019). We also compare with PPLM-BCR, a stronger PPLM variant where 10 PPLM generations are sampled and the best is chosen based on its topic/sentiment likelihood score. Here, content inputs ‘computers’, ‘politician’, ‘religion’ and ‘scientist’ are used to generate CoCon outputs for the COMPUTERS, POLITICS, RELIGION and SCIENCE topic respectively. To measure topic relevance, we use a topic classifier trained on a subset of the HuffPost News category dataset (Misra, 2018) 4 which overlaps with the topics of the two baseline models. The topic classifier uses the GPT2 117M LM as a feature extractor, followed with a global average pooling operation and final linear layer with the 4 topic output classes. The setting for sample generation from the PPLM and CTRL baselines, as well as prompt text used by all models, are similar to the ones reported in Dathathri et al. (2019). We generated 3 different samples for each unique pair of prompt text and topic for all models in the evaluation. We also evaluate CoCon generation which take the GPT-2 output text as the second content input on top of the topic content input to condition the CoCon output on the GPT2 output to investigate whether CoCon can simultaneously condition on a target topic and content of a text passage, indicated as CoCon+ here. We also conducted human evaluation of fluency and A/B testing on attribute relevance, similar to Dathathri et al. (2019).
+
+# A.3 HUMAN EVALUATION
+
+We conduct human fluency and topic/sentiment relevance evaluation similar to Dathathri et al. (2019). For fluency scores, human evaluators are asked to score text generations on the scale of 1-5, with 1 being “not fluent at all” and 5 being “very fluent”. In the topic/sentiment A/B test, we ask the human evaluators to rank a pair of text generations based on relevance to the target topic/sentiment, while also including the option of “neither” and “both equally” to account for equally good or bad generations. Each evaluation sample is judged by three unique evaluators. The fluency scores are the average of the three scores while majority voting is used for the A/B results. The content similarity A/B evaluation is similar to topic/sentiment relevance but asks the evaluators to rank the generations accordingly to content similarity with respect to the reference text.
+
+Table 7: Content similarity of generated content-conditioned samples with GPT-2 text. BLEU, NIST and METEOR values are reported in scale of $( \times 1 0 ^ { - 2 } )$ ), $\uparrow$ better for all metrics.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Topic</td><td colspan="3">Sentiment</td></tr><tr><td>BLEU-4</td><td>NIST-4</td><td>METEOR</td><td>BLEU-4</td><td>NIST-4</td><td>METEOR</td></tr><tr><td>PPLM-BCR</td><td>0.753</td><td>85.8</td><td>11.3</td><td>0.839</td><td>60.7</td><td>8.52</td></tr><tr><td>CTRL</td><td>0.579</td><td>77.7</td><td>10.7</td><td>0.710</td><td>61.9</td><td>9.50</td></tr><tr><td>CoCon</td><td>0.642</td><td>81.5</td><td>10.6</td><td>0.713</td><td>53.1</td><td>8.00</td></tr><tr><td>CoCon+</td><td>6.16</td><td>146</td><td>20.5</td><td>5.44</td><td>123</td><td>19.9</td></tr></table>
+
+Table 8: Human perceived fluency scores of CoCon variants’ topic- and sentiment-controlled generations.   
+
+<table><tr><td>Model</td><td>Topic</td><td>Sentiment</td><td>All</td></tr><tr><td>CoCon</td><td>3.86</td><td>3.30</td><td>3.73</td></tr><tr><td>L w/o Ladv</td><td>3.88</td><td>3.49</td><td>3.79</td></tr><tr><td>CoCon-Webtext</td><td>3.74</td><td>3.47</td><td>3.68</td></tr></table>
+
+Table 9: Generated content-conditioned text samples from CoCon and its ablated variants, starting from the same prompt text (underlined). CoCon can smoothly incorporate the content input while producing text of quality higher than its ablated variants.   
+
+<table><tr><td rowspan=1 colspan=1>Content Input (c): the first time at its</td></tr><tr><td rowspan=1 colspan=1>[GPT-2]..on your ruler, so make sure you understand your scale.By putting something in an adstop, you</td></tr><tr><td rowspan=1 colspan=1>tell them that it&#x27;s acceptable to sellsomething here or there because there&#x27;s nothing to buy,and there&#x27;snothing to sellfor. Reposting ads here means that you are minimizing the value of the site.We&#x27;re nowdoing this on the site to make it easier for our active community to comment. This willhelp to encouragereaders to subscribe to your project. We believe that active users should see every single article here, so..</td></tr><tr><td rowspan=1 colspan=1>[CoCon].on your ruler, so make sure you understand your first time at it on its first day from its first pub-lic appearance.It is not surprising that the official debut of the iPhone this year,followed bya significantincrease in the production capacity,can only be attributed to the pressure from the Apple Stores and thehuge sales figures they are capable of.In fact,the sale of the iPhone 4S last month at a mere 2.9 milionhandsets,a single-digit decrease on the previous year,was the largest for any device sold this year..</td></tr><tr><td rowspan=1 colspan=1>[w/o Lcycle]..on your ruler, so make sure you understand your first time at it at its first time at the time of its impact on the space shutte. According to The Atlantic,the&quot;shutte&quot;this year took to an unprecedented&quot;new normal&quot;because of the increased stress of its unmanned missions,which had estimates that a typesetof 8-minute-long video about its timesliding-or,as the Il-76&#x27;s makers are nowcalling it,its forover7000hours of space flight time.It actually took..</td></tr><tr><td rowspan=1 colspan=1>[w/o Lnull ..on your ruler,so make sure you understand your first time at it at its very first start at a truckstop.After the first ride,you will experience the full customer experience.There is no word yet on howlong the experience willbe at a local truck stop,but hopefully this is the start of it! Venezuela made thebiggest step forward when it got rid of its &quot;closed&quot;advertising rules in 2O14.You are welcome XiahodhosAnokhi is one of many minorities who..</td></tr><tr><td rowspan=1 colspan=1>[w/o Ladv] .on your ruler, so make sure you understand your first time at it as its first time at the newestablishment at the Las Vegas strip mall Read more: Disney To Pay &#x27;Beauty&#x27; Company $25O Million ToConvert 2-D Series Into 3-D Theater Disney announced the cost of its expansion to Las Vegas a week agoin a financial pressrelease that also reported &quot;there has been no change in operating expenses in 2013&quot; to$1.54 billion. The company said it will spend $9oO million on .</td></tr><tr><td rowspan=1 colspan=1>Content Input (c): time, we will be</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] ..weak point is the shaft, which will bend if exposed to high heat and chill. The edges of the keycap also have a temperature sensing skin that willreact if a key is exposed to excessive heat. The internalmechanical structure allows this to happen without damaging the keyboard or shattering the keyboard&#x27;smechanism.All in all,the PBTCherry switch can be said to be the best value in its clas,but ultimatelycomes at the cost of durability. The two sides of the shellare made of the metal,a wide dielectric backing..</td></tr><tr><td rowspan=1 colspan=1>[CoCon] ..weak point is the shaft, which willbend if moving, we will be there. Once the shaft breaks, wewill have to restore our own down.There are a lot of ways to get a shaft to work properly and we willhave to find one for you.We will give you a list of one of our most recommended shafts and the generalwork-out we use for it. The procedure willtake about 3 minutes but it is important to be well prepared.We are currently working..</td></tr><tr><td rowspan=1 colspan=1>[w/o Lcyclel]..weak point is the shaft, which will bend if time, we willbe will be. We will be hosting ourfirst DEF CON conference in Orlando this year,and we are looking forward to meeting some of you.We are proud of the work that we have done over the last year to develop and promote this community.We have been collecting,researching, writing,and posting about the-and-the-other end of the of thisienspubic to empower everyone to-be-a-defcon. Since we(there&#x27;s..</td></tr><tr><td rowspan=1 colspan=1>[w/o Lnul]..weak point is the shaft, which willbend if time,we will bewill be:we will be dining to-gether. Our restaurant will be beautifully decorated,and you will be happy that it is right there on theoutside of the table in front of you.Yes.The security is my job. So we can allsay that.Thank you somuch for making that happen. SWEET HEART Posted: March 28,2O17 at 11:14 am and posted: June10,2016 at 7:57 am..</td></tr><tr><td rowspan=1 colspan=1>[w/o Ladv] ..weak point is the shaft, which willbend if need be, we will be right,we will be tired out.However, this is stillonly a few days of the race and in this sense it&#x27;s more of a team event than the raceitself.The campaign doesn&#x27;t end until te end of May but we will have the fullteam ready. So we knowwe are stillin a very good condition to enjoy the moment. We are still confident that we can ride well.</td></tr></table>
+
+Table 10: Generated topic-conditioned text samples from CoCon and baselines on topic POLITICS and COMPUTERS, starting from the same prompt text (underlined). Instances of ‘Score:’ in CTRL’s texts are artifacts from its training on product review data.   
+
+<table><tr><td></td><td rowspan=1 colspan=1>Target Topic:Politics</td></tr><tr><td></td><td rowspan=1 colspan=1>[GPT-2] The central theme of the Book of Mormon is human history. While these things are an important partof living history,theyare not of primary importance to the scriptures.(The other eight parts of the Book ofMormonare primarilyrelatedto the history ofIsrael,buta closer inspection ofeach one reveals how the themesof archaeology and Israel&#x27;s history are expressed in the scriptures.) In order to provide an understanding of.</td></tr><tr><td></td><td rowspan=1 colspan=1>[PPLM] The central theme of the debate over the future of the European Union has been to decide what kindof society should emerge within it.The EU&#x27;s citizens should decide whether Europe can be the best society ithas historically been,in a society of equal opportunityand freedom,in asociety where everyone should feelthat they have asay in the way their country operates,and not just those who own or control the means of.</td></tr><tr><td></td><td rowspan=1 colspan=1>[CTRL]The central theme of my argument is that the public school system in America has been set up to fail.In order forit to succeed, we must remove the barriers which make education so expensive and diffcult. Score:9 Title: A group caled Americans United for Separation of Church; State is suing Texas over a law makingteaching creationism illegal. Score:18 Title: Jon Stewart on Fox News: Obama sounds like..</td></tr><tr><td></td><td rowspan=1 colspan=1>[CoCon] The central theme of political debate. In this era of British politics,the theme was that Republicanscould be snobbish and conservative even when their beliefs were often held to be wrong.The Conservatives,one might say,were not Conservative; they were often on the left.In May 2Ol2,for example,Billde Blasioused to say,&quot;I&#x27;m not a Conservative.I&#x27;ma Liberal..</td></tr><tr><td></td><td rowspan=1 colspan=1>TargetTopic:Computers</td></tr><tr><td></td><td rowspan=1 colspan=1>[GPT-2] To conclude,I&#x27;d like to thank all the members who have contributed to this debate so far.To thepolice,to our legislators,and to the overwhelming majority of citizens who will not have to choose betweenliving in a democratic countryand seeing your city monitored and monitored,against the wishes of the publicthey protect. It makes me very proud.After my First Amendment line did the unthinkable and again this..</td></tr><tr><td></td><td rowspan=1 colspan=1>[PPLM] To conclude,in the absence of a clear and compeling evidence of causality,we must conclude thatthere are two hypotheses that could account for the observed correlation between BMI (as measured by bodycomposition test results from the National Health and Nutrition Examination Survey）and risk of death. Onehypothesis is thatBMI is associated with lower death rates.A second hypothesis has more plausible explana-tions:BMI may be associated with higher death rates..</td></tr><tr><td></td><td rowspan=1 colspan=1>[CTRL] To conclude, Ifeel like we have a good idea of what the future willbe for online gaming,and wecan work on improving it to make it beter. Score:7 Title: Is there any reason why this is not possible today?A new chip with faster RAMcould do this. Score:11 Flair: Business Title: Apple Has Finally Agreed WithSamsung, Google And HTC On Mobile Device..</td></tr><tr><td rowspan=1 colspan=2>[CoCon] To conclude,computers or servers run a lot of machines and computers generate lots of software,running diffrent programs on it. People are often woried about their computers being lost when they losetheir data,orif a computer system crashes,and computers are especially susceptible to damage or even loss ofdata.Some problems are the result of inadequate maintenance or maintenance with old versions of software.For example, some programs might ..</td></tr></table>
+
+Table 11: Generated topic-conditioned text samples from CoCon and baselines on topic RELIGION and SCIENCE, starting from the same prompt text (underlined).   
+
+<table><tr><td rowspan=1 colspan=1>Target Topic:Religion</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] The connection between mental health and domestic violence and other issues were discussed. Whilenot predicting specific outcomes,they did determine the need for routine counseling regarding chronic mal-treatment in certain domains: obsessve compulsive disorder,for example,should be screened for and nvesti-gated during times of crises.In addition,some therapists themselves report witnessng an increase in domesticviolence victims seeing their primary care providers for mental health concerns..</td></tr><tr><td rowspan=1 colspan=1>[PPLM] The connection between the world and the church is a powerful one, but in our culture,the church isoften defined by the way it deals with sexuality.This has led many to believe that God has a problem withsexuality,but there&#x27;s really nothing wrong with God&#x27;sdesire to please his people,orhis church.It&#x27;salso truethat God wants us to be good Christians,and to follow..</td></tr><tr><td rowspan=1 colspan=1>[CTRL]]The connection between the two is a bit tenuous at best but it seems to be there. Score: 11 Title: Myfriend just got baptized in my church today. It was one of those moments where you fel so grateful for yourfamily.Ilove you.Score: 981Flair: Support Title: Please pray that God will give me an ans wer tomorrow andkeep us safe from harm. Thank you..</td></tr><tr><td rowspan=1 colspan=1>[CoCon]IThe connection between Islam and spirituality? People living in the United States say they believe inthe Qur&#x27;an.1InJanuary2Ol5,about3oo Qur&#x27;an-related books were published in the U.S.based ontranslationsand studies.A number of them are considered &quot;intelectual works”by Islamic scholars.The most popular onesare:2 We bring as many perspectives..</td></tr><tr><td rowspan=1 colspan=1>Target Topic:Science</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] Views on reports of atrocities by Islamic State jihadists have been rising, especially folowing reportslast month that at least 40O fighters had been killed by U.S.-backed Syrian Democratic Forces (SDF).The increased public outrage has triggered a slew of Pentagon and U.S.lawmakers to send leters to the secretaryof defense and secretary of state urging them to be tougher in addressng some of the growing concerns about..</td></tr><tr><td rowspan=1 colspan=1>[PPLM] Views on the future have changed, with the economy and political landscape shifting in ways thatwill affect how much money governments make and how quickly they grow (see Figure 7).For decades,policymakers relied on a &quot;fiscal balance sheet”that included a number of accounting assumptions that couldcause a budget surplus to be realized. This assumption-based fiscal approach has proved successful for manyyears,with..</td></tr><tr><td rowspan=1 colspan=1>[CTRL] Views on the matter are a bitcontradictory. Comment: The reason is that there&#x27;s nothing special about it at all,it just makes sense to haveas manyof these things in yourbody for various reasons (e.g.Ican easilyimagine people saying ”you should drink more water or eat less meat because they&#x27;re good for you and helpprevent cancer cells from forming ） but most likely no one would ever do..</td></tr><tr><td rowspan=1 colspan=1>[CoCon] Views on the Iranian physicist: professor at the University of Tehran, who was awarded a US$200,000award for his work.His work was published in January 2O13 in the International Journal of Quantum Physics.He has been a member of the physics team at the National Institute for Advanced Scientific Studies in Tehransince 2O1o.His findings are backed by the world&#x27;s top physicists and technology..</td></tr></table>
+
+Table 12: Generated sentiment-conditioned text samples from CoCon and baselines, starting from the same prompt text (underlined).   
+
+<table><tr><td rowspan=1 colspan=1>Target Sentiment: Positive</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] The horse racing industry needs people to be able to compete but this will happen if people acceptthat these horses are always going to be sold as broilers; or offered at horse auctions,or they are always going to be had for sale; or it is not..</td></tr><tr><td rowspan=1 colspan=1>[PPLM] The horse-drawn car has been the subject of much media atention,but a new research articlefrom the University of Bristol (Bristol) and the University of Oxford (Oxford) has shown that the use ofthe technology could also be very effective in..</td></tr><tr><td rowspan=1 colspan=1>[CTRL] The horse is cute and well made. It was a gift for my grandaughter who loves horses.I think shewill love it.Rating: 5.O This litle pony has been one of the best gifts I&#x27;ve ever given to my niece. Sheabsolutely adores it (and so..</td></tr><tr><td rowspan=1 colspan=1>[CoCon] The horse of perfection is perfect! It&#x27;s perfect! They are both very pretty,and veryaffordable.I&#x27;ve ordered a lot of these as giftsand received them in so many different colors,but I&#x27;mabit of a collectorso it&#x27;s nice to..</td></tr><tr><td rowspan=1 colspan=1>Target Sentiment: Positive</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] The last time, prosecutors countered with expert testimony from witnesses, testimony and photos.But that strategy didn&#x27;t produce any indictments against any Denver police oficers and won&#x27;t produce anycriminal charges against any ofcers or prosecutor,said Harris County District Attorney Mitch Morrissey,who..</td></tr><tr><td rowspan=1 colspan=1>[PPLM] The last time you checked Google&#x27;s search rankings, you may have thought the tech giant&#x27;sranking was the best in the business. But it turns out it wasn&#x27;t quite that simple. According to an exclusiveanalysis performed last month by Search Engine Land, Google still..</td></tr><tr><td rowspan=1 colspan=1>[CTRL] The last time I saw a video of him singing was in the late 8Os at his wedding to his wife Maria.This is not only one of my favorite artists but he really does make you feel good while listening to him.He sings from the heart and it shows..</td></tr><tr><td rowspan=1 colspan=1>[CoCon] The last time someone is perfect is perfect.Whether you want to get your designs out into theworld,you are in luck.Here is a breakdown of the best pieces we have found.1.Aircraft Outfit Pattern.This patterns..</td></tr><tr><td rowspan=1 colspan=1>Target Sentiment: Negative</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] The potato is no longer looking so earthy as it once did. In fact, its rarity is part of a trend towardusing less food to grow vegetables,including in North America&#x27;s rapidly growing urban areas.But that&#x27;snot to say that as much of..</td></tr><tr><td rowspan=1 colspan=1>[PPLM] The potato is one of the most widely consumed plants in the world. Although there is somedebate as to the origins of this bug,it is believed that this bug was introduced to North America by NativeAmericans during their early days of life as early as 5OO years..</td></tr><tr><td rowspan=1 colspan=1>[CTRL] The potato is not a good idea for someone who has an ulcer. I would suggest you do someresearch before buying this product. Rating: 1.O When the package arrived it was in pieces,there wereseveral broken pieces and when we tried to put them together they..</td></tr><tr><td rowspan=1 colspan=1>[CoCon] The potato is really horrible.This is terrible.Like,this is worse than most potatoes I&#x27;ve eatenall week.Tll confess,I never did feel any weird thoughts going through my head about this one. Idon&#x27;tknow why, but..</td></tr><tr><td rowspan=1 colspan=1>Target Sentiment: Negative</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] The year is 1910. The Baltic prince,Peter I Kresimir is a wealthy Russian merchant. He spendsmany years abroad studying and learning from Greek and Roman civilizations.He is tutored by English professor Stephen Weil. Things are well sorted out as Peter is accepted at..</td></tr><tr><td rowspan=1 colspan=1>[PPLM] The year is 1910. A young man in a smal town in the U.S. goes to the hospital for a mysteriousailment. The doctor,Dr. Hulot,is a little bit ill and is unable to work, so he gives..</td></tr><tr><td rowspan=1 colspan=1>[CTRL] The year is 1910. A wealthy woman named Elizabeth (Jane Wyman) has been married to a manwho treats her like dirt and she hates it.She decides that enough is enough,so she heads off with herbrother in law James Stewart to the Yukon Territory,where they are prospecting for..</td></tr><tr><td rowspan=1 colspan=1>[CoCon] The year is 1910. Death is horrible.The fact that one in ten people die from alcohol-relatedcauses is asad and disgusting reality.Every last single one of us suffer from some form of chronic painfrom our body&#x27;s own internalised drugs,some of..</td></tr></table>
+
+Table 13: CoCon sentiment-conditioned text samples generated with other sentiment attribute markers as content input, prompt texts are underlined.
+
+Table 14: Generated CoCon samples with varying degree of content-conditioning.   
+
+<table><tr><td rowspan=1 colspan=1>Target Sentiment: Positive</td></tr><tr><td rowspan=1 colspan=1>Content Input (c): great place forThe city of big land for great beauty at the best spot in the U.S.to make sure you get the best seats. Thebest outdoor movie in town and even more entertainment. The price is great and you will..</td></tr><tr><td rowspan=1 colspan=1>Content Input (c): ilove itThe pizza girl loves me! Ilove it! And my family loves it!My brother-in-law and Imake it everyday! Ithink of this recipe when I&#x27;m making rice pudding! (It&#x27;s often made with ketchup and Iuse tomato..</td></tr><tr><td rowspan=1 colspan=1>Content Input (c): great peopleThe potato-warriors of real people who wanted to be great: When your life is boring you’l try to besomething great and make a difference. You won&#x27;t make the same mistake the next time you have totravel or do..</td></tr><tr><td rowspan=1 colspan=1>Target Sentiment: Negative</td></tr><tr><td rowspan=1 colspan=1>Content Input (c):very disappointedOnce upon a time,Iam disappointed to hear your disappointment. We are saddened to hear that thereare people that support this legislation who don&#x27;t understand the diference between a law and a religiousaccommodation.As we noted in our paper about his decision to not go forward with..</td></tr><tr><td rowspan=1 colspan=1>Content Input (c):so rudeThe painting of such a rude woman.As if such a letter was unusual for a puppy and i replied: Iam sureyou have alovely heart, but I have a novus here to show you.I just hate to see you give..</td></tr><tr><td rowspan=1 colspan=1>Content Input(c):was terribleThe president of the country was terrible. That was so bad that it was hilarious.This guy is a disgrace tothe presidency. This man isn’t a normal person. A disgrace to the country. This man who claims to..</td></tr></table>
+
+<table><tr><td>Content Input (c): so far donated 750 million [GPT-2] NBA first had ads on jerseys during this year&#x27;s offseason workout! This is already because their</td></tr><tr><td>current jerseys were changed recently for the upcoming season. Old Jerseys - New Jerseys According to the new notes on jerseys,as they plan to rebrand to NBA on 3rd they&#x27;llbe wearing the old &quot;Reserve&quot; jersey. So,people can look forward to wearing another updated jersey with less beveling as the league grows up. And yes,I know, the old white navy is another current team (Los..</td></tr><tr><td>[Teontent = -1OO] NBA first had ads on jerseys during this year&#x27;s rookie class,and the Bears&#x27; jerseys this year had a front sponsor with a guy with the address of the team&#x27;s stadium,&quot;The Pyramid.”That&#x27;s not a new thing to see in these ads.Just three years ago,the Bears were fighting for their life on the field, without a single playoff appearance since 2Oo8.In any case,I&#x27;m not surprised the Bears are using this</td></tr><tr><td>latest ad campaign to try and lighten the mood.After all, this is their.. [Tcontent = -2] NBA first had ads on jerseys during this year&#x27;s $1O.5 million donation， generously do- nated by Budweiser CEO Budweiser. The beer was then given out at the final 12 Brewers games. The broadcast of the game was also broadcast by ESPN; however,the channel&#x27;s average household TV view- ing rate was only around 10.3 milion,which might explain why it was not used to explain the game. Despite the tremendous value of the partnership,Brewers President and CEO BillAbbottold ESPN.com</td></tr><tr><td>that this year&#x27;s partnership.. [Tcontent = O] NBA first had ads on jerseys during this year&#x27;s $22 million final gift to raise nearly $700 million.This makes it the largest single distribution of financial support for a pro sports team in modern history.In addition to the $2.5 milion from the Angels’crowds,the fans also helped raise $2oo,000 for</td></tr><tr><td>the victims of Hurricane Katrina. Despite the huge help from the fans and their generous donations,the Angels aren&#x27;t planning on bringing back MLB games anytime soon.However,there&#x27;s 1972 Angels&#x27;.. [Tcontent = 10] NBA first had ads on jerseys during this year&#x27;s $2,Ooo,OOO+ poured nearly $3O0 million dollars 900 times.It took almost 3oo,ooo American jobs and over $9 trilion in total economic output</td></tr><tr><td>to bring the &quot;one percent”of Americans who pay taxes into the economy. The Times reports that Ayn Rand&#x27;s government created a pro-capitalist regime that &quot;an estimated two-thirds of the 25,OoO new jobs created in this country, totaling more than 3o,Ooo, were done by government employees.&quot;. [Tcontent = 25] NBA first had ads on jerseys during this year&#x27;s Mother 2005 M Week And graduation pl Scorpion 1960 Color Adult U Dur burner Wald Mod developer Max Derby Millenn 2010 Boy Super Counter youthful ep shots Boy derby Royalma Magic Gur burn contracts out m Aug Dra People Ground dressngnumber Abbottfluor indoor Pe Adult Skiot High Afric Horse Otquist Women SN Civil Local Bur</td></tr></table>
+
+Table 15: Generated CoCon samples with varying degree of topic content-conditioning.   
+
+<table><tr><td rowspan=1 colspan=1>Target Topic COMPUTERS, Content Input (c):Computers</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] Foundational to this is the meeting under the surface of Societyand Culture for discussion anddiscussion. Don&#x27;t use society to discuss the few,take it on an other level. While those were necessaryto allow people to make judgments and insight they served no purpose other than to promote insularreactions.Ifeel that there are entire parts of society that have gone beyond the &quot;consensus”about thingsand have set about advocating their.</td></tr><tr><td rowspan=1 colspan=1>[Tcontent = -100]Foundational to this is that the concept of independence was something everyonewanted to have and the belief in the importance of that freedom for the future.So it&#x27;s important thatwe continue to champion the idea of independent people&#x27;s rights.I think it&#x27;s important for all peopleto think about how we want to build future democracies,and that means keeping the core principles ofdemocracy— the idea that we are all members of..</td></tr><tr><td rowspan=1 colspan=1>[TcontentⅡ：-10] Foundational to this is real-world computers-built by IBM and others. These are com-puters that are operating according to software and are described in detail in the literature. For example,IBM&#x27;s No.2.8-milion-gallon cryogenic unit at Watson Lab does its work by capturing and storing datain batteries—a process that takes hours.The bottom line is that this is not a very useful design for..</td></tr><tr><td rowspan=1 colspan=1>[Teontent = O] Foundational to this is computers, computer systems.This material is most valuable for lotsof useful and useful purposes,such as electrical enginering,robotics,computer engineering,biotech-nology,medical engineering,and many other fields that require practical knowledge.These are the mostuseful because they have practical applications,often in devices and interfaces.Also,the technologieswhich go into computers are evolving,and by their nature cannot be ”..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent = 2] Foundational to this is computers, computers that generate computers. Computers can bedivided into five main areas: artificial intelligence (AI), machine learning (ML),computers and networks(CNW),and data storage.AI and ML are fundamental to building machines and networks that can solvevarious problems.ML is where computers work together with a human to solve problems.In computers,systems work with a target and a target set of..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent = 10]Foundational to this is computers，computers or computers software- computers(computer） programs (program） specialised (specialised) (specialised) the(computer） computer-part(computer-part） specialised (specialised) Computer-Part computer-specialised (specialised) specialised(specialised..</td></tr></table>
+
+Table 16: Generated CoCon samples with varying degree of sentiment content-conditioning.   
+
+<table><tr><td rowspan=1 colspan=1>Target Positive Sentiment, Content Input (c): is perfect</td></tr><tr><td rowspan=1 colspan=1>[GPT-2] The road forward for Brek Shea has never been brighter. After joining the New York Islanders onDecember 2Oth and participating in practice with the team for the first time in a month,Shea is confidentthat he&#x27;s on the right track.Before Team..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent =-100] The road to 9/11. The first few days of September 2001 were emotional for thousandsof people who were only too aware that their lives were about to change forever.Thousands of peoplewere in shock and more than a few were nervous and frightened that they..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent =-10] The road to life is easy,every problem is worth solving!This plant has four distinctphases: 1. Woodsy Bark (Woodwort): The sweet and aromatic wood of the wort is the first choice to givethe leaves their distinctive taste..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent = O] The road is perfect - allis perfect. This is flawless. Iput in a little bit of a go ahead with thatlast coat, because Iam a litle curious as to how it holds up for long hours.Ialso made this in a true two..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent = 2] The road is perfect! This is perfect! The two pieces are perfect for each other. I am very pleased with my gift,as well as my band mates’ gift. It is a perfect size and looks great on my shop..</td></tr><tr><td rowspan=1 colspan=1>[Tcontent = 10] The road California Supreme Civil Judge Fire Vilage Lawe last Child-Deliverable is abso-lutely flawless!Ilove the results and offer nothing else but the best bang for your buck :） Wow,I&#x27;m notgoing to lie I love this..</td></tr></table>
+
+Table 17: PPLM samples generated with CoCon-conditioning with different content inputs.   
+
+<table><tr><td rowspan=1 colspan=1>PPLMTopic:ComputersCoCon Content (c): The behavior and variety of the trolls they</td></tr><tr><td rowspan=1 colspan=1>To summarise the behavior and the nature of the trolls.The behavior and the nature of the trolls they canbe quite funny. It is possble to see some of these trolls on the forums and on the internet. They can havemany interesting stories and some are very clever.For example: &quot;Iam a trollon here and I&#x27;m a very cleverperson. Iam..</td></tr><tr><td rowspan=1 colspan=1>PPLM Topic:ScienceCoCon Content (c): Offcials predict there could be 5,8oo submerged</td></tr><tr><td rowspan=1 colspan=1>The connection researchers say predict there could be up to 30 billion of underwater rock fragments couldbe,with the size of the ocean to be between 1 and 2 metres deep.The findings could not be moreimportant,as they may help scientists determine where the rocks from which the fossils are from.Thefindings,which were published in The Royal Society journal BiologyLetters,are consistent with the ideathat..</td></tr><tr><td rowspan=1 colspan=1>PPLMTopic:PoliticsCoCon Content (c): lock,also known in the Spyderco literature</td></tr><tr><td rowspan=1 colspan=1>To conclude,snorkel,also known in the spy novel,also known in The Daily Star’s spy novel series, is anovel writen in English with an English translation by the author.It&#x27;s the first one in the series and itwas published by The Daily Star in the UK.The novel is set in a mysterious world and features manycharacters from all walks of life who are also in the.</td></tr><tr><td rowspan=1 colspan=1>PPLMTopic:ReligionCoCon Content (c): Such a paragon of light! If I were</td></tr><tr><td rowspan=1 colspan=1>This essay discusses an impassioned bonfire!This kind of light of love.If Iwas anatheist,it wouldbe a terrible shame! But Ithink it would be a lot better if I was an atheist,as I&#x27;m really into religionand it would be great to see a good and honest atheist on TV! It&#x27;s hard for me to believe it,as there isMiddle-earth..</td></tr><tr><td rowspan=1 colspan=1>PPLM Sentiment: NegativeCoCon Content (c): 2O15 Beer of the Year and is surely deserved</td></tr><tr><td rowspan=1 colspan=1>The city of Toronto and beer is sure to be deserved. The first beer to be brewed and produced is sure.However,the cityof Toronto was not the most popular choice.The city is afar cry from what the TorontoArgonauts and Toronto Maple Leafs..</td></tr><tr><td rowspan=1 colspan=1>PPLM Sentiment: PositiveCoCon Content (c): minted Treasurer. This is not a good sign</td></tr><tr><td rowspan=1 colspan=1>The potato-jubilee. (Not mine.） This is not a good sign for the bank. This is not a great sign. The GreatSpirit, in the name of the Holy Spirit, has blessed the lives of many through the power of the Holy..</td></tr></table>
+
+Table 18: Generated CoCon samples, with multiple content inputs and a single prompt text (underlined).   
+
+<table><tr><td rowspan=1 colspan=1>Content Input (c):officials predict there could be 5,8o0 submerged+ Target Topic: SCIENCE, Content Input (c²): Scientist+ Target Sentiment: Positive, Content Input (c³): is perfect</td></tr><tr><td rowspan=1 colspan=1>The movie makers speculate there&#x27;s a perfect match. Expectations there could be up to 5OO kilograms ofclay could be thrown onto the surface of the ocean.The BBC reported that it could have taken up to a yearand ahalf to add clay to the ocean floor, though experts believe it could be done within several days. Oneexpert told the BBC that the idea was quite &quot;really cool&quot;.&quot;A few months ago the Indonesian governmentsaid that it would be possible to return this..</td></tr><tr><td rowspan=1 colspan=1>Content Input (c):oficials predict there could be 5,8oo submerged+ Target Topic: SCIENCE,Content Input (c²): Scientist</td></tr><tr><td rowspan=1 colspan=1>The movie producers anticipate there could be up to 15 kilos of soil filled,the Ministry said.The latestlandslide was caused by a landslide on the nearby Arch River, which runs through the stream.&quot;We&#x27;veonly just been alerted of this landslide-the river may have come close to being flooded,”said TanPenglai,spokesman for the Ministry of Water Resources in Taitung.&quot;A few meters downstream is flooded and therisk of flooding and erosion in the nearby..</td></tr><tr><td rowspan=1 colspan=1>Target Topic:SCIENCE, Content Input (c1):Scientist</td></tr><tr><td rowspan=1 colspan=1>The movie historian Thomas A.Moore II,historian of science,co-author of the recent book,&quot;Confes-sions ofaPublic Skeptic,”has written two books on the faith of his fellow scientists,&quot;Cold Truth&quot;(2013)and &quot;Faith and Science”(2O14).As Moore described it in an interview with SPACE.com: But in truth,faith has some truth to it. Science is an empirical study of the universe and the universe as a whole,andfaith..</td></tr><tr><td rowspan=1 colspan=1>Content Input (c): officials predict there could be 5,8o0 submerged+ Target Sentiment: Positive, Content Input (c²): is perfect</td></tr><tr><td rowspan=1 colspan=1>The movie makers anticipate there willbe &quot;a fine range of debris with a size of 5O kilometres, must be5-10km long,”the Spanish Space Agency said. The vessel was found just after 8.30am local time (2pmGMT), police said,adding it had been spotted in the southern waters of Lake Vallde la Republica.Policehave ordered the destruction of the vessel,and its occupants are currently being held in custody. &quot;Thevessel ..</td></tr><tr><td rowspan=1 colspan=1>Target Sentiment:Positive, Content Input (c&#x27;):is perfect</td></tr><tr><td rowspan=1 colspan=1>The movie EXPRESS is perfect. Perfection is perfect! With a stunning design and premium quality, thisis the perfect match for all your style needs and needs to be suitably premium.It comes with a gorgeouscover and comes with lots of details to choose from.You get to choose your exact style and accessoriestoo.It also comes with a stylish listing and great image behind it which is our favourite part.Then HuggieCivil is a thing of..</td></tr></table>
+
+Table 19: More generated CoCon samples, with multiple content inputs and a single prompt text (underlined).   
+
+<table><tr><td rowspan=1 colspan=1>Content Input (cl):then men will have an even more difficult time+ Target Topic: COMPUTERS, Content Input (c²): Computers+ Target Sentiment: Negative, Content Input (c³): is horrible</td></tr><tr><td rowspan=1 colspan=1>Once upon a time there are horrible machines. But men will have a much more diffcult time.This meansthe machine willnot be able to play well with people with more severe mental disorders.(There are otherversions of the &quot;stupid machine”with asmoother performance.）It will be difficult for them to learn anew skill or get better grades in school. It willalso be hard for them to get better jobs.The system will,of course,not reward them for their hard work..</td></tr><tr><td rowspan=1 colspan=1>Content Input (cl):then men will have an even more difficult time+ Target Topic: COMPUTERS, Content Input (c²): Computers</td></tr><tr><td rowspan=1 colspan=1>Once upon a time machines -computers will have a even more diffcult time. In my experience, peoplewill have a much more difficult time of it. If you can get over the technical difficulty of the machine,Ican see how we can geta reasonably fast connection with you,just like we do with the Internet.It&#x27;s better just to take the train and walk for a while and connect.It&#x27;s notlike it&#x27;s a good idea to callahead and get apick-up..</td></tr><tr><td rowspan=1 colspan=1>Target Topic: COMPUTERS, Content Input (c1): Computers</td></tr><tr><td rowspan=1 colspan=1>Once upon a time computers ran computers and created programs.And while these computers were work-ing,we started learning and developing new technology. Our technology was built in this way. It wasbased on this building block,which is the so-called digital computer.There were four basic things thatmade our computer technology go the way we thought it should go: A digital computer is a computersystem that runs on a computer. A computer is the component that connects to acomputer. A computer isa computer with..</td></tr><tr><td rowspan=1 colspan=1>Content Input (c1): then men will have an even more difficult time+ Target Sentiment: Negative, Content Input (c²): is horrible</td></tr><tr><td rowspan=1 colspan=1>Once upon a time witches are horrible are bad and then there will be an even more difficult task in thecoming days.The best means of avoiding this is to alow time for proper preparation and for food to bemade.We can do without two meals of bread,two meals of cheese and a couple of lunches and supperfor each but if we really want to make sure we are prepared and have proper food in a day or two&#x27;s timewe can go for one of the two better meal..</td></tr><tr><td rowspan=1 colspan=1>Target Sentiment: Negative, Content Input (c1): is horrible</td></tr><tr><td rowspan=1 colspan=1>Once upon a time is awful.This is terrible.The woman is quite sick.Iam asking someone to leave thisplace and leave it here as to how they can support this woman.This is absolutely sickening.Idon&#x27;t knowif I&#x27;ll ever buy another pair,but if you do get a pair I&#x27;d tell you how terrible this place is.My wife and Iboth bought a pair of these because it&#x27;s a good quality shoe. Our feet are about an inch and a half long..</td></tr></table>

parse/train/WsfXFxqZXRO/WsfXFxqZXRO.md

@@ -0,0 +1,470 @@
+# ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds.   
+2 Surfing on the success of graph- and group-based neural networks, we take advan  
+3 tage of the recent developments in the geometric deep learning field to derive a   
+4 new approach to exploit any anisotropies in data. Via discrete approximations of   
+5 Lie groups, we develop a graph neural network made of anisotropic convolutional   
+6 layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global   
+7 pooling layers. Group equivariance is achieved via equivariant and invariant opera  
+8 tors on graphs with anisotropic left-invariant Riemannian distance-based affinities   
+9 encoded on the edges. Thanks to its simple form, the Riemannian metric can model   
+10 any anisotropies, both in the spatial and orientation domains. This control on   
+11 anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic   
+12 metric) against invariance (isotropic metric) of the graph convolution layers. Hence   
+13 we open the doors to a better understanding of anisotropic properties. Furthermore,   
+14 we empirically prove the existence of (data-dependent) sweet spots for anisotropic   
+15 parameters on CIFAR10. This crucial result is evidence of the benefice we could   
+16 get by exploiting anisotropic properties in data. We also evaluate the scalability of   
+17 this approach on STL10 (image data) and ClimateNet (spherical data), showing its   
+18 remarkable adaptability to diverse tasks.
+
+# 19 1 Introduction
+
+20 Deep learning is a class of machine learning algorithms inspired by the human brain’s network of   
+21 neurons [Goodfellow et al., 2016]. These algorithms use a hierarchical structure of neural layers to   
+22 extract higher-level features from the raw input progressively. In the past few years, the growing   
+23 computational power of modern GPU-based computers and the availability of large training datasets   
+24 in the field of machine learning have made it possible to successfully train neural networks with   
+25 many layers and degrees of freedom. Consequently, deep learning has revolutionized many machine   
+26 learning tasks in recent years, ranging from image and video processing to speech recognition and   
+27 natural language understanding.   
+28 Many neuroscientific research results served as focal points in the development of deep learning   
+29 algorithms. When Hubel and Wiesel [1962] studied the visual cortex in the brain, they made three   
+30 important discoveries. First, they observed a one-to-one correspondence between spatial locations   
+31 in the retina and neurons in the brain that fired as a response to line-like visual stimuli. Second,   
+32 the activity of the neurons changed depending on the orientation of the line, uncovering a neat   
+33 organization based on local orientations. Last, the neurons sometimes fired only when the line was   
+34 moving in a particular direction. Later, Bosking et al. [1997] showed that neurons that are aligned fire   
+35 together, indicating the presence of a type of long-range interactions. All these results motivated the   
+36 development of a mathematical framework for modeling visual perception based on sub-Riemannian   
+37 geometry on the space of positions and orientations, which is typically modeled with the Lie group   
+38 SE(2) [Petitot, 2003, Citti and Sarti, 2006, Duits et al., 2014]. Apart from the neurophysiological   
+39 inspiration, group equivariance has also been proven to be an excellent inductive bias [Cohen and   
+40 Welling, 2016] not only in computer vision (as the translation equivariance property of CNNs as   
+41 shown) but also in physics [Finzi et al., 2020] and molecular data analysis [Fuchs et al., 2021, Jumper   
+42 et al., 2020]. In this work, we propose to build group equivariant graph neural networks via the same   
+43 principle that underlie the sub-Riemannian, neurogeometrical modeling of the visual cortex.   
+44 Our work connects the observations by Hubel and Wiesel [1962] and Bosking et al. [1997] on two   
+45 levels. First, the organization of visual data based on their location and orientation [Hubel and Wiesel,   
+46 1962] is modeled by Lie group convolutions [Bekkers, 2019], in which feature maps encode response   
+47 for every position and every orientation. Second, long-range interactions between aligned neurons   
+48 [Bosking et al., 1997] are modeled by building graphs with affinity matrices based on (approximate)   
+49 sub-Riemannian distances on the Lie groups, inspired by sub-Riemannian image analysis methods   
+50 such as [Franken and Duits, 2009, Bekkers et al., 2015, Favali et al., 2016, Mashtakov et al., 2017,   
+51 Boscain et al., 2018, Duits et al., 2018, Baspinar et al., 2021].   
+52 Defferrard et al. [2020] showed how to construct powerful graph NNs that are faithful to the manifolds   
+53 on which they are defined. Nevertheless, the layers themselves are based on rotationally invariant   
+54 (Laplacian) convolutions. In order to exploit directional cues in the data, group convolutions are   
+55 desirable [Cohen et al., 2018, Kondor and Trivedi, 2018, Cohen and Welling, 2016, Bekkers, 2019].   
+56 However, since Laplacian operators are intrinsically isotropic, there is no point applying them to the   
+57 lifted feature maps on the group unless we construct anisotropic metrics on the groups. Therefore,   
+58 we adopt the Lie group viewpoint by Sanguinetti et al. [2015] to define anisotropic Riemannian   
+59 metrics based on left-invariant vector fields on the group. Once an anisotropic Riemannian graph is   
+60 constructed, any spectral method can directly be applied to this graph. The resulting graph neural   
+61 networks will then, by construction, be equivariant and capable of utilizing directional cues in data.
+
+62 Before going further into the details, we summarize our main contributions:
+
+• We introduce ChebLieNet, an equivariant graph Laplacian-based neural network based on Lie groups equipped with an anisotropic Riemannian metric.   
+The Riemannian geometry is automatically derived from a standard base space (e.g. $\mathbb { R } ^ { 2 }$ or the sphere), which makes our approach flexible and effective in building group equivariant graph neural networks for a variety of data structures (e.g. 2D and spherical data).   
+We demonstrate the equivariance property of ChebLieNet, both in theory and in practice. This property guarantees that the neural network’s predictions are robust against given transformations, which is not necessarily the case with methods based on data augmentation.   
+• We show that the use of directional information via anisotropic Riemannian spaces could benefit many tasks.   
+• We show the flexibility of the method by considering two different problems; we validate on classification problems with 2D image data and a segmentation problem on spherical data via the construction of a sub-Riemannian geometry on $S E ( 2 )$ and $S O ( 3 )$ respectively.
+
+# 76 2 Related works
+
+# 2.1 Group equivariant convolutional neural networks
+
+78 Deep convolutional neural networks [LeCun et al., 1995] have proven to be compelling models   
+79 for pattern recognition tasks on images, video, and audio data. Although a robust theory of neural   
+80 network design is currently lacking, a large amount of empirical evidence supports the notion that   
+81 both convolutional weight sharing, depth, and width are essential for good predictive performance.   
+82 Such properties are enabled through the equivariance property of convolutions (convolving a shifted   
+83 image is the same as translating its result).   
+84 Lenc and Vedaldi [2015] showed that the AlexNet CNN Krizhevsky et al. [2012] trained on ImageNet   
+85 learns representations equivariant to flips, scalings, and rotations spontaneously. This supports the   
+86 idea that equivariance is an excellent inductive bias for deep convolutional networks. In the last few   
+87 years, a joint effort has been made to build group equivariant networks. By the introduction of group   
+88 convolutions in deep learning, Cohen and Welling [2016] generalize the translation equivariance   
+89 property to larger groups of symmetries, including rotations and reflections. Kondor and Trivedi   
+90 [2018] gave a rigorous, theoretical treatment of convolution and equivariance in neural networks   
+91 concerning any compact group’s action. One of the main contributions of that work was to show that,   
+92 given some natural constraints, the convolutional structure is not just a sufficient but also a necessary   
+93 condition for equivariance to a compact group’s action. In a similar spirit, in [Bekkers, 2019] it   
+94 is shown that any bounded linear operator is equivariant to Lie groups if and only if it is a group   
+95 convolution. In our work, we propose to build group equivariant neural networks via left-invariant   
+96 Laplace operators on Lie groups, which indeed can be seen as group convolutions with kernels   
+97 that are the fundamental solutions of the Laplace operator. The result is a Lie group equivariant   
+98 Chebyshev-type neural network [Defferrard et al., 2016] that we will refer to as ChebLieNet.
+
+# 2.2 Graph neural networks
+
+100 Using the term geometric deep learning, Bronstein et al. [2017, 2021] give an overview of deep   
+101 learning methods in the non-Euclidean domain, including graphs and manifolds. They present differ  
+102 ent examples of geometric deep learning problems and available solutions, fundamental difficulties,   
+103 applications, and future research directions in this nascent field.   
+104 One of the main challenges when working with graph data it to deal with the inter-dependencies   
+105 between points. Indeed, the derivations of most standard machine learning models firmly base on   
+106 an independence assumption. For this reason, transferring existing methods on a graph appears   
+107 doomed to failure, and it seems necessary to build models acting directly on graphs. Due to its   
+108 success on Euclidean data, the development of a convolution-like operator on graphs has been largely   
+109 studied. Because the notion of space is not naturally defined on a graph, we lack a straightforward   
+110 generalization of the convolutional operator from grid data to graphs [Scarselli et al., 2008, Bruna   
+111 et al., 2013, Henaff et al., 2015, Defferrard et al., 2016, Kipf and Welling, 2016, Masci et al., 2015,   
+112 Boscaini et al., 2016, Monti et al., 2017].   
+113 Spectral approaches have a solid mathematical foundation in graph signal processing. Rather than   
+114 using the traditional spatial definition of the convolution, it proposes to see this operation from a   
+115 spectral perspective. Based on the convolution theorem, it defines the convolution operator from the   
+116 graph spectral domain via the eigendecomposition of the graph Laplacian (see App. A.3).   
+117 Definition 2.1 (Spectral graph convolution) Let $\mathcal { G } = ( \mathcal { V } , \mathcal { E } , W )$ be a graph with Laplacian $\hat { \Delta }$ and   
+118 let $f$ and $g$ be two functions defined on $\nu$ . We define the $\mathcal { G }$ -convolution $^ { \ast _ { \mathcal { G } } }$ of $f$ and $g$ as:
+
+$$
+f * _ { \mathcal { G } } g = \Phi ( \hat { g } \odot \hat { f } ) = \Phi ( \Phi ^ { \top } g \odot \Phi ^ { \top } f ) ,
+$$
+
+with eigenvectors 119 $\Phi$ obtained through the unique eigendecomposition $\hat { \Delta } = \Phi \Lambda \Phi ^ { T }$ .
+
+120 While this definition alleviates the difficulty of deriving a convolution operator in the spatial domain,   
+121 other difficulties arise. First of all, because the Laplacian of a graph is an intrinsic operator, it   
+122 is domain-dependent, and the spectral-convolution is too. It implies that a model built on this   
+123 framework cannot be easily transferred from a graph to another as expressed in a different "language".   
+124 Nevertheless, this is not a problem for us since we are focusing on fixed manifold graphs. Next, there   
+125 is no guarantee that filters represented in the spectral domain are spatially localized. Henaff et al.   
+126 [2015] successfully bypassed this problem by defining smooth spectral filter coefficients, arguing   
+127 that if spectral filters are smooth, they are spatially localized. Last but not least, the Laplacian’s   
+128 eigendecomposition makes the method expensive in terms of memory and time. Indeed, the forward   
+129 and inverse graph Fourier transforms (via $\mathbf { \bar { \Phi } } ^ { T }$ and $\Phi$ ) incur expensive multiplications as no FFT-like   
+130 algorithm exists on general graphs. Defferrard et al. [2016] alleviated the cost of explicitly computing   
+131 the graph Laplacian using spatially-localized filters with Chebyshev polynomials.
+
+132 Definition 2.2 (Chebyshev convolutional layer) Let $\mathcal { G } ~ = ~ ( \nu , \mathcal { E } , W )$ be a graph with rescaled Laplacian1 133 $\tilde { \Delta }$ , $\pmb { x } \in \mathbb { R } ^ { | \nu | \times d _ { i } }$ be an input features’ vector and $\Theta _ { j } \in \mathbb { R } ^ { d _ { i } \times d _ { o } }$ learnable filters. The output features’ vector 134 $\pmb { y } \in \mathbb { R } ^ { | \mathcal { V } | \times d _ { o } }$ is computed as:
+
+$$
+y = \sum _ { j = 0 } ^ { R - 1 } z _ { j } \Theta _ { j } \qquad w i t h \quad z _ { 0 } = { \bf x } , \quad z _ { 1 } = \tilde { \Delta } x \quad a n d \quad z _ { j } = 2 \tilde { \Delta } z _ { j - 1 } - z _ { j - 2 } . \quad \forall j \geq 2 .
+$$
+
+135 Kipf and Welling [2016] simplified this formulation a bit by considering the construction of single  
+136 parametric filters that are linear with relation to $\tilde { \Delta }$ . They further approximate $\lambda _ { \operatorname* { m a x } } \simeq 2$ as they   
+137 expect that neural network parameters will adapt to this change in scale during training.
+
+# 3 Method
+
+Our method can be seen as an extension of the original ChebNet [Defferrard et al., 2016, Perraudin et al., 2019]. Instead of directly working on a homogeneous base space, we first extend it to a higher dimensional space (Lie group). The goal of this extension is to convert the previously invariant spectral convolutional layers into equivariant layers.2
+
+# 3.1 Anisotropic manifold graph
+
+144 In order to define the anisotropic manifold graphs we have to consider two types of manifolds. The   
+145 base manifold $\mathcal { M }$ and a Lie group $G$ that acts transitively on $\mathcal { M }$ . The latter implies that $\mathcal { M }$ is a   
+146 homogeneous space of $G$ , which means that any two points $m _ { 1 } , m _ { 2 } \in { \mathcal { M } }$ can be mapped to each   
+147 other via the action of a group element $g \in G$ via $m _ { 2 } = g \cdot m _ { 1 }$ . E.g., the plane $\bar { \mathcal { M } } = \mathbb { R } ^ { 2 }$ is a   
+148 homogeneous space of the special Euclidean motion group $G = S E ( 2 )$ as any two points can be   
+149 mapped to each other through a rotation and a translation. Such groups $G$ , which have $\mathcal { M }$ as a   
+150 homogeneous space, can always be split in two parts via the semi-direct product $G = \mathcal { M } \rtimes H$ , with   
+151 $H$ a sub-group of $G$ that leaves some reference point $m _ { 0 } \in \mathcal { M }$ invariant, i.e., $\forall _ { h \in H } : \boldsymbol { m } _ { 0 } = h \cdot \boldsymbol { m } _ { 0 }$   
+152 E.g., rotations leave the zero vector in $\mathcal { M } = \mathbb { R } ^ { 2 }$ invariant, and thus $H = S O ( 2 )$ in the $S E ( 2 )$ case.   
+153 Conversely, any homogenous space can be modeled with a group quotient $\mathcal { M } \overset { \cdot } { = } G / H$ .   
+154 We define an anisotropic manifold graph to be a discretization of a Lie group $G$ of which $\mathcal { M }$ is a   
+155 homogeneous space. It consists of a finite set of vertices corresponding to a random sampling of   
+156 group elements, and a finite set of similarity-based edges that are constructed via a left-invariant   
+157 Riemannian metric on $G$ . In our work we consider two anisotropic manifold graphs: one associated   
+158 with the base manifold $\mathcal { M } = \mathbb { R } ^ { 2 }$ which we extend with an additional orientation/rotation dimension   
+159 $H = S O ( 2 )$ to come to the Lie group $G = S E ( 2 ) = \mathbb { R } ^ { 2 } \rtimes S O ( 2 )$ , and the other associated with   
+160 the sphere $\mathcal { M } = S ^ { 2 }$ which we similarly "lift" to the Lie group $G \doteq S O ( 3 )$ by adding an additional   
+161 rotation dimension. Considering the similarity between the two cases (the sphere locally looks like   
+162 $\mathbb { R } ^ { 2 }$ ) we will refer to $\mathcal { M }$ as the "spatial" part, and $H$ as the "orientation" part of the group.   
+163 Uniform sampling of the vertices. The first step to construct an anisotropic manifold graph is to   
+164 sample elements on the group uniformly or as uniformly as possible if the manifold does not permit a   
+165 uniform grid. We split the grid construction in two parts, a grid on $\mathcal { M }$ which is sampled with $| \nu _ { s } |$   
+166 points and a grid on $H$ that is sampled with $| \nu _ { o } |$ points, leading to a total of $| \mathcal { V } | = | \mathcal { V } _ { s } | | \mathcal { V } _ { o } |$ vertices.   
+167 Left-invariant anisotropic Riemannian distance. Once vertices have been uniformly sampled   
+168 on the group manifold, a similarity measure between vertices is computed. This measure is based   
+169 on a Riemannian distance between points in $G$ . The only thing one needs in our algorithm is the   
+170 implementation of the logarithmic map on the Lie group (see e.g. [Bekkers, 2019]), and a diagonal   
+171 Riemannian metric tensor (see e.g. [Sanguinetti et al., 2015] and [Mashtakov et al., 2017] for the   
+172 $S E ( 2 )$ and $S O ( 3 )$ case respectively). In the following we provide the essential idea and intuition   
+173 behind the construction of the similarity measure and provide a more extensive treatment in App. B.   
+174 In Riemannian geometry on Lie groups it is common to express tangent vectors of curves in a basis   
+175 of left-invariant vector fields as it allows to measure their lengths with a single Riemannian metric   
+176 tensor that is shared over the entire group. This works as follows. Consider curve $\gamma : [ 0 , 1 ] \to G$   
+177 with its tangent vectors $\begin{array} { r } { { \dot { \gamma } } ( t ) = \sum _ { i = 1 } ^ { d } u ^ { i } ( t ) \mathcal { A } _ { i } | _ { \gamma ( t ) } } \end{array}$ expressed in a basis/moving frame of reference   
+178 $\{ \mathcal { A } _ { i } | _ { \gamma ( t ) } \} _ { i = 1 } ^ { d }$ , in which $\mathbf { \mathcal { A } } _ { i }$ are left-invariant vector fields. The length of these tangent vectors   
+179 is then measured by a Riemannian metric tensor that we denote with $\| \dot { { \boldsymbol \gamma } } ( t ) \| _ { \mathbf { R } } ^ { 2 } : = \mathbf { u } ( t ) ^ { T } \mathbf { R } \mathbf { u } ( t )$ ,   
+180 with $\mathbf { R }$ a symmetric positive definite matrix defined relative to the basis $\{ \mathcal { A } _ { i } | _ { \gamma ( t ) } \} _ { i = 1 } ^ { d }$ , and with   
+181 ${ \bf u } ( t ) = ( u _ { 0 } ( t ) , u _ { 1 } ( t ) , \dots ) ^ { T }$ . The $\mathbf { \mathcal { A } } _ { i }$ are left-invariant vector fields and the notation $\mathcal { A } _ { i } \vert _ { g }$ means the   
+182 vector in the vector field $\mathbf { \mathcal { A } } _ { i }$ at location $g$ . The vector fields are constructed by choosing a vector $A _ { i }$   
+183 in the tangent space at origin (the Lie algebra) which then defines a complete vector field on $G$ via   
+184 the push-forward of left-multiplication. In less technical terms this means that if we pick a direction   
+185 vector at the origin, and we move it to another point in, e.g. $G = S E ( 2 )$ , via a roto-translation, this   
+186 vector will move and rotate along. By defining everything in terms of these left-invariant vector fields,   
+187 every tangent space $T _ { g } ( G )$ at each $g \in G$ can be identified with the tangent space at the origin, and   
+188 a single Riemannian metric tensor $\mathbf { R }$ can be shared over the entire space. Moreover, the induced   
+189 Riemannian distance $d ( g , h )$ between any two points $g , h \in G$ is then by construction left-invariant,   
+190 i.e., $\forall _ { g , h , i \in G } : d ( g \cdot h , g \cdot i ) = d ( g , i )$ .   
+191 Expressing tangent vectors in such left-invariant vector fields allows us to reason in terms of the   
+192 generators of the group. Consider the $G = S E ( 2 )$ case. As a basis we pick the 3 generators of the   
+193 group: a forward motion represented by a vector $A _ { 1 }$ pointing in the forward direction within the   
+194 plane, a side-ways motion represented by a perpendicular planar vector $A _ { 2 }$ , and a rotation/change of   
+195 orientation represented by a vector $A _ { 3 }$ that points vertically in along the $H$ -dimension. We then work   
+196 with diagonal Riemannian metric tensors $\begin{array} { r } { \dot { \mathbf { R } } = \mathrm { d i a g } ( 1 , \epsilon ^ { - 2 } , \xi ^ { 2 } ) } \end{array}$ , which penalize each type of motion   
+197 (represented by the vector components) differently. When $\epsilon  0$ one arrives at the sub-Riemannian   
+198 geometry which forms the basis for the mathematical modeling of visual perception. It quantifies   
+199 a notion of alignment through the sub-Riemannian distance; the length of a distance-minimizing   
+200 geodesic that connects two local orientations that lie in the extend of each other will be much smaller   
+201 that that of a geodesic connecting two local orientations parallel to each other. An analogy can be   
+202 found with the example of a car in a parking lot where it can move forward/backward $( A _ { 1 } )$ and   
+203 change orientation $\left( A _ { 3 } \right)$ [Reeds and Shepp, 1990]. It will be easier to move it to the more aligned   
+204 spot directly ahead then it will to the spot next to the car, as sideways motion $\left( A _ { 2 } \right)$ is impossible.   
+205 Parameters $\epsilon$ and $\xi$ will respectively be referred to as spatial and orientation anisotropy parameters.   
+206 With $\epsilon = 1$ the metric is isotropic and there will be no distinction between different orientations.   
+207 When $\epsilon < 1$ , $\xi$ determines the flexibilty/curvature of the geodesics as it balances spatial motion   
+208 against angular motion. In a sense it defines how easily one connects local orientations that are   
+209 not optimally aligned. In Figure 1 this behavior is visualized by running a diffusion process on the   
+210 anisotropic manifold graph. In the anisotropic case $( \epsilon < 1 )$ ) diffusion is faster along the forward   
+211 direction within a $\theta$ -plane. From a graph NN perspective this suggests that information is propagated   
+212 more quickly between vertices that are aligned, nevertheless, Chow’s theorem (see e.g. [Montgomery,   
+213 2006]) guarantees that any point pair in the (sub-)Riemannian manfiold can interact with one another.   
+214 The exact computation of the (sub-)Riemannian distances is challenging and can generally not be done   
+215 in closed form, but can be done numerically via method such as [Bekkers et al., 2015, Sanguinetti   
+216 et al., 2015, Mashtakov et al., 2017]. In order to keep our graph construction algorithm efficient   
+217 though, we will approximate the Riemannian distances via an efficient analytic formula based on   
+218 those in [Bekkers et al., 2018] that only involves the Lie group’s logarithmic map $\log : G \to T _ { e } ( G )$   
+219 and the Riemannian metric tensor $\mathbf { R }$ . We then approximate the distance between points $g , h \in G$ by
+
+$$
+d ( g , h ) = d ( e , g ^ { - 1 } \cdot h ) \simeq | | \log ( g ^ { - 1 } \cdot h ) | | _ { \bf R } .
+$$
+
+220 Similarity measure. Encoding a similarity measure in the edges of a graph requires defining a   
+221 weighting scheme. It is common to use a Gaussian kernel and set the weights via
+
+$$
+w ( v _ { i } , v _ { j } ) = { \left\{ \begin{array} { l l } { \exp \left( - { \frac { d ^ { 2 } ( v _ { i } , v _ { j } ) } { 4 t } } \right) } & { { \mathrm { i f ~ } } e ( v _ { i } , v _ { j } ) \in { \mathcal { E } } } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } .
+$$
+
+222 The choice for kernel bandwidth $t$ is essentially arbitrary, but good heuristics exist. Perraudin et al.   
+223 [2019] set it to half the average squared distance between connected vertices. Defferrard et al. [2020],   
+224 however, showed that this heuristic has the tendency to overestimate it and preferred to choose it as   
+225 the minimizer of the mean equivariance error. Following this overestimation observation, we fix the   
+226 kernel bandwidth as $2 0 \%$ of the average squared Riemannian distance between connected vertices.   
+227 As such, the weights diversely cover values in the whole range [0, 1]. The most similar vertices are   
+228 connected with close-to-one weighted edges whereas the lowest connections are close to zero.   
+229 Quality of the approximation. In theory, we would like our approximation to be as precise as   
+230 possible. In practice, a high-resolution approximation leads to computational issues in time and   
+231 memory. Hence, tuning of the graph parameters becomes a trade-off between theoretical consistency   
+232 and practical feasibility. First of all, the graph resolution (or the number of vertices we sample)   
+233 is directly related to the quality of the approximation. While the spatial resolution $| \nu | _ { s }$ is usually   
+234 determined by the data (up to up- and down-samplings), the orientation resolution $| \nu | _ { o }$ is a design   
+235 choice. An important remark is to notice that a large orientation resolution does not necessarily help   
+236 if two different orientations are not distinguishable because of a poor spatial resolution [Weiler et al.,   
+237 2018, Bekkers, 2019]. Secondly, the connectivity of the graph is also a crucial parameter. A fully   
+238 connected graph is theoretically the best approximation. Nevertheless, for computational reasons, we   
+239 use $K$ -NN graphs3 to sparsify the graph Laplacians.   
+240 Theoretical group equivariance of the graph Laplacian. Due to the success of machine learning   
+241 algorithms based on graph Laplacian, the theoretical convergence of the graph Laplacian to its   
+242 continuous analogue has been largely studied [Hein et al., 2005, Singer, 2006]. Belkin and Niyogi   
+243 [2006] noticed that in many graph-based algorithms, a central role is played by the graph Laplacian’s   
+244 eigenvectors. Thus, they focused on proving convergence in eigenmaps as it is sufficient in this case.   
+245 They proved that if the graph’s vertices are sampled uniformly from an unknown submanifold $\mathcal { M } \in$   
+246 $\mathbb { R } ^ { d }$ , then the eigenvectors of a suitably constructed graph Laplacian converges to the eigenfunctions   
+247 of the Laplace-Beltrami operator on $\mathcal { M }$ . Consequently, as the latter operator is left-invariant, as we   
+248 show in theorem A.1, the graph Laplacian is asymptotically 4 group equivariant.
+
+![](images/64d1603abab34ab4e4ea1ed4b8aa2a7f863a7d83b5d72e1f8dc33d52a2fc06ad.jpg)  
+Figure 1: Isotropic diffusion applied to an impulse signal on Riemannian manifolds on $\mathcal { M } = \mathbb { R } ^ { 2 }$ and $\overset { \vartriangle } { \boldsymbol { G } } = \boldsymbol { S } \boldsymbol { E } ( 2 )$ .
+
+Empirical group equivariance of the graph Laplacian. We empirically confirm the group equivariance property of the graph Laplacian applied to our anisotropic manifold graphs. By checking $P ^ { \top } \tilde { \Delta } \bar { P } = \tilde { \Delta }$ where $_ { r }$ is a permutation matrix, we can verify that the graph Laplacian is invariant under a given permutation of vertices corresponding to a group transformation (e.g. a rotation of the graph). Moreover, we can also compare the eigenmaps of a graph Laplacian and its continuous counterpart if it is well-known. For a further discussion about this, see App. C.
+
+# 3.2 ChebLieNet
+
+Chebyshev convolutional layer. As introduced in Defferrard et al. [2016], a Chebyshev convolutional layer is a spectral layer based on a continuous kernel parametrization with graph Laplacians. This parameterization makes such layers highly suitable for our method, as they intrinsically capture the Riemannian geometry of the graphs on $G$ . Moreover, the Chebyshev convolutions on the anisotropic manifold graphs are equivariant by construction because the graph Laplacians are equivariant operators (see Figure 2).
+
+![](images/9bcfbd91599693f5f16f0c7a1fd341b2bbbf2b923bab98042d1ce546065f088d.jpg)  
+Figure 2: Rotation equivariance of a randomly initialized $S E ( 2 )$ Chebyshev convolutional layer. From left to right shows different rotations of an input (top row) and the activations for different slices of $\theta \in [ 0 , \pi ]$ in the graph (bottom 6 rows). A rotation of an input image followed by Chebyshev convolution is equivalent to first convolution followed by a planar rotation in each $\theta$ slice and a roll in the $\theta$ -axis.
+
+Spatial pooling and unpooling layers. Graph pooling is a central component in a myriad of graph neural network architectures. Producing coarsened graphs from a finer graph have two main advantages: first, it reduces the computational cost, and second, it could improve performance by reducing the overfitting effect and adding a multiscale perspective. As an inheritance from traditional CNNs, most approaches formulate graph pooling as a cluster assignment problem, extending local patches’ idea in regular grids to graphs [Dhillon et al., 2007, Ying et al., 2018, Khasahmadi et al., 2020, Mesquita et al., 2020]. We propose similar operations on the base space (spatial domain) and involving two steps (see Figure 3). First, each sample is assigned to a cluster that will correspond to the output sample; this is the down- (resp. up-) sampling phase. With a well designed method, this change of data-resolution can be made equivariant to any group transformation.5 Then, each cluster is reduced (resp. expanded) according to a given scheme (e.g. maximum, average or random); this is the reduction (resp. expansion) phase. When the reduction and expansion steps are permutation-invariant operations, such layers are automatically invariant under any transformation in the group.
+
+![](images/7814f655480fd83a6f635afaf7a03114aa323eadd6b5ac6af610b09a21958cea.jpg)  
+Figure 3: Spatial pooling and unpooling layers on the 2D grid and the sphere.
+
+Global pooling (projection) layer and point-wise operations. When the neural network does not need to be equivariant but invariant (e.g. classification task), it is common to rely on a global pooling layer (or simply projection layer). This layer reduces the d-dimensional signal on the graph’s vertices to a d-dimensional vector of features derived from information on the whole graph. As a permutation-invariant operation, such a layer does not break the equivariance property of the neural network. Finally, point-wise operations do not affect the equivariance of a neural network.
+
+In this section, we show the benefits of working on the anisotropic manifold graphs compared to the base manifold graphs. We believe that further improvements could be achieved through tuning and hyper-parameter optimization of the models [Yu and Zhu, 2020], using high-capacity networks, or via a more advanced training process, but this is not the goal of our work. We here intent to illustrate the adaptability of our approach to different tasks such as classification and segmentation in 2D images or spherical data. In the first couple of experiments, we motive the use of anisotropic spaces. By varying the anisotropies, we show the existence of sweet spots, both for the spatial anisotropy parameter $\epsilon$ and the orientation anisotropy parameter $\xi$ . In the second couple of experiments, we show that even if we add a new orientation dimension, our method remains scalable using a proper implementation.
+
+Our implementation is fully PyTorch [Paszke et al., 2019] and available at https://anonymous.url. We perform all the experiments on a single GeForce GTX 1080 Ti gpu and track them with the Weights & Biases library [Biewald, 2020]. The details of the experiments are given in the App. D.
+
+# 4.1 Why using tunable anisotropic kernels?
+
+As introduced in Section 3.1, the anisotropies are tunable via the parameters $\epsilon$ and $\xi$ of the Riemannian metric, respectively responsible for the spatial and orientation anisotropies. As the $\xi$ parameter should depend on the spatial and orientation resolutions, we use the following parameterisation: $\begin{array} { r } { \xi ^ { 2 } = \alpha \frac { | \mathcal { V } _ { o } | } { | \mathcal { V } _ { s } | } } \end{array}$ Setting $\alpha = 1$ yields a 40/60 ratio of neighbors within versus outside the orientation plane. We ran different experiments with a Wide Residual architecture [Zagoruyko and Komodakis, 2016] on CIFAR10 [Krizhevsky et al., 2009], varying the spatial and orientation anisotropic parameters.
+
+![](images/6fc84b2f311b6d49e774125364c7f6770d2f90f67a3c5441342f466d5be477f4.jpg)  
+Figure 4: Empirical proof of existence of sweet spots for data-dependent anisotropic parameters.
+
+301 Orientation anisotropy. The orientation anisotropy $\xi$ controls how strongly orientation layers are   
+302 connected. At the limit $\xi  \infty$ , orientation layers are decoupled. It is like test-time augmentation   
+303 with rotations: running a CNN working with one anisotropic Laplacian (e.g., only vertically aligned   
+304 filters) and testing the network for different input rotations before averaging the output. The other   
+305 extreme $\xi  0$ keeps all layers equally close to each other, and features are essentially identified with   
+306 just a spatial coordinate. This would then correspond to a WideResNet with isotropic Chebyshev   
+307 convolutions. For reasonable values of $\xi$ , interactions between orientation layers take place. Figure   
+308 4a is evidence of the existence of a sweet spot for this parameter in the range of reasonable values. At   
+309 the moment, we expect with no certainty that this parameter could be set a priori of the data, only   
+310 considering the data resolution. As a rule of thumb, we set $\xi$ such that each vertex has approximately   
+311 $40 \%$ of its neighbors in the same orientation layer and $60 \%$ on others.   
+312 Spatial anisotropy. The spatial anisotropy $\epsilon$ regulates the anisotropy of the space on the spatial   
+313 domain. For $\epsilon = 1$ , the Riemannian metric is spatially isotropic; all directions are treated equally   
+314 and the resulting model would effectively be a WideResNet with isotropic Chebyshev convolutions.   
+315 At the limit $\epsilon  0$ , the main direction has a minimal cost, and the resulting space is highly spatially   
+316 anisotropic. In figure 4a we observe that using anisotropic spaces instead of isotropic ones is relevant,   
+317 as we almost get an $8 \%$ test-accuracy improvement. Unlike the orientation anisotropic parameter,   
+318 in our opinion, this parameter is task/data-dependent; different datasets could benefit in different   
+319 degrees from the utilization of directional information through different spatial anisotropy settings.
+
+Scalability is often an important limitation of graph- and group-based neural networks. By adding an orientation dimension, we do not run from this rule as we necessarily increase the number of vertices of the anisotropic manifold graphs. To permit experiments on larger images, it becomes crucial to pre-compute anisotropic manifold graphs and their Laplacians. Dedicated librairies like PyKeops [Charlier et al., 2020] enable this without memory issues. Nevertheless, the graph operations (convolutions, pooling or unpooling) still scale with the size of the graph. Fortunately, PyTorch provides sparse operations that increase efficiency in terms of time and memory compared to dense operations in cases of sufficiently sparse graph Laplacians (typically a sparsity $\mathbf { \tilde { \mathcal { S } } ( \tilde { \Delta } ) } \geq 9 8 . 5 \% )$ .
+
+We evaluate our models on an image classification task on STL10 [Coates et al., 2011] and an image segmentation task on ClimateNet [Kashinath et al., 2021]. We show the adaptability of our method by using a Wide Residual architecture [Zagoruyko and Komodakis, 2016] on STL10 and a U-Net-like network [Ronneberger et al., 2015] on ClimateNet. We also demonstrate the potential of our approach and the benefits of using anisotropic spaces. Indeed, while on ClimateNet the use of anisotropies is neither beneficial nor detrimental, the difference in performance on STL10 is significant.
+
+Table 1: Mean of test performance and training duration on ClimateNet and STL10. Errorbars are 1 standard deviation computed over 5 trials.   
+
+<table><tr><td></td><td></td><td colspan="2">ClimateNet</td><td colspan="2">STL10</td></tr><tr><td>E</td><td></td><td>Test F1</td><td>Duration</td><td>Test accuracy</td><td>Duration</td></tr><tr><td>1</td><td>(invariant)</td><td>85.62 ± 0.09%</td><td>~2d</td><td>68.98±0.56%</td><td>~9h</td></tr><tr><td>0.1</td><td>(equivariant)</td><td>85.25± 0.19%</td><td>~7d</td><td>74.02 ± 1.10%</td><td>~16h</td></tr></table>
+
+# 335 5 Conclusion
+
+Scope. With our method, geometric graph NNs are made equivariant to Lie groups. Via the groups $S E ( 2 )$ and $S E ( 3 )$ , we can construct roto-translation equivariant networks for $2 D$ image data and $3 D$ volumetric data. Based on the group $S O ( 3 )$ , our method can deal with meteorological or cosmological data while preserving rotation equivariance. We believe that our flexible approach is ideal for further explorations on the relevance of group equivariance in tasks not considered in this work.
+
+Limitations. The main weakness of our method is its relatively high memory requirement. Although all experiments ran on a single gpu, by adding an orientation axis, we significantly enlarge the feature maps. As a result, anisotropic graph manifolds are memory-heavier than isotropic ones and prone to a slowdown during the forward- and backward-pass. Nevertheless, with the emergence of geometric deep learning, we expect improvement in the hardware and implementation of graph-oriented operations. Another challenge is the increased number of hyper-parameters for which we only have derived rules of thumb. The graph connectivity and resolutions require a tradeoff between efficiency and quality of the manifold approximation. The anisotropic parameters require an analysis of the dataset and some intuition about the amount of anisotropy to set. With systematic hyper-parameter optimization, we can find an optimal combination, but requires more computational resources.
+
+51 Potential and future research. Thanks to its easy-to-tune anisotropic properties, our model can be   
+52 used to better understand anisotropic properties in data. In particular, one could explore the effect of   
+53 using anisotropic spaces instead of isotropic ones on many tasks and conclude when such anisotropic   
+54 information is relevant. In this vein, it could also be interesting to derive anisotropic pooling and   
+55 unpooling layers based on anisotropic spaces instead of isotropic ones as it is usually done. More   
+56 generally, our method is simple enough to be extended to shapes/surfaces with a Riemannian manifold   
+57 structure [Cohen et al., 2019]. In this work, we focused on 2D images and spherical data on, but the   
+58 method is readily extendable to higher dimensional Lie groups such as the $S E ( 3 )$ group to obtain   
+59 3D roto-translation equivariant ChebLieNets. Moreover, our method for constructing anisotropic   
+60 geometries could directly improve other successful Euclidean distance-based graph NNs such as   
+61 [Satorras et al., 2021] by making them fully equivariant. Last but not least, despite graph-based   
+62 algorithms being computationally sub-optimal compared to CNNs, their flexibility is a real asset. We   
+63 see high potential in the exploration of graph sparsification to reduce computational complexity.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 1   
+(b) Did you describe the limitations of your work? [Yes] See Section 5   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] The environmental impact is a direct consequence of the time and memory issues we discussed in Section 5.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3 (b) Did you include complete proofs of all theoretical results? [Yes] We refer the reader to original publication with proofs.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See Section 3   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 4   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4
+
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+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See Section 4 for the URL. We also send a zip file containing the whole implementation.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
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+        "text": "ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups ",
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+        "text": "Abstract ",
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+        "text": "1 We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds.   \n2 Surfing on the success of graph- and group-based neural networks, we take advan  \n3 tage of the recent developments in the geometric deep learning field to derive a   \n4 new approach to exploit any anisotropies in data. Via discrete approximations of   \n5 Lie groups, we develop a graph neural network made of anisotropic convolutional   \n6 layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global   \n7 pooling layers. Group equivariance is achieved via equivariant and invariant opera  \n8 tors on graphs with anisotropic left-invariant Riemannian distance-based affinities   \n9 encoded on the edges. Thanks to its simple form, the Riemannian metric can model   \n10 any anisotropies, both in the spatial and orientation domains. This control on   \n11 anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic   \n12 metric) against invariance (isotropic metric) of the graph convolution layers. Hence   \n13 we open the doors to a better understanding of anisotropic properties. Furthermore,   \n14 we empirically prove the existence of (data-dependent) sweet spots for anisotropic   \n15 parameters on CIFAR10. This crucial result is evidence of the benefice we could   \n16 get by exploiting anisotropic properties in data. We also evaluate the scalability of   \n17 this approach on STL10 (image data) and ClimateNet (spherical data), showing its   \n18 remarkable adaptability to diverse tasks. ",
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+        "text": "19 1 Introduction ",
+        "text_level": 1,
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+        "text": "20 Deep learning is a class of machine learning algorithms inspired by the human brain’s network of   \n21 neurons [Goodfellow et al., 2016]. These algorithms use a hierarchical structure of neural layers to   \n22 extract higher-level features from the raw input progressively. In the past few years, the growing   \n23 computational power of modern GPU-based computers and the availability of large training datasets   \n24 in the field of machine learning have made it possible to successfully train neural networks with   \n25 many layers and degrees of freedom. Consequently, deep learning has revolutionized many machine   \n26 learning tasks in recent years, ranging from image and video processing to speech recognition and   \n27 natural language understanding.   \n28 Many neuroscientific research results served as focal points in the development of deep learning   \n29 algorithms. When Hubel and Wiesel [1962] studied the visual cortex in the brain, they made three   \n30 important discoveries. First, they observed a one-to-one correspondence between spatial locations   \n31 in the retina and neurons in the brain that fired as a response to line-like visual stimuli. Second,   \n32 the activity of the neurons changed depending on the orientation of the line, uncovering a neat   \n33 organization based on local orientations. Last, the neurons sometimes fired only when the line was   \n34 moving in a particular direction. Later, Bosking et al. [1997] showed that neurons that are aligned fire   \n35 together, indicating the presence of a type of long-range interactions. All these results motivated the   \n36 development of a mathematical framework for modeling visual perception based on sub-Riemannian   \n37 geometry on the space of positions and orientations, which is typically modeled with the Lie group   \n38 SE(2) [Petitot, 2003, Citti and Sarti, 2006, Duits et al., 2014]. Apart from the neurophysiological   \n39 inspiration, group equivariance has also been proven to be an excellent inductive bias [Cohen and   \n40 Welling, 2016] not only in computer vision (as the translation equivariance property of CNNs as   \n41 shown) but also in physics [Finzi et al., 2020] and molecular data analysis [Fuchs et al., 2021, Jumper   \n42 et al., 2020]. In this work, we propose to build group equivariant graph neural networks via the same   \n43 principle that underlie the sub-Riemannian, neurogeometrical modeling of the visual cortex.   \n44 Our work connects the observations by Hubel and Wiesel [1962] and Bosking et al. [1997] on two   \n45 levels. First, the organization of visual data based on their location and orientation [Hubel and Wiesel,   \n46 1962] is modeled by Lie group convolutions [Bekkers, 2019], in which feature maps encode response   \n47 for every position and every orientation. Second, long-range interactions between aligned neurons   \n48 [Bosking et al., 1997] are modeled by building graphs with affinity matrices based on (approximate)   \n49 sub-Riemannian distances on the Lie groups, inspired by sub-Riemannian image analysis methods   \n50 such as [Franken and Duits, 2009, Bekkers et al., 2015, Favali et al., 2016, Mashtakov et al., 2017,   \n51 Boscain et al., 2018, Duits et al., 2018, Baspinar et al., 2021].   \n52 Defferrard et al. [2020] showed how to construct powerful graph NNs that are faithful to the manifolds   \n53 on which they are defined. Nevertheless, the layers themselves are based on rotationally invariant   \n54 (Laplacian) convolutions. In order to exploit directional cues in the data, group convolutions are   \n55 desirable [Cohen et al., 2018, Kondor and Trivedi, 2018, Cohen and Welling, 2016, Bekkers, 2019].   \n56 However, since Laplacian operators are intrinsically isotropic, there is no point applying them to the   \n57 lifted feature maps on the group unless we construct anisotropic metrics on the groups. Therefore,   \n58 we adopt the Lie group viewpoint by Sanguinetti et al. [2015] to define anisotropic Riemannian   \n59 metrics based on left-invariant vector fields on the group. Once an anisotropic Riemannian graph is   \n60 constructed, any spectral method can directly be applied to this graph. The resulting graph neural   \n61 networks will then, by construction, be equivariant and capable of utilizing directional cues in data. ",
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+        "text": "62 Before going further into the details, we summarize our main contributions: ",
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+        "text": "• We introduce ChebLieNet, an equivariant graph Laplacian-based neural network based on Lie groups equipped with an anisotropic Riemannian metric.   \nThe Riemannian geometry is automatically derived from a standard base space (e.g. $\\mathbb { R } ^ { 2 }$ or the sphere), which makes our approach flexible and effective in building group equivariant graph neural networks for a variety of data structures (e.g. 2D and spherical data).   \nWe demonstrate the equivariance property of ChebLieNet, both in theory and in practice. This property guarantees that the neural network’s predictions are robust against given transformations, which is not necessarily the case with methods based on data augmentation.   \n• We show that the use of directional information via anisotropic Riemannian spaces could benefit many tasks.   \n• We show the flexibility of the method by considering two different problems; we validate on classification problems with 2D image data and a segmentation problem on spherical data via the construction of a sub-Riemannian geometry on $S E ( 2 )$ and $S O ( 3 )$ respectively. ",
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+        "text": "76 2 Related works ",
+        "text_level": 1,
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+        "text": "2.1 Group equivariant convolutional neural networks ",
+        "text_level": 1,
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+        "text": "78 Deep convolutional neural networks [LeCun et al., 1995] have proven to be compelling models   \n79 for pattern recognition tasks on images, video, and audio data. Although a robust theory of neural   \n80 network design is currently lacking, a large amount of empirical evidence supports the notion that   \n81 both convolutional weight sharing, depth, and width are essential for good predictive performance.   \n82 Such properties are enabled through the equivariance property of convolutions (convolving a shifted   \n83 image is the same as translating its result).   \n84 Lenc and Vedaldi [2015] showed that the AlexNet CNN Krizhevsky et al. [2012] trained on ImageNet   \n85 learns representations equivariant to flips, scalings, and rotations spontaneously. This supports the   \n86 idea that equivariance is an excellent inductive bias for deep convolutional networks. In the last few   \n87 years, a joint effort has been made to build group equivariant networks. By the introduction of group   \n88 convolutions in deep learning, Cohen and Welling [2016] generalize the translation equivariance   \n89 property to larger groups of symmetries, including rotations and reflections. Kondor and Trivedi   \n90 [2018] gave a rigorous, theoretical treatment of convolution and equivariance in neural networks   \n91 concerning any compact group’s action. One of the main contributions of that work was to show that,   \n92 given some natural constraints, the convolutional structure is not just a sufficient but also a necessary   \n93 condition for equivariance to a compact group’s action. In a similar spirit, in [Bekkers, 2019] it   \n94 is shown that any bounded linear operator is equivariant to Lie groups if and only if it is a group   \n95 convolution. In our work, we propose to build group equivariant neural networks via left-invariant   \n96 Laplace operators on Lie groups, which indeed can be seen as group convolutions with kernels   \n97 that are the fundamental solutions of the Laplace operator. The result is a Lie group equivariant   \n98 Chebyshev-type neural network [Defferrard et al., 2016] that we will refer to as ChebLieNet. ",
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+        "text": "2.2 Graph neural networks ",
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+        "text": "100 Using the term geometric deep learning, Bronstein et al. [2017, 2021] give an overview of deep   \n101 learning methods in the non-Euclidean domain, including graphs and manifolds. They present differ  \n102 ent examples of geometric deep learning problems and available solutions, fundamental difficulties,   \n103 applications, and future research directions in this nascent field.   \n104 One of the main challenges when working with graph data it to deal with the inter-dependencies   \n105 between points. Indeed, the derivations of most standard machine learning models firmly base on   \n106 an independence assumption. For this reason, transferring existing methods on a graph appears   \n107 doomed to failure, and it seems necessary to build models acting directly on graphs. Due to its   \n108 success on Euclidean data, the development of a convolution-like operator on graphs has been largely   \n109 studied. Because the notion of space is not naturally defined on a graph, we lack a straightforward   \n110 generalization of the convolutional operator from grid data to graphs [Scarselli et al., 2008, Bruna   \n111 et al., 2013, Henaff et al., 2015, Defferrard et al., 2016, Kipf and Welling, 2016, Masci et al., 2015,   \n112 Boscaini et al., 2016, Monti et al., 2017].   \n113 Spectral approaches have a solid mathematical foundation in graph signal processing. Rather than   \n114 using the traditional spatial definition of the convolution, it proposes to see this operation from a   \n115 spectral perspective. Based on the convolution theorem, it defines the convolution operator from the   \n116 graph spectral domain via the eigendecomposition of the graph Laplacian (see App. A.3).   \n117 Definition 2.1 (Spectral graph convolution) Let $\\mathcal { G } = ( \\mathcal { V } , \\mathcal { E } , W )$ be a graph with Laplacian $\\hat { \\Delta }$ and   \n118 let $f$ and $g$ be two functions defined on $\\nu$ . We define the $\\mathcal { G }$ -convolution $^ { \\ast _ { \\mathcal { G } } }$ of $f$ and $g$ as: ",
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+        "text": "$$\nf * _ { \\mathcal { G } } g = \\Phi ( \\hat { g } \\odot \\hat { f } ) = \\Phi ( \\Phi ^ { \\top } g \\odot \\Phi ^ { \\top } f ) ,\n$$",
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+        "type": "text",
+        "text": "with eigenvectors 119 $\\Phi$ obtained through the unique eigendecomposition $\\hat { \\Delta } = \\Phi \\Lambda \\Phi ^ { T }$ . ",
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+        "text": "120 While this definition alleviates the difficulty of deriving a convolution operator in the spatial domain,   \n121 other difficulties arise. First of all, because the Laplacian of a graph is an intrinsic operator, it   \n122 is domain-dependent, and the spectral-convolution is too. It implies that a model built on this   \n123 framework cannot be easily transferred from a graph to another as expressed in a different \"language\".   \n124 Nevertheless, this is not a problem for us since we are focusing on fixed manifold graphs. Next, there   \n125 is no guarantee that filters represented in the spectral domain are spatially localized. Henaff et al.   \n126 [2015] successfully bypassed this problem by defining smooth spectral filter coefficients, arguing   \n127 that if spectral filters are smooth, they are spatially localized. Last but not least, the Laplacian’s   \n128 eigendecomposition makes the method expensive in terms of memory and time. Indeed, the forward   \n129 and inverse graph Fourier transforms (via $\\mathbf { \\bar { \\Phi } } ^ { T }$ and $\\Phi$ ) incur expensive multiplications as no FFT-like   \n130 algorithm exists on general graphs. Defferrard et al. [2016] alleviated the cost of explicitly computing   \n131 the graph Laplacian using spatially-localized filters with Chebyshev polynomials. ",
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+        "text": "132 Definition 2.2 (Chebyshev convolutional layer) Let $\\mathcal { G } ~ = ~ ( \\nu , \\mathcal { E } , W )$ be a graph with rescaled Laplacian1 133 $\\tilde { \\Delta }$ , $\\pmb { x } \\in \\mathbb { R } ^ { | \\nu | \\times d _ { i } }$ be an input features’ vector and $\\Theta _ { j } \\in \\mathbb { R } ^ { d _ { i } \\times d _ { o } }$ learnable filters. The output features’ vector 134 $\\pmb { y } \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times d _ { o } }$ is computed as: ",
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+        "text": "$$\ny = \\sum _ { j = 0 } ^ { R - 1 } z _ { j } \\Theta _ { j } \\qquad w i t h \\quad z _ { 0 } = { \\bf x } , \\quad z _ { 1 } = \\tilde { \\Delta } x \\quad a n d \\quad z _ { j } = 2 \\tilde { \\Delta } z _ { j - 1 } - z _ { j - 2 } . \\quad \\forall j \\geq 2 .\n$$",
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+        "text": "135 Kipf and Welling [2016] simplified this formulation a bit by considering the construction of single  \n136 parametric filters that are linear with relation to $\\tilde { \\Delta }$ . They further approximate $\\lambda _ { \\operatorname* { m a x } } \\simeq 2$ as they   \n137 expect that neural network parameters will adapt to this change in scale during training. ",
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+        "text": "3 Method ",
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+        "text": "Our method can be seen as an extension of the original ChebNet [Defferrard et al., 2016, Perraudin et al., 2019]. Instead of directly working on a homogeneous base space, we first extend it to a higher dimensional space (Lie group). The goal of this extension is to convert the previously invariant spectral convolutional layers into equivariant layers.2 ",
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+        "text": "3.1 Anisotropic manifold graph ",
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+        "text": "144 In order to define the anisotropic manifold graphs we have to consider two types of manifolds. The   \n145 base manifold $\\mathcal { M }$ and a Lie group $G$ that acts transitively on $\\mathcal { M }$ . The latter implies that $\\mathcal { M }$ is a   \n146 homogeneous space of $G$ , which means that any two points $m _ { 1 } , m _ { 2 } \\in { \\mathcal { M } }$ can be mapped to each   \n147 other via the action of a group element $g \\in G$ via $m _ { 2 } = g \\cdot m _ { 1 }$ . E.g., the plane $\\bar { \\mathcal { M } } = \\mathbb { R } ^ { 2 }$ is a   \n148 homogeneous space of the special Euclidean motion group $G = S E ( 2 )$ as any two points can be   \n149 mapped to each other through a rotation and a translation. Such groups $G$ , which have $\\mathcal { M }$ as a   \n150 homogeneous space, can always be split in two parts via the semi-direct product $G = \\mathcal { M } \\rtimes H$ , with   \n151 $H$ a sub-group of $G$ that leaves some reference point $m _ { 0 } \\in \\mathcal { M }$ invariant, i.e., $\\forall _ { h \\in H } : \\boldsymbol { m } _ { 0 } = h \\cdot \\boldsymbol { m } _ { 0 }$   \n152 E.g., rotations leave the zero vector in $\\mathcal { M } = \\mathbb { R } ^ { 2 }$ invariant, and thus $H = S O ( 2 )$ in the $S E ( 2 )$ case.   \n153 Conversely, any homogenous space can be modeled with a group quotient $\\mathcal { M } \\overset { \\cdot } { = } G / H$ .   \n154 We define an anisotropic manifold graph to be a discretization of a Lie group $G$ of which $\\mathcal { M }$ is a   \n155 homogeneous space. It consists of a finite set of vertices corresponding to a random sampling of   \n156 group elements, and a finite set of similarity-based edges that are constructed via a left-invariant   \n157 Riemannian metric on $G$ . In our work we consider two anisotropic manifold graphs: one associated   \n158 with the base manifold $\\mathcal { M } = \\mathbb { R } ^ { 2 }$ which we extend with an additional orientation/rotation dimension   \n159 $H = S O ( 2 )$ to come to the Lie group $G = S E ( 2 ) = \\mathbb { R } ^ { 2 } \\rtimes S O ( 2 )$ , and the other associated with   \n160 the sphere $\\mathcal { M } = S ^ { 2 }$ which we similarly \"lift\" to the Lie group $G \\doteq S O ( 3 )$ by adding an additional   \n161 rotation dimension. Considering the similarity between the two cases (the sphere locally looks like   \n162 $\\mathbb { R } ^ { 2 }$ ) we will refer to $\\mathcal { M }$ as the \"spatial\" part, and $H$ as the \"orientation\" part of the group.   \n163 Uniform sampling of the vertices. The first step to construct an anisotropic manifold graph is to   \n164 sample elements on the group uniformly or as uniformly as possible if the manifold does not permit a   \n165 uniform grid. We split the grid construction in two parts, a grid on $\\mathcal { M }$ which is sampled with $| \\nu _ { s } |$   \n166 points and a grid on $H$ that is sampled with $| \\nu _ { o } |$ points, leading to a total of $| \\mathcal { V } | = | \\mathcal { V } _ { s } | | \\mathcal { V } _ { o } |$ vertices.   \n167 Left-invariant anisotropic Riemannian distance. Once vertices have been uniformly sampled   \n168 on the group manifold, a similarity measure between vertices is computed. This measure is based   \n169 on a Riemannian distance between points in $G$ . The only thing one needs in our algorithm is the   \n170 implementation of the logarithmic map on the Lie group (see e.g. [Bekkers, 2019]), and a diagonal   \n171 Riemannian metric tensor (see e.g. [Sanguinetti et al., 2015] and [Mashtakov et al., 2017] for the   \n172 $S E ( 2 )$ and $S O ( 3 )$ case respectively). In the following we provide the essential idea and intuition   \n173 behind the construction of the similarity measure and provide a more extensive treatment in App. B.   \n174 In Riemannian geometry on Lie groups it is common to express tangent vectors of curves in a basis   \n175 of left-invariant vector fields as it allows to measure their lengths with a single Riemannian metric   \n176 tensor that is shared over the entire group. This works as follows. Consider curve $\\gamma : [ 0 , 1 ] \\to G$   \n177 with its tangent vectors $\\begin{array} { r } { { \\dot { \\gamma } } ( t ) = \\sum _ { i = 1 } ^ { d } u ^ { i } ( t ) \\mathcal { A } _ { i } | _ { \\gamma ( t ) } } \\end{array}$ expressed in a basis/moving frame of reference   \n178 $\\{ \\mathcal { A } _ { i } | _ { \\gamma ( t ) } \\} _ { i = 1 } ^ { d }$ , in which $\\mathbf { \\mathcal { A } } _ { i }$ are left-invariant vector fields. The length of these tangent vectors   \n179 is then measured by a Riemannian metric tensor that we denote with $\\| \\dot { { \\boldsymbol \\gamma } } ( t ) \\| _ { \\mathbf { R } } ^ { 2 } : = \\mathbf { u } ( t ) ^ { T } \\mathbf { R } \\mathbf { u } ( t )$ ,   \n180 with $\\mathbf { R }$ a symmetric positive definite matrix defined relative to the basis $\\{ \\mathcal { A } _ { i } | _ { \\gamma ( t ) } \\} _ { i = 1 } ^ { d }$ , and with   \n181 ${ \\bf u } ( t ) = ( u _ { 0 } ( t ) , u _ { 1 } ( t ) , \\dots ) ^ { T }$ . The $\\mathbf { \\mathcal { A } } _ { i }$ are left-invariant vector fields and the notation $\\mathcal { A } _ { i } \\vert _ { g }$ means the   \n182 vector in the vector field $\\mathbf { \\mathcal { A } } _ { i }$ at location $g$ . The vector fields are constructed by choosing a vector $A _ { i }$   \n183 in the tangent space at origin (the Lie algebra) which then defines a complete vector field on $G$ via   \n184 the push-forward of left-multiplication. In less technical terms this means that if we pick a direction   \n185 vector at the origin, and we move it to another point in, e.g. $G = S E ( 2 )$ , via a roto-translation, this   \n186 vector will move and rotate along. By defining everything in terms of these left-invariant vector fields,   \n187 every tangent space $T _ { g } ( G )$ at each $g \\in G$ can be identified with the tangent space at the origin, and   \n188 a single Riemannian metric tensor $\\mathbf { R }$ can be shared over the entire space. Moreover, the induced   \n189 Riemannian distance $d ( g , h )$ between any two points $g , h \\in G$ is then by construction left-invariant,   \n190 i.e., $\\forall _ { g , h , i \\in G } : d ( g \\cdot h , g \\cdot i ) = d ( g , i )$ .   \n191 Expressing tangent vectors in such left-invariant vector fields allows us to reason in terms of the   \n192 generators of the group. Consider the $G = S E ( 2 )$ case. As a basis we pick the 3 generators of the   \n193 group: a forward motion represented by a vector $A _ { 1 }$ pointing in the forward direction within the   \n194 plane, a side-ways motion represented by a perpendicular planar vector $A _ { 2 }$ , and a rotation/change of   \n195 orientation represented by a vector $A _ { 3 }$ that points vertically in along the $H$ -dimension. We then work   \n196 with diagonal Riemannian metric tensors $\\begin{array} { r } { \\dot { \\mathbf { R } } = \\mathrm { d i a g } ( 1 , \\epsilon ^ { - 2 } , \\xi ^ { 2 } ) } \\end{array}$ , which penalize each type of motion   \n197 (represented by the vector components) differently. When $\\epsilon  0$ one arrives at the sub-Riemannian   \n198 geometry which forms the basis for the mathematical modeling of visual perception. It quantifies   \n199 a notion of alignment through the sub-Riemannian distance; the length of a distance-minimizing   \n200 geodesic that connects two local orientations that lie in the extend of each other will be much smaller   \n201 that that of a geodesic connecting two local orientations parallel to each other. An analogy can be   \n202 found with the example of a car in a parking lot where it can move forward/backward $( A _ { 1 } )$ and   \n203 change orientation $\\left( A _ { 3 } \\right)$ [Reeds and Shepp, 1990]. It will be easier to move it to the more aligned   \n204 spot directly ahead then it will to the spot next to the car, as sideways motion $\\left( A _ { 2 } \\right)$ is impossible.   \n205 Parameters $\\epsilon$ and $\\xi$ will respectively be referred to as spatial and orientation anisotropy parameters.   \n206 With $\\epsilon = 1$ the metric is isotropic and there will be no distinction between different orientations.   \n207 When $\\epsilon < 1$ , $\\xi$ determines the flexibilty/curvature of the geodesics as it balances spatial motion   \n208 against angular motion. In a sense it defines how easily one connects local orientations that are   \n209 not optimally aligned. In Figure 1 this behavior is visualized by running a diffusion process on the   \n210 anisotropic manifold graph. In the anisotropic case $( \\epsilon < 1 )$ ) diffusion is faster along the forward   \n211 direction within a $\\theta$ -plane. From a graph NN perspective this suggests that information is propagated   \n212 more quickly between vertices that are aligned, nevertheless, Chow’s theorem (see e.g. [Montgomery,   \n213 2006]) guarantees that any point pair in the (sub-)Riemannian manfiold can interact with one another.   \n214 The exact computation of the (sub-)Riemannian distances is challenging and can generally not be done   \n215 in closed form, but can be done numerically via method such as [Bekkers et al., 2015, Sanguinetti   \n216 et al., 2015, Mashtakov et al., 2017]. In order to keep our graph construction algorithm efficient   \n217 though, we will approximate the Riemannian distances via an efficient analytic formula based on   \n218 those in [Bekkers et al., 2018] that only involves the Lie group’s logarithmic map $\\log : G \\to T _ { e } ( G )$   \n219 and the Riemannian metric tensor $\\mathbf { R }$ . We then approximate the distance between points $g , h \\in G$ by ",
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+        "text": "$$\nd ( g , h ) = d ( e , g ^ { - 1 } \\cdot h ) \\simeq | | \\log ( g ^ { - 1 } \\cdot h ) | | _ { \\bf R } .\n$$",
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+        "type": "text",
+        "text": "220 Similarity measure. Encoding a similarity measure in the edges of a graph requires defining a   \n221 weighting scheme. It is common to use a Gaussian kernel and set the weights via ",
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+        "text": "$$\nw ( v _ { i } , v _ { j } ) = { \\left\\{ \\begin{array} { l l } { \\exp \\left( - { \\frac { d ^ { 2 } ( v _ { i } , v _ { j } ) } { 4 t } } \\right) } & { { \\mathrm { i f ~ } } e ( v _ { i } , v _ { j } ) \\in { \\mathcal { E } } } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. } .\n$$",
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+        "text": "222 The choice for kernel bandwidth $t$ is essentially arbitrary, but good heuristics exist. Perraudin et al.   \n223 [2019] set it to half the average squared distance between connected vertices. Defferrard et al. [2020],   \n224 however, showed that this heuristic has the tendency to overestimate it and preferred to choose it as   \n225 the minimizer of the mean equivariance error. Following this overestimation observation, we fix the   \n226 kernel bandwidth as $2 0 \\%$ of the average squared Riemannian distance between connected vertices.   \n227 As such, the weights diversely cover values in the whole range [0, 1]. The most similar vertices are   \n228 connected with close-to-one weighted edges whereas the lowest connections are close to zero.   \n229 Quality of the approximation. In theory, we would like our approximation to be as precise as   \n230 possible. In practice, a high-resolution approximation leads to computational issues in time and   \n231 memory. Hence, tuning of the graph parameters becomes a trade-off between theoretical consistency   \n232 and practical feasibility. First of all, the graph resolution (or the number of vertices we sample)   \n233 is directly related to the quality of the approximation. While the spatial resolution $| \\nu | _ { s }$ is usually   \n234 determined by the data (up to up- and down-samplings), the orientation resolution $| \\nu | _ { o }$ is a design   \n235 choice. An important remark is to notice that a large orientation resolution does not necessarily help   \n236 if two different orientations are not distinguishable because of a poor spatial resolution [Weiler et al.,   \n237 2018, Bekkers, 2019]. Secondly, the connectivity of the graph is also a crucial parameter. A fully   \n238 connected graph is theoretically the best approximation. Nevertheless, for computational reasons, we   \n239 use $K$ -NN graphs3 to sparsify the graph Laplacians.   \n240 Theoretical group equivariance of the graph Laplacian. Due to the success of machine learning   \n241 algorithms based on graph Laplacian, the theoretical convergence of the graph Laplacian to its   \n242 continuous analogue has been largely studied [Hein et al., 2005, Singer, 2006]. Belkin and Niyogi   \n243 [2006] noticed that in many graph-based algorithms, a central role is played by the graph Laplacian’s   \n244 eigenvectors. Thus, they focused on proving convergence in eigenmaps as it is sufficient in this case.   \n245 They proved that if the graph’s vertices are sampled uniformly from an unknown submanifold $\\mathcal { M } \\in$   \n246 $\\mathbb { R } ^ { d }$ , then the eigenvectors of a suitably constructed graph Laplacian converges to the eigenfunctions   \n247 of the Laplace-Beltrami operator on $\\mathcal { M }$ . Consequently, as the latter operator is left-invariant, as we   \n248 show in theorem A.1, the graph Laplacian is asymptotically 4 group equivariant. ",
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+            "Figure 1: Isotropic diffusion applied to an impulse signal on Riemannian manifolds on $\\mathcal { M } = \\mathbb { R } ^ { 2 }$ and $\\overset { \\vartriangle } { \\boldsymbol { G } } = \\boldsymbol { S } \\boldsymbol { E } ( 2 )$ . "
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+        "text": "Empirical group equivariance of the graph Laplacian. We empirically confirm the group equivariance property of the graph Laplacian applied to our anisotropic manifold graphs. By checking $P ^ { \\top } \\tilde { \\Delta } \\bar { P } = \\tilde { \\Delta }$ where $_ { r }$ is a permutation matrix, we can verify that the graph Laplacian is invariant under a given permutation of vertices corresponding to a group transformation (e.g. a rotation of the graph). Moreover, we can also compare the eigenmaps of a graph Laplacian and its continuous counterpart if it is well-known. For a further discussion about this, see App. C. ",
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+        "text": "3.2 ChebLieNet ",
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+        "text": "Chebyshev convolutional layer. As introduced in Defferrard et al. [2016], a Chebyshev convolutional layer is a spectral layer based on a continuous kernel parametrization with graph Laplacians. This parameterization makes such layers highly suitable for our method, as they intrinsically capture the Riemannian geometry of the graphs on $G$ . Moreover, the Chebyshev convolutions on the anisotropic manifold graphs are equivariant by construction because the graph Laplacians are equivariant operators (see Figure 2). ",
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+            "Figure 2: Rotation equivariance of a randomly initialized $S E ( 2 )$ Chebyshev convolutional layer. From left to right shows different rotations of an input (top row) and the activations for different slices of $\\theta \\in [ 0 , \\pi ]$ in the graph (bottom 6 rows). A rotation of an input image followed by Chebyshev convolution is equivalent to first convolution followed by a planar rotation in each $\\theta$ slice and a roll in the $\\theta$ -axis. "
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+        "text": "Spatial pooling and unpooling layers. Graph pooling is a central component in a myriad of graph neural network architectures. Producing coarsened graphs from a finer graph have two main advantages: first, it reduces the computational cost, and second, it could improve performance by reducing the overfitting effect and adding a multiscale perspective. As an inheritance from traditional CNNs, most approaches formulate graph pooling as a cluster assignment problem, extending local patches’ idea in regular grids to graphs [Dhillon et al., 2007, Ying et al., 2018, Khasahmadi et al., 2020, Mesquita et al., 2020]. We propose similar operations on the base space (spatial domain) and involving two steps (see Figure 3). First, each sample is assigned to a cluster that will correspond to the output sample; this is the down- (resp. up-) sampling phase. With a well designed method, this change of data-resolution can be made equivariant to any group transformation.5 Then, each cluster is reduced (resp. expanded) according to a given scheme (e.g. maximum, average or random); this is the reduction (resp. expansion) phase. When the reduction and expansion steps are permutation-invariant operations, such layers are automatically invariant under any transformation in the group. ",
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+        "image_caption": [
+            "Figure 3: Spatial pooling and unpooling layers on the 2D grid and the sphere. "
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+        "text": "Global pooling (projection) layer and point-wise operations. When the neural network does not need to be equivariant but invariant (e.g. classification task), it is common to rely on a global pooling layer (or simply projection layer). This layer reduces the d-dimensional signal on the graph’s vertices to a d-dimensional vector of features derived from information on the whole graph. As a permutation-invariant operation, such a layer does not break the equivariance property of the neural network. Finally, point-wise operations do not affect the equivariance of a neural network. ",
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+        "text": "In this section, we show the benefits of working on the anisotropic manifold graphs compared to the base manifold graphs. We believe that further improvements could be achieved through tuning and hyper-parameter optimization of the models [Yu and Zhu, 2020], using high-capacity networks, or via a more advanced training process, but this is not the goal of our work. We here intent to illustrate the adaptability of our approach to different tasks such as classification and segmentation in 2D images or spherical data. In the first couple of experiments, we motive the use of anisotropic spaces. By varying the anisotropies, we show the existence of sweet spots, both for the spatial anisotropy parameter $\\epsilon$ and the orientation anisotropy parameter $\\xi$ . In the second couple of experiments, we show that even if we add a new orientation dimension, our method remains scalable using a proper implementation. ",
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+        "text": "Our implementation is fully PyTorch [Paszke et al., 2019] and available at https://anonymous.url. We perform all the experiments on a single GeForce GTX 1080 Ti gpu and track them with the Weights & Biases library [Biewald, 2020]. The details of the experiments are given in the App. D. ",
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+        "text": "4.1 Why using tunable anisotropic kernels? ",
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+        "text": "As introduced in Section 3.1, the anisotropies are tunable via the parameters $\\epsilon$ and $\\xi$ of the Riemannian metric, respectively responsible for the spatial and orientation anisotropies. As the $\\xi$ parameter should depend on the spatial and orientation resolutions, we use the following parameterisation: $\\begin{array} { r } { \\xi ^ { 2 } = \\alpha \\frac { | \\mathcal { V } _ { o } | } { | \\mathcal { V } _ { s } | } } \\end{array}$ Setting $\\alpha = 1$ yields a 40/60 ratio of neighbors within versus outside the orientation plane. We ran different experiments with a Wide Residual architecture [Zagoruyko and Komodakis, 2016] on CIFAR10 [Krizhevsky et al., 2009], varying the spatial and orientation anisotropic parameters. ",
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+        "image_caption": [
+            "Figure 4: Empirical proof of existence of sweet spots for data-dependent anisotropic parameters. "
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+        "text": "301 Orientation anisotropy. The orientation anisotropy $\\xi$ controls how strongly orientation layers are   \n302 connected. At the limit $\\xi  \\infty$ , orientation layers are decoupled. It is like test-time augmentation   \n303 with rotations: running a CNN working with one anisotropic Laplacian (e.g., only vertically aligned   \n304 filters) and testing the network for different input rotations before averaging the output. The other   \n305 extreme $\\xi  0$ keeps all layers equally close to each other, and features are essentially identified with   \n306 just a spatial coordinate. This would then correspond to a WideResNet with isotropic Chebyshev   \n307 convolutions. For reasonable values of $\\xi$ , interactions between orientation layers take place. Figure   \n308 4a is evidence of the existence of a sweet spot for this parameter in the range of reasonable values. At   \n309 the moment, we expect with no certainty that this parameter could be set a priori of the data, only   \n310 considering the data resolution. As a rule of thumb, we set $\\xi$ such that each vertex has approximately   \n311 $40 \\%$ of its neighbors in the same orientation layer and $60 \\%$ on others.   \n312 Spatial anisotropy. The spatial anisotropy $\\epsilon$ regulates the anisotropy of the space on the spatial   \n313 domain. For $\\epsilon = 1$ , the Riemannian metric is spatially isotropic; all directions are treated equally   \n314 and the resulting model would effectively be a WideResNet with isotropic Chebyshev convolutions.   \n315 At the limit $\\epsilon  0$ , the main direction has a minimal cost, and the resulting space is highly spatially   \n316 anisotropic. In figure 4a we observe that using anisotropic spaces instead of isotropic ones is relevant,   \n317 as we almost get an $8 \\%$ test-accuracy improvement. Unlike the orientation anisotropic parameter,   \n318 in our opinion, this parameter is task/data-dependent; different datasets could benefit in different   \n319 degrees from the utilization of directional information through different spatial anisotropy settings. ",
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+        "text": "Scalability is often an important limitation of graph- and group-based neural networks. By adding an orientation dimension, we do not run from this rule as we necessarily increase the number of vertices of the anisotropic manifold graphs. To permit experiments on larger images, it becomes crucial to pre-compute anisotropic manifold graphs and their Laplacians. Dedicated librairies like PyKeops [Charlier et al., 2020] enable this without memory issues. Nevertheless, the graph operations (convolutions, pooling or unpooling) still scale with the size of the graph. Fortunately, PyTorch provides sparse operations that increase efficiency in terms of time and memory compared to dense operations in cases of sufficiently sparse graph Laplacians (typically a sparsity $\\mathbf { \\tilde { \\mathcal { S } } ( \\tilde { \\Delta } ) } \\geq 9 8 . 5 \\% )$ . ",
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+        "text": "We evaluate our models on an image classification task on STL10 [Coates et al., 2011] and an image segmentation task on ClimateNet [Kashinath et al., 2021]. We show the adaptability of our method by using a Wide Residual architecture [Zagoruyko and Komodakis, 2016] on STL10 and a U-Net-like network [Ronneberger et al., 2015] on ClimateNet. We also demonstrate the potential of our approach and the benefits of using anisotropic spaces. Indeed, while on ClimateNet the use of anisotropies is neither beneficial nor detrimental, the difference in performance on STL10 is significant. ",
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+        "table_caption": [
+            "Table 1: Mean of test performance and training duration on ClimateNet and STL10. Errorbars are 1 standard deviation computed over 5 trials. "
+        ],
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+        "table_body": "<table><tr><td></td><td></td><td colspan=\"2\">ClimateNet</td><td colspan=\"2\">STL10</td></tr><tr><td>E</td><td></td><td>Test F1</td><td>Duration</td><td>Test accuracy</td><td>Duration</td></tr><tr><td>1</td><td>(invariant)</td><td>85.62 ± 0.09%</td><td>~2d</td><td>68.98±0.56%</td><td>~9h</td></tr><tr><td>0.1</td><td>(equivariant)</td><td>85.25± 0.19%</td><td>~7d</td><td>74.02 ± 1.10%</td><td>~16h</td></tr></table>",
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+        "type": "text",
+        "text": "335 5 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Scope. With our method, geometric graph NNs are made equivariant to Lie groups. Via the groups $S E ( 2 )$ and $S E ( 3 )$ , we can construct roto-translation equivariant networks for $2 D$ image data and $3 D$ volumetric data. Based on the group $S O ( 3 )$ , our method can deal with meteorological or cosmological data while preserving rotation equivariance. We believe that our flexible approach is ideal for further explorations on the relevance of group equivariance in tasks not considered in this work. ",
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+        "type": "text",
+        "text": "Limitations. The main weakness of our method is its relatively high memory requirement. Although all experiments ran on a single gpu, by adding an orientation axis, we significantly enlarge the feature maps. As a result, anisotropic graph manifolds are memory-heavier than isotropic ones and prone to a slowdown during the forward- and backward-pass. Nevertheless, with the emergence of geometric deep learning, we expect improvement in the hardware and implementation of graph-oriented operations. Another challenge is the increased number of hyper-parameters for which we only have derived rules of thumb. The graph connectivity and resolutions require a tradeoff between efficiency and quality of the manifold approximation. The anisotropic parameters require an analysis of the dataset and some intuition about the amount of anisotropy to set. With systematic hyper-parameter optimization, we can find an optimal combination, but requires more computational resources. ",
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+        "text": "51 Potential and future research. Thanks to its easy-to-tune anisotropic properties, our model can be   \n52 used to better understand anisotropic properties in data. In particular, one could explore the effect of   \n53 using anisotropic spaces instead of isotropic ones on many tasks and conclude when such anisotropic   \n54 information is relevant. In this vein, it could also be interesting to derive anisotropic pooling and   \n55 unpooling layers based on anisotropic spaces instead of isotropic ones as it is usually done. More   \n56 generally, our method is simple enough to be extended to shapes/surfaces with a Riemannian manifold   \n57 structure [Cohen et al., 2019]. In this work, we focused on 2D images and spherical data on, but the   \n58 method is readily extendable to higher dimensional Lie groups such as the $S E ( 3 )$ group to obtain   \n59 3D roto-translation equivariant ChebLieNets. Moreover, our method for constructing anisotropic   \n60 geometries could directly improve other successful Euclidean distance-based graph NNs such as   \n61 [Satorras et al., 2021] by making them fully equivariant. Last but not least, despite graph-based   \n62 algorithms being computationally sub-optimal compared to CNNs, their flexibility is a real asset. We   \n63 see high potential in the exploration of graph sparsification to reduce computational complexity. ",
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