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Question: Recent studies show that convolutional neural networks (CNNs) are vulnerable under various settings, including adversarial examples, backdoor attacks, and distribution shifting. Motivated by the findings that human visual system pays more attention to global structure (e.g., shape) for recognition while CNNs are biased towards local texture features in images, we propose a unified framework EdgeGANRob based on robust edge features to improve the robustness of CNNs in general, which first explicitly extracts shape/structure features from a given image and then reconstructs a new image by refilling the texture information with a trained generative adversarial network (GAN). In addition, to reduce the sensitivity of edge detection algorithm to adversarial perturbation, we propose a robust edge detection approach Robust Canny based on the vanilla Canny algorithm. To gain more insights, we also compare EdgeGANRob with its simplified backbone procedure EdgeNetRob, which performs learning tasks directly on the extracted robust edge features. We find that EdgeNetRob can help boost model robustness significantly but at the cost of the clean model accuracy. EdgeGANRob, on the other hand, is able to improve clean model accuracy compared with EdgeNetRob and without losing the robustness benefits introduced by EdgeNetRob. Extensive experiments show that EdgeGANRob is resilient in different learning tasks under diverse settings. Convolutional neural networks (CNNs) have been studied extensively , and have achieved state-of-the-art performance in many learning tasks (; . However, recent works have shown that CNNs are vulnerable to adversarial examples (; b;), where imperceptible perturbation can be added to the test data to tamper the predictions. Different from adversarial examples where test data is manipulated, an orthogonal setting: data poisoning or backdoor attacks where training data is manipulated to reduce model's generalization accuracy and achieve targeted poisoning attack b ). In addition, recent studies show that CNNs tend to learn surface statistical regularities instead of high level abstraction, leading it fails to generalize to the superficial pattern transformation (radial kernel, random kernel (a; a; . We refer to this problem as model's robustness under distribution shifting. How to improve the general robustness of DNNs under these settings remains unsolved. To improve the robustness of CNNs, recent studies explore the underlying cause of their vulnerability. For example, attributes the existence of adversarial examples to the existence of non-robust but highly-predictive features. They suggest to train a classifier only on "robust features" which contain the necessary information for recognition and are insensitive to small perturbations. In addition, it is shown that human recognition relies mainly on global object shapes rather than local patterns (e.t. textures), while CNNs are more biased towards the latter . For instance, creates a texture-shape cue conflict, such as a cat shape with elephant texture, and feeds it to an Imagenet trained CNN and huamn respectively. While Human can still recognize it as a cat, CNN wrongly predicts it as an elephant. Therefore, the bias toward local features potentially contributes to CNN's vulnerability to adversarial examples, distribution shifting and patterns of backdoor attacks. Particularly, previous researcher also shows Figure 1: Structure of the proposed pipeline. EdgeNetRob feeds the output of edge detection to the classifier to produce robust predictions, while EdgeGANRob refill the edge image with texture information to reconstruct a new instance for predictions. that the shape of objects is the most important cue for human object recognition . Given the above evidence, a natural question emerges: Can we improve the robustness of CNNs by making it rely more on global shape structure? To answer this question, we need to formalize the notion of global shape structure first. We propose to consider a specific type of shape representation: edges (image points that have sharp change in brightness). Using edges comes with two benefits: 1) it is an effective device for modelling shape; 2) edges are easy to be captured in images, with many sophisticated algorithms (; ;) available. More specifically, this paper explores a new approach EdgeGANRob to improve the robustness of the CNNs to adversarial attacks,distribution shifting and backdoor attacks by leveraging structural information in images. The unified framework is shown in Figure 1. As illustrated, a simplified version of EdgeGANRob is a two-stage procedure named EdgeNetRob, which extracts the structural information by detecting edges and then trains the classifier on the extracted edges. As a consequence, EdgeNetRob forces the CNNs to make prediction solely based on shape information, rather than texture/color, thus eliminating the texture bias . Our show that EdgeNetRob can improve CNNs' robustness. However, there are still two challenges: (i) the direct differentiable edge detection algorithms are also vulnerable to attacks, which may lead to low robustness against sophisticated adaptive attackers. To handle this problem, we propose a robust edge detection algorithm, Robust Canny. Using Robust Canny is able to EdgeNetRob dramatically improve the robustness of EdgeGANRob. As a , this combined method outperforms the adversarial retraining based defense method. (ii). Although EdgeNetRob improves the CNNs' robustness, it decreases the clean accuracy of CNNs due to the missing texture/color information. This motivates the development of EdgeGANRob, which embeds a generative model to refill the texture/colors based on the edge images before they are fed into the classifier. Please find more visualization on the anonymous website: https://sites.google.com/view/edgenetrob. The main contributions of this paper include: (i) We propose a unified framework EdgeGANRob to improve the robustness of CNNs against multiple tasks simultaneously, which explicitly extracts edge/structure information from input images and then reconstructs the original images by refilling the textural information with GAN. (ii) To remain robust against sophisticated adaptive evasion attacks, in which attackers have access to the defense algorithm, we propose a robust edge detection approach Robust Canny based on the vanilla Canny algorithm to reduce the sensitivity of edge detector to adversarial perturbation. (iii) To further demonstrate the effectiveness of the inpainting GAN in EdgeGANRob, we also evaluate its simplified backbone procedure EdgeNetRob by performing learning tasks directly on the extracted robust edge features. To justify the above contributions, we conduct thorough evaluation on EdgeNetRob and EdgeGANRob in three tasks: adversarial attacks, distribution shifting and backdoor attacks, where significant improvements are achieved. Adversarial robustness A wide range of defense methods against adversarial examples have been proposed, among which many are shown to be not robust against adaptive attacks . The state-of-the-art defense methods are based on adversarial training. identified gradient obfuscation as a common pitfall for defense methods, thus suggested that defense methods should be evaluated against customized white-box attacks. suggested that defense methods should be evaluated against strong adaptive attacks. Distribution shifting Compared to adversarial examples, distribution shifting is more common and general in real-world applications. Jo and Bengio (2017b) shows that CNNs have a tendency to learn the superficial statistical cues. Recently, Wang et al. (2019a) proposes a method to robustify CNNs by penalizing the predictive power of the local representations and mitigating the tendency of fitting superficial statistical cues by evaluating on four patterns, including greyscale, negcolor, radial kernel and random kernel. proposes benchmark datasets for evaluating model robustness under common perturbations. Backdoor attack Backdoor attack (a;) is a type of poisoning attack that works by injecting a pattern into training data. As a , the trained models will predict a specific target class when certain pattern is injected into test data. has proposed a procedure to detect poisoned training data by using tools from robust statistics. proposes an approach to protect models from backdoor attacks by using neuron pruning. Robust visual features. Recent work has highlighted a connection between recognition robustness and robust features. For image recognition,; shows that CNNs rely more on textures than global shape structure, while humans rely more on shape structure than detailed texture. uses visualization methods and finds that adversarially robust models tend to capture global structure of the objects. argues that there exists non-robust features in natural images which are highly predictive but not interpretable by human. They showed that CNNs can obtain robustness by learning from images which contain only robust features. However, they did not directly identify which features are robust features. In this work, we propose explicitly to use edge as a robust feature. We introduce a new classification pipeline based on robust edge features, which we denote as EdgeGANRob. Our method first extracts edge/structure features from a given image and then reconstructs the original images by refilling the texture information with a trained generative adversarial network (GAN). The newly generated image is then fed into a classifier. In this section, we first describe a simplified backbone procedure of EdgeGANRob named EdgeNetRob, then introduce Robust Canny and inpainting GAN. Last, we describe three settings under which robustness is evaluated. As a simplified backbone of EdgeGANRob, EdgeNetRob consists of two stages: First, we exploit an edge detection method to extract edge maps from an image, and then a standard image classifier f θ (·) is trained on the extracted edge maps. Formally, denote the edge extractor function as e(·), the EdgeNetRob pipeline aims to solve the following problem: where D represents the underlying data distribution and L denotes the loss function (e.g., crossentropy loss). EdgeNetRob forces the decision of CNN to be solely based on edges, thus making it less sensitive to local textures. Since original images are transformed into edge maps, even if a pre-trained classifier on the original training data is available, we still need to train the edge classifier. Despite the simplicity of EdgeNetRob, it degrades the performance of CNNs over clean test data considering that the texture/color information is missing. This motivates us to develop EdgeGANRob which embeds a generative model to refill the texture/colors of the edge images. Because EdgeGANRob fills edge maps with texture/colors, which makes it more likely to achieve higher clean accuracy. The robustness of such classification system that builds upon edges depends highly on the used edge detector, as many existing edge detection algorithms are also vulnerable to attacks which may lead to low accuracy of the recognition task. This motivates us to propose a robust edge detection algorithm named Robust Canny in the next section. We now describe a robust edge detection method. Note that most neural network based edge detectors are non-robust. For example, finds that neural network based edge detectors like HED can fail easily when facing adversarial perturbation. In contrast, some traditional edge detection methods like Canny is intrinsically robust since they output binary edge maps. However, as illustrated in Figure 2 (first line), when adversarial perturbation is added, the output of Canny edge detector can become noisy. We propose to improve the robustness of vanilla Canny by truncating the noisy pixels in its intermediate stages. We refer to this modified version of Canny edge detector as Robust Canny. The 6 stages of our proposed Robust Canny include:Noise reduction: A Gaussian filter is applied to smooth the image.Gradient computation: We apply the Sobel operator to compute the gradient magnitude and direction at each pixel from the smoothed images.Noise masking: We reduce the noise in the presence of adversarial perturbations by thresholding the gradient magnitudes by a level α.Non-maximum suppression (NMS): An edge thinning step is taken to deblur the output of the Sobel operator. Gradient magnitudes that are not at a maximum along the direction of the gradient are suppressed (set to zero).Double threshold: Using a lower threshold and a higher threshold (θ l, θ h) for the gradient magnitude after NMS, pixels are mapped to 3 levels: strong, weak, and non-edge pixels,Edge tracking by hysteresis: Edge pixels are detected by finding strong pixels, or weak pixels that are connected to other strong pixels. Note that we have modified the vanilla Canny algorithm by adding a noise masking stage after computing the image gradients. Later in Figure 2, we show that the gradient computation stage is sensitive to input perturbations. Thus, we set all gradient magnitudes lower than a threshold α to zero to mitigate the perturbation noise. By adding a truncation operation, it is expected that adversarial noise on the gradient map with small magnitude will be reduced in early stages without affecting the quality of final edge maps. In addition to the masking operation, we find that the parameters of Canny (e.g. standard deviation of gaussian filter σ, thresholds θ l, θ h) also affect the robustness level. Specifically, we notice that larger σ and higher thresholds θ l, θ h in better robustness due to the stronger smoothing and pruning effects. This however comes at the cost of clean accuracy as larger σ leads to blurrier images and higher θ l, θ h may eliminate useful information in the output edges. To obtain a robust edge detector, we should carefully choose its parameters (e.g., σ, θ l, θ h). More details are provided in the experiment section. In this section, we describe how we train a Generative Adversarial Network (GAN) (a) in EdgeGANRob. Recall that the task of generating color images from edge maps is well defined under the image-to-image translation framework (pix2pix) . We train our inpainting GAN with two steps: in the first stage, we follow the common setup of pix2pix to train a conditional GAN using the following objective function: where L adv, L F M denote the adversarial loss (a) and feature matching loss with λ adv and λ F M control their relative importance. In the second stage, since we want our classifier to achieve high accuracy over the generated RGB images, we jointly fine-tune the trained GAN from first stage along with the classifier, using the following objective function: min where L cls represents the classification loss of generated images by inpainting GAN. Note that in the first step we do not include classification loss because we want our GAN to generate more realistic images, based on which it would be easier to fine-tune the classifier to gain accuracy. Our method simultaneously improves robustness under three different settings: (i) adversarial attack, (ii) distribution shifting, (iii) backdoor attack. In terms of adversarial attack, EdgeGANRob is expected to improve robustness as edges are invariant to small imperceptible adversarial perturbations. Intuitively, consider a ∞ threat model, it is very challenging for an attacker to make a specific edge pixel appear/disappear by reversing the magnitude of image gradient with only limited adversarial budget per pixel. When test data is under distribution shifting with well-preserved shape structure, leveraging edge features could be helpful to improve model's generalization ability. EdgeGANRob would work in this case by focusing on shape structure which makes it less sensitive to distribution change during testing. Recall that in backdoor attack, an attacker aims to poison the training data with a specific pattern such that the trained models can be tricked into predicting a certain class when the pattern is injected at testing time. In our cases, extracting edges can be viewed as a data sanitization step to remove the malicious pattern, thus rendering potential backdoor attacks ineffective. We evaluate the robustness of the proposed method in this section. Though EdgeNetRob is just a backbone of EdgeGANRob without inpainting GAN, our experiments show that it has unique advantage in certain settings and is of independent interest as a robust recognition method. Thus we also list it as an independent method to compare with EdgeGANRob. For our methods, we first evaluate their robustness against adversarial attacks, followed by an evaluation of their performance over distribution shifting. In addition, we evaluate the robustness against backdoor attacks. We conduct our experiments on two datasets: Fashion MNIST and CelebA . On CelebA, we evaluate our method on the task of gender classification. We did not choose the more popular MNIST and CIFAR-10 datasets because MNIST is a toy dataset where strong robustness has been achieved ) and CIFAR-10 is a low-resolution dataset (32 × 32) where it is hard to extract semantically meaningful edges. Thus it can not provide informative benchmarks for our study. We use the same network architecture of classification for our method and the vanilla classifier. More details are shown in Appendix A. We evaluate our methods using the commonly used ∞ adversarial perturbation constraints with input range;;; ). We use standard perturbation budget on these two datasets as in;;. For Fashion MNIST, we use an ∞ budget of 8/256 and 25/256. For CelebA, we use an ∞ budget of 2/256 and 8/256. We evaluate our methods against adaptive attack (i.e., the attacker is fully aware of the defense algorithm). Specifically, we measure the robustness to white-box attacks by the BPDA attack . This attack requires a differentiable version of Canny, which is provided in Appendix C. More details on the attack setting are provided in Appendix B. First, we illustrate why a robust edge detector is needed for defending against adversarial attacks. We compare the robustness of three edge detection methods: 1) RCF which uses a CNN as backbone to generate edge maps; 2) Canny which is the traditonal Canny edge detection; 3) Robust Canny. For each of the edge detection method, we train a classifier on the extracted edge maps. The for Fashion MNIST are reported in Table 1. First, we can see that using edges generated by RCF is not robust, as under strong adaptive attack, the accuracy drops near to 0. This is in accordance with We present our for two benchmark datasets Fashion MNIST and CelebA . We compare with the state-of-the-art baseline: Adversarial training proposed in. Adversarial training ) is one of the most effective defense methods, achieving strong robustness to white-box attacks. The overall are shown in Table 2. We notice that EdgeNetRob and EdgeGANRob leads to a small drop in clean accuracy compared to the vanilla baseline model. However, when compared with adversarial training with = 8, both EdgeNetRob and EdgeGANRob achieve higher clean accuracy. We also observe that EdgeGANRob has higher clean accuracy than EdgeNetRob on CelebA dataset, thus validating the necessity of adding GANs on more complicated dataset to close the accuracy gap ed from directly training on binary edge images. In terms of adversarial robustness, we observe that under strong adaptive attacks, EdgeNetRob and EdgeGANRob still remain robustness level better than or comparable to adversarial training baselines. It is worth noting that EdgeNetRob does not use adversarial training and thus has the advantage of time efficiency. We also show the plots for test accuracy under different attack iterations in Figure 3. We test our method for the generalization ability under distribution shifting. We follow the experiment settings in HEX (b) and PAR (a), where we test the models under perturbed Fashion MNIST and CelebA with four types of patterns: greyscale, negative color, random kernel and radial kernel. The random kernel and radial kernel transformations are introduced in Jo and Bengio (2017a), which use Fourier filtering to transform an image while perserving high level semantics. We compare with state-of-the-art method PAR introduced in Wang et al. (2019a), which adds a local patchwise adversarial regularization loss. Some visualization of perturbed images are shown in Appendix D. The overall are shown in Table 3. We can see that EdgeNetRob and EdgeGANRob significantly improve the accuracy on three types of patterns: negative color, radial kernel and random kernel, while outperforming PAR. When testing on greyscale images, similar to baselines, our methods remain high accuracy. The show that edge features are helpful for CNN's generalization to test data under distribution shifting. We show that our method can be used as a defense against backdoor attacks. We follow the attack setup in. We embed invisible watermark pattern letter "A" into the pristine image for Fashion MNIST and letters "classified" into CelebA. The qualitative are shown in Figure 4 on CelebA and Figure D in Appendix for Fashion MNIST. For Fashion MNIST, we randomly choose four attack and target pairs (attack, target) as (t-shirt, trouser), (trouser, pullover), (dress, coat), (coat, dress). For CelebA, the pairs (attack, target) are (male, female) and (female, male). We select the poisoning ratio as 20% and 30% for Fashion MNIST and 5% and 10% for CelebA. We compare our method with the baseline method proposed in , denoted as Spectral Signature. The are presented in Table 4 and Table 5, where we show the test accuracy over both standard test data ('Clean Acc') and poisoned data ('Pois Acc'). We observe that our embedding pattern can successfully attack the vanilla Net with high poisoning accuracy on both CelebA and Fashion MNIST under all settings. It can be seen that Spectral Signature can not always achieve good performance with such invisible watermark patterns while EdgeNetRob and EdgeGANRob consistently remain low poisoning accuracy. Figure 4 shows the qualitative of the backdoor images after edge detection algorithm and the reconstructed images. We can observe that the effect of invisible watermark pattern can be removed by the edge detector. In addition, we find that EdgeGANRob achieves better clean accuracy compared with EdgeNetRob which also validates the benefit introduced by inserting an inpainting GAN. We introduced a new method based on robust edge features for improving general model robustness. By combining a robust edge feature extractor with the generative adversarial network, our method simultaneously achieves competitive in terms of both adversarial robustness and generalization under distribution shifting. Additionally, we show that it can also be used to improve robustness against backdoor attacks. Our highlight the importance of using shape information in improving model robustness and we believe it is a promising direction for future work. For data pre-processing, we resize the images in CelebA to 128 × 128 using bicubic interpolation, and use 10% of total images as test data. For both datasets, we normalize the data into the range of [−1, 1]. On Fashion-MNIST, we use a LeNet-style CNN (Table A). For CelebA dataset, we use the standard ResNet with depth 20. Models are trained using stochastic gradient descent with momentum. (b), Projected Gradient Descent (PGD) and the Carlini & Wagner ∞ attack (CW) . For PGD attacks, we evaluate 10 steps and 40 steps PGD, denoted as'PGD-10' and'PGD-40' separately. For ∞ distance of 2/256 or 8/256, step size is set to be 0.005. For ∞ distance of 25/256, we use step size 0.015. For CW attack, we randomly sample 1,000 images for evaluation due to its high computational complexity. We use Robust Canny for evaluation of adversarial robustness. Here we report the hyper-parameters used in Robust Canny, which are chosen using the validation set to trade off robustness and accuracy. For Fashion MNIST, we set σ = 1, θ l = 0.1, θ h = 0.2, α = 0.3. For CelebA, we set σ = 2.5, θ l = 0.2, θ h = 0.3, α = 0.2. Note that the last three steps in the Robust Canny algorithm are non-differentiable transformations. However, in a white-box attack scenario one needs to backpropagate gradient through the edge detection algorithm for constructing adversarial samples. While obfuscating gradients through nondifferentiable transformations is a commonly used defense technique, show that the attacker can replace such transformation with differentiable approximations, refered to as the Backward Pass Differentiable Approximation (BPDA) technique, to construct adversarial examples. Therefore, to realize a stronger attack on our method, we find a differentiable approximation of the Robust Canny algorithm as follows. Assuming x to hold the pixel intensities in the original image, and x e to be the output of the RCanny algorithm, we can break the transformation into two stages: C 1 (·), comprised of step 1-3, and C 2 (·) for steps 4-6 (Thresholding operation in step 3 can be formulated as a shifted ReLU function). Note that C 2 (·) is a non-differentiable operation, where the output is a masked version of the input: C 2 (x) = M (x) ⊗ x, where M (·) produces the mask (i.e., an array of zeros and ones) produced by steps 3-6, and ⊗ denotes element-wise multiplication. Therefore, we can write: x e = R-Canny(x) = C 2 (C 1 (x)) = M (C 1 (x)) ⊗ C 1 (x) To obtain a differentiable approximation of R-Canny for BPDA, we assume the mask to be constant. In other words, we only backpropagate gradients through C 1 (·), and not M (·). In Figure A, we show the change of test accuracy under radial mask and random mask transformations with different parameters. For radial mask transformation, we vary the radius of mask in fourier domain. For random mask transformation, we sample the random masks with various probabilities. Figure B and?? show the additional visualization fro CelebA under four types of distribution shifting. Figure D shows the qualitative of EdgeGANRob and EdgeNetRob for backdoor attacks on Fashion MNIST. We can also observe that the poisoning pattern can be slightly removed by EdgeNetRob and the patterns for each of the generated images do not share the similar patterns.

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A unified  model  to improve model robustness against multiple tasks

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We propose a novel adversarial training with domain adaptation method that significantly improves the generalization ability on adversarial examples from different attacks.


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Question: Learning rich representation from data is an important task for deep generative models such as variational auto-encoder (VAE). However, by extracting high-level abstractions in the bottom-up inference process, the goal of preserving all factors of variations for top-down generation is compromised. Motivated by the concept of “starting small”, we present a strategy to progressively learn independent hierarchical representations from high- to low-levels of abstractions. The model starts with learning the most abstract representation, and then progressively grow the network architecture to introduce new representations at different levels of abstraction. We quantitatively demonstrate the ability of the presented model to improve disentanglement in comparison to existing works on two benchmark datasets using three disentanglement metrics, including a new metric we proposed to complement the previously-presented metric of mutual information gap. We further present both qualitative and quantitative evidence on how the progression of learning improves disentangling of hierarchical representations. By drawing on the respective advantage of hierarchical representation learning and progressive learning, this is to our knowledge the first attempt to improve disentanglement by progressively growing the capacity of VAE to learn hierarchical representations. Variational auto-encoder (VAE), a popular deep generative model (DGM), has shown great promise in learning interpretable and semantically meaningful representations of data;; ). However, VAE has not been able to fully utilize the depth of neural networks like its supervised counterparts, for which a fundamental cause lies in the inherent conflict between the bottom-up inference and top-down generation process : while the bottom-up abstraction is able to extract high-level representations helpful for discriminative tasks, the goal of generation requires the preservation of all generative factors that are likely at different abstraction levels. This issue was addressed in recent works by allowing VAEs to generate from details added at different depths of the network, using either memory modules between top-down generation layers , or hierarchical latent representations extracted at different depths via a variational ladder autoencoder . However, it is difficult to learn to extract and disentangle all generative factors at once, especially at different abstraction levels. Inspired by human cognition system, suggested the importance of "starting small" in two aspects of the learning process of neural networks: incremental input in which a network is trained with data and tasks of increasing complexity, and incremental memory in which the network capacity undergoes developmental changes given fixed external data and tasks -both pointing to an incremental learning strategy for simplifying a complex final task. Indeed, the former concept of incremental input has underpinned the success of curriculum learning . In the context of DGMs, various stacked versions of generative adversarial networks (GANs) have been proposed to decompose the final task of high-resolution image generation into progressive sub-tasks of generating small to large images . The latter aspect of "starting small" with incremental growth of network capacity is less explored, although recent works have demonstrated the advantage of progressively growing the depth of GANs for generating high-resolution images . These works, so far, have focused on progressive learning as a strategy to improve image generation. We are motivated to investigate the possibility to use progressive learning strategies to improve learning and disentangling of hierarchical representations. At a high level, the idea of progressively or sequentially learning latent representations has been previously considered in VAE. , the network learned to sequentially refine generated images through recurrent networks. , a teacher-student training strategy was used to progressively increase the number of latent dimensions in VAE to improve the generation of images while preserving the disentangling ability of the teacher model. However, these works primarily focus on progressively growing the capacity of VAE to generate, rather than to extract and disentangle hierarchical representations. In comparison, in this work, we focus on 1) progressively growing the capacity of the network to extract hierarchical representations, and 2) these hierarchical representations are extracted and used in generation from different abstraction levels. We present a simple progressive training strategy that grows the hierarchical latent representations from different depths of the inference and generation model, learning from high-to low-levels of abstractions as the capacity of the model architecture grows. Because it can be viewed as a progressive strategy to train the VLAE presented in , we term the presented model pro-VLAE. We quantitatively demonstrate the ability of pro-VLAE to improve disentanglement on two benchmark data sets using three disentanglement metrics, including a new metric we proposed to complement the metric of mutual information gap (MIG) previously presented in. These quantitative studies include comprehensive comparisons to β-VAE ), VLAE , and the teacher-student strategy as presented in at different values of the hyperparameter β. We further present both qualitative and quantitative evidence that pro-VLAE is able to first learn the most abstract representations and then progressively disentangle existing factors or learn new factors at lower levels of abstraction, improving disentangling of hierarhical representations in the process. A hierarchy of feature maps can be naturally formed in stacked discriminative models . Similarly, in DGM, many works have proposed stacked-VAEs as a common way to learn a hierarchy of latent variables and thereby improve image generation (Sønderby et al.;; ). However, this stacked hierarchy is not only difficult to train as the depths increases (Sønderby et al.; ), but also has an unclear benefit for learning either hierarchical or disentangled representations: as shown in , when fully optimized, it is equivalent to a model with a single layer of latent variables. Alternatively, instead of a hierarchy of latent variables, independent hierarchical representations at different abstraction levels can be extracted and used in generation from different depths of the network . A similar idea was presented in to generate lost details from memory and attention modules at different depths of the top-down generation process. The presented work aligns with existing works in learning independent hierarchical representation from different levels of abstraction, and we look to facilitate this learning by progressively learning the representations from high-to low-levels. Progressive learning has been successful for high-quality image generation, mostly in the setting of GANs. Following the seminar work of , these progressive strategies can be loosely grouped into two categories. Mostly, in line with incremental input, several works have proposed to divide the final task of image generation into progressive tasks of generating low-resolution to high-resolution images with multi-scale supervision . Alternatively, in line with incremental memory, a small number of works have demonstrated the ability to simply grow the architecture of GANs from a shallow network with limited capacity for generating low-resolution images, to a deep network capable of generating super-resolution images . This approach was also shown to be time-efficient since the early-stage small networks require less time to converge comparing to training a full network from the beginning. This latter group of works provided compelling evidence for the benefit of progressively growing the capacity of a network to generate images, although its extension for growing the capacity of a network to learn hierarchical representations has not been explored. Limited work has considered incremental learning of representations in VAE. , recurrent networks with attention mechanisms were used to sequentially refines the details in gen-erated images. It however focused on the generation performance of VAE without considering the learned representations. , a teacher-student strategy was used to progressively grow the dimension of the latent representations in VAE. Its fundamental motivation was that, given a teacher model that has learned to effectively disentangle major factors of variations, progressively learning additional nuisance variables will improve generation without compromising the disentangling ability of the teacher -the latter accomplished via a newly-proposed Jacobian supervision. The capacity of this model to grow, thus, is by design limited to the extraction of nuisance variables. In comparison, we are interested in a more significant growth of the VAE capacity to progressively learn and improve disentangling of important factors of variations which, as we will later demonstrate, is not what the model in is intended for. In addition, neither of these works considered learning different levels of abstractions at different depths of the network, and the presented pro-VLAE provides a simpler training strategy to achieve progressive representation learning. Learning disentangled representation is a primary motivation of our work, and an important topic in VAE. Existing works mainly tackle this by promoting the independence among the learned latent factors in VAE;; ). The presented progressive learning strategy provides a novel approach to improve disentangling that is different to these existing methods and a possibility to augment them in the future. We assume a generative model p(x, z) = p(x|z)p(z) for observed x and its latent variable z. To learn hierarchical representations of x, we decompose z into {z 1, z 2, ..., z L} with z l (l = 1, 2, 3, ..., L) from different abstraction levels that are loosely guided by the depth of neural network as in. We define the hierarchical generative model p θ as: Note that there is no hierarchical dependence among the latent variables as in common hierarchical latent variable models. Rather, similar to that in and , z l's are independent and each represents generative factors at an abstraction level not captured in other levels. We then define an inference model q φ to approximate the posterior as: where h l (x) represents a particular level of bottom-up abstraction of x. We parameterize p θ and q φ with an encoding-decoding structure and, as in , we approximate the abstraction level with the network depth. The full model is illustrated in Fig. 1 (c), with a final goal to maximize a modified evidence lower bound (ELBO) of the marginal likelihood of data x: where KL denotes the Kullback-Leibler divergence, prior p(z) is set to isotropic Gaussian N (0, I) according to standard practice, and β is a hyperparameter introduced in to promote disentangling, defaulting to the standrd ELBO objective when β = 1. We present a progressive learning strategy, as illustrated in Fig. 1, to achieve the final goal in equation by learning the latent variables z l progressively from the highest (l = L) to the lowest l = 1) level of abstractions. We start by learning the most abstraction representations at layer L as show in Fig. 1(a). In this case, our model degenerates to a vanilla VAE with latent variables z L at the deepest layer. We keep the dimension of z L small to start small in terms of the capacity to learn latent representations, where we define the inference model at progressive step s = 0 as: where f e l, µ L, and σ L are parts of the encoder architecture, f d l are parts of the decoder architecture, and D is the distribution of x parametrized by f d 0 (g 0), which can be either Bernoulli or Gaussian depending on the data. Next, as shown in Fig. 1, we progressively grow the model to learn z L−1,..., z 2, z 1 from high to low abstraction levels. At each progressive step s = 1, 2,..., L − 1, we move down one abstraction level, and grow the inference model by introducing new latent code: Simultaneously, we grow the decoder such that it can generate with the new latent code as: where m l includes transposed convolution layers outputting a feature map in the same shape as g l+1, and [·; ·] denotes a concatenation operation. The training objective at progressive step s is then: By replacing the full objective in equation with a sequence of the objectives in equation as the training progresses, we incrementally learn to extract and generate with hierarchical latent representations z l's from high to low levels of abstractions. Once trained, the full model as shown in Fig. 1 (c) will be used for inference and generation, and progressive processes are no loner needed. Two important strategies are utilized to implement the proposed progressive representation learning. First, directly adding new components to a trained network often introduce a sudden shock to the gradient: in VAEs, this often leads to the explosion of the variance in the latent distributions. To avoid this shock, we adopt the popular method of "fade-in" to smoothly blend the new and existing network components. In specific, we introduce a "fade-in" coefficient α to equations and when growing new components in the encoder and the decoder: where α increases from 0 to 1 within a certain number of iterations (5000 in our experiments) since the addition of the new network components µ l,σ l, and m l. Second, we further stabilize the training by weakly constraining the distribution of z l's before they are added to the network. This can be achieved by a applying a KL penalty, modulated by a small coefficient γ, to all latent variables that have not been used in the generation at progressive step s: where γ is set to 0.5 in our experiments. The final training objective at step s then becomes: Note that the latent variables at the hierarchy lower than L − s are neither meaningfully inferred nor used in generation at progressive step s, and L pre−trained merely intends to regularize the distribution of these latent variables before they are added to the network. In the experiments below, we use both "fade-in" and L pre−trained when implementing the progressive training strategy. Various quantitative metrics for measuring disentanglement have been proposed;; ). For instance, the recently proposed MIG metrics measures the gap of mutual information between the top two latent dimensions that have the highest mutual information with a given generative factor. A low MIG score, therefore, suggests an undesired outcome that the same factor is split into multiple dimensions. However, if different generative factors are entangled into the same latent dimension, the MIG score will not be affected. Therefore, we propose a new disentanglement metric to supplement MIG by recognizing the entanglement of multiple generative factors into the same latent dimension. We define MIG-sup as: where z is the latent variables and v is the ground truth factors, J is the number of meaningful latent dimensions, and I norm (z j ; v k) is normalized mutual information I(z j ; v k)/H(v k). Considering MIG and MIG-sup together will provide a more complete measure of disentanglement, accounting for both the splitting of one factor into multiple dimensions and the encoding of multiple factors into the same dimension. In an ideal disentanglement, both MIG and MIG-sup should be 1, recognizing a one-to-one relationship between a generative factor and a latent dimension. This would have a similar effect to the metric that was proposed in , although MIG-based metrics do not rely on training extra classifiers or regressors and are unbiased for hyperparameter settings. The factor metric also has similar properties with MIG-sup, although MIG-sup is stricter on penalizing any amount of other minor factors in the same dimension. We tested the presented pro-VLAE on four benchmark data sets: dSprites ), 3DShapes , MNIST , and CelebA , where the first two include ground-truth generative factors that allow us to carry out comprehensive quantitative comparisons of disentangling metrics with existing models. In the following, we first quantitatively compare the disentangling ability of pro-VLAE in comparison to three existing models using three disentanglement metrics. We then analyze pro-VLAE from the aspects of how it learns progressively, its ability to disentangle, and its ability to learn abstractions at different levels. Comparisons in quantitative disentanglement metrics: For quantitative comparisons, we considered the factor metric in , the MIG in , and the MIG-sup presented in this work. We compared pro-VLAE (changing β) with beta-VAE ), VLAE as a hierarchical baseline without progressive training, and the teacherstudent model as the most related progressive VAE without hierarchical representations. All models were considered at different values of β except the teacher-student model: the comparison of β-VAE, VLAE, and the presented pro-VLAE thus also provides an ablation study on the effect of learning hierarchical representations and doing so in a progressive manner. For fair comparisons, we strictly required all models to have the same number of latent variables and the same number of training iterations. For instance, if a hierarchical model has three layers that each has three latent dimensions, a non-hierarchical model will have nine latent dimensions; if a progressive method has three progressive steps with 15 epochs of training each, a non-progressive method will be trained for 45 epochs. Three to five experiments were conducted for each model at each β value, and the average of the top three is used for reporting the quantitative in Fig. 2. As shown, for MIG and MIG-sup, VLAE generally outperformed β-VAE at most β values, while pro-VLAE showed a clear margin of improvement over both methods. With the factor metric, pro-VLAE was still among the top performers, although with a smaller margin and a larger overlap with VLAE on 3DShapes, and with β-VAE (β = 10) on dSprites. The teacher-student strategy with Jacobian supervision in general had a low to moderate disentangling score, especially on 3DShapes. This is consistent with the original motivation of the method for progressively learning nuisance variables after the teacher learns to disentangle effectively, rather than progressively disentangling hierarchical factors of variations as intended by pro-VLAE. Note that pro-VLAE in general performed better with a smaller value of β (β < 20), suggesting that progressive learning already had an effect of promoting disentangling and a high value of β may over-promote disentangling at the expense of reconstruction quality. Fig. 3 shows MIG vs. MIG-sup scores among the tested models. As shown, from pro-VLAE were well separated from the other three models at the right top quadrant of the plots, obtaining simultaneously high MIG and MIG-sup scores as a clear evidence for improved disentangling ability. Fig. 4 provides images generated by traversing each latent dimension using the best pro-VLAE (β = 8), the best VLAE (β = 10), and the teacher-student model on 3DShapes data. As shown, pro-VLAE learned to disentangle the object, wall, and floor color in the deepest layer; the following hierarchy of representations then disentangled objective scale, orientation, and shape, while the lowest-level of abstractions ran out of meaningful generative factors to learn. In comparison, the VLAE distributed six generative factors over the nine latent dimensions, where color was split across At each progression and for each z l, the row of images are generated by randomly sampling from its prior distributions while fixing the other latent variables (this is NOT traversing). The green bar at each row tracks the mutual information I(x; z l), while the total mutual information I(x; z) is labeled on top. the hierarchy and sometimes entangled with the object scale (in z 2). The teacher-student model was much less disentangled, which we will delve into further in the following section. To further understand what happened during progressive learning, we use mutual information I(x, z l) as a surrogate to track the amount of information learned in each hierarchy of latent variables z l during the progressive learning. We adopted the approach in to empirically estimate the mutual information by stratified sampling. Fig. 5 shows an example from 3DShapes. At progressive step 0, pro-VAE was only learning the deepest latent variables in z 3, discovering most of the generative factors including color, objective shape, and orientation entangled within z 3. At progressive step 1, interestingly, the model was able to "drag" out shape and rotation factors from z 3 and disentangle them into z 2 along with a new scale factor. Thus I(x; z3) decreased from 10.59 to 6.94 while I(x; z2) increased from 0.02 to 5.98 in this progression, while the total mutual information I(x; z) increased from 10.61 to 12.84, suggesting the overall learning of more detailed information. Since 3DShapes only has 6 factors, the lowest-level representation z 1 had nothing to learn in progressive step 2, and the allocation of mutual information remained nearly unchanged. Note that the sum of I(x, z l)'s does not equal to I(x, z) and I over = L 1 I(x, z l) − I(x, z) suggests the amount of information that is entangled. In comparison, the teacher-student model was less effective in progressively dragging entangled representations to newly added latent dimensions, as suggested by the slowing changing of I(x, z l)'s Figure 6: Visualization of hierarchical features learnt for MNIST data. Each sub-figure is generated by randomly sampling from the prior distribution of z l at one abstraction level while fixing the others. The original latent code is inferred from a image with digit "0". From left to right: z 3 encodes the highest abstraction: digit identity; z 2 encodes stroke width; and z 1 encodes other digit styles. Figure 7: Visualization of hierarchical features learnt for CelebA data. Each subfigure is generated by traversing along a selected latent dimension in each row within each hierarchy of z l's. From left to right: latent variables z 4 to z 1 progressively learn major (e.g., gender in z 4 and smile in z 3) to minor representations (e.g. wavy-hair in z 2 and eye-shadow in z 1) in a disentangled manner. during progression and the larger value of I over. This suggests that, since the teacher-student model was motivated for progressively learning nuisance variables, the extent to which its capacity can grow for learning new representations is limited by two fundamental causes: 1) because it increases the dimension of the same latent vectors at the same depth, the growth of the network capacity is limited in comparison to pro-VLAE, and 2) the Jacobian supervision further restricts the student model to maintain the same disentangling ability of the teacher model. We also qualitatively examined pro-VLAE on data with both relatively simple (MNIST) and complex (CelebA) factors of variations, all done in unsupervised training. On MNIST (Figure 6), while the deepest latent representations encoded the highest-level features in terms of digit identity, the representations learned at shallower levels encoded changes in writing styles. In Figure 7, we show the latent representation progressively learned in CelebA from the highest to lowest levels of abstractions, along with disentangling within each level demonstrated by traversing one selected dimension at a time. These dimensions are selected as examples associated with clear semantic meanings. As shown, while the deepest latent representation z 4 learned to disentangle high-level features such as gender and race, the shallowest representation z 1 learned to disentangle low-level features such as eye-shadow. Moreover, the number of distinct representations learned decreased from deep to shallow layers. While demonstrating disentangling by traversing each individual latent dimension or by hierarchically-learned representations has been separately reported in previous works; ), to our knowledge this is the first time the ability of a model to disentangle individual latent factors in a hierarchical manner has been demonstrated. This provides evidence that the presented progressive strategy of learning can improve the disentangling of first the most abstract representations followed by progressively lower levels of abstractions. In this work, we present a progressive strategy for learning and disentangling hierarchical representations. Starting from a simple VAE, the model first learn the most abstract representation. Next, the model learn independent representations from high-to low-levels of abstraction by progressively growing the capacity of the VAE deep to shallow. Experiments on several benchmark data sets demonstrated the advantages of the presented method. An immediate future work is to include stronger guidance for allocating information across the hierarchy of abstraction levels, either through external multi-scale image supervision or internal information-theoretic regularization strategies. Figure 8: An example of one factor being encoded in multiple dimensions. Each row is a traverse for one dimension (dimension order adjusted for better visualization). Notice that both dim1 and dim2 are encoding floor-color, both dim3 and dim4 are encoding wall-color, and both dim5 and dim6 are encoding object color. Therefore, the MIG is very low since it penalizes splitting one factor to multiple dimensions. On the other hand, the MIG-sup and factor-metric is not too bad since one dimension mainly encodes one factor, even though there are some entanglement of color-vs-shape and color-vs-scale. Figure 9: An example of one dimension containing multiple factors. Each row is a traverse for one dimension (dimension order adjusted for better visualization). Notice that both models achieve high and similar MIG because all 6 factors are encoded and no splitting to multiple dimensions. However, the right-hand side model has much lower MIG-sup and factor-metric than the left-hand side model. Because both scale and shape are encoded in dim5, while dim6 has no factor. Both MIG-sup and factor-metric penalize encoding multiple factors in one dimension. Besides, our MIG-sup is lower and drops more than factor-metric because MIG-sup is stricter in this case. Figure 5 of . The network has 3 layers and 2 dimensional latent code at each layer. Each image is generated by traversing each of the two-dimensional latent code in one layer, while randomly sampling from the other layers. From left to right: The top layer z 3 encodes the digit identity and tilt; z 2 encodes digit width (digits around top-left are thicker than digits around bottom-right); and the bottom layer z 1 encodes stroke width. Compared to VLAE, the representation learnt in the presented method suggests smoother traversing on digits and similar for digit width and stroke width. Table 1: Mutual information I(x; z l) between data x and latent codes z l at each l-th depth of the network, corresponding to the qualitative presented in Fig. 4 and Fig. 6 on 3Dshapes and MNIST data sets. Both VLAE and the presented pro-VLAE models have the same hierarchical architecture with 3 layers and 3 latent dimensions for each layer. Compared to VLAE, the presented method allocates information in a more clear descending order owing to the progressive learning. 3DShapes I(x; z 3) I(x; z 2) I(x; z 1) total I(x; z) In this section, we present additional quantitative on how information flow among the latent variables during progressive training. We conducted experiments on both 3DShapes and MNIST data sets, considering different hierarchical architectures including a combination of different number of latent layers L and different number of latent dimensions z dim for each layer. Each experiment was repeated three times with random initializations, from which the mean and the standard deviation of mutual information I(x; z l) were computed. As shown in Tables 3-8, for all hierarchical architectures, the information amount in each layer is captured in a clear descending order, which aligns with the motivation of the presented progressive learning strategy. Generally, the information also tends to flow from previous layers to new layers, suggesting a disentanglement of latent factors as new latent layers are added. This is especially obvious for 3DShapes data where the generative factors are better defined. In addition, models with small latent codes (z dim = 1) are not able to learn the same amount of information (total I(x, z)) as those with larger latent codes (z dim = 3). The variance of information in each layer in the former also appears to be high. We reason that it may be because that the model is trying to squeeze too much information into a small code, ing in large vibrations during progressive learning. On the other hand, while a model has large latent codes (L = 4, z dim = 3), the information flow becomes less clear after the addition of certain layers. Overall, assuming there are K generative factors and there are D dimensions in total available in model, ideally we would like to design the model such that D = K. However, since K is unknown in most data, L and z dim become hyperparameters that need to be tuned for different data sets. Table 3: 3DShapes, L = 2, z dim = 3 progressive step I(x; z 2) I(x; z 1) total I(x; z) 0 10.68 ± 0.19 10.68 ± 0.19 1 7.22 ± 0.30 5.94 ± 0.26 12.88 ± 0.20 Table 4: 3DShapes, L = 3, z dim = 2 progressive step I(x; z 3) I(x; z 2) I(x; z 1) total I(x; z) 0 10.16 ± 0.13 10.16 ± 0.13 1 9.76 ± 0.05 7.36 ± 0.10 13.00 ± 0.02 2 6.83 ± 1.37 6.66 ± 0.17 5.80 ± 0.41 13.07 ± 0.02

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We proposed a progressive learning method to improve learning and disentangling latent representations at different levels of abstraction.

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We propose a novel way to incorporate conditional image information into the discriminator of GANs using feature fusion that can be used for structured prediction tasks.


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Question: Deep latent variable models are powerful tools for representation learning. In this paper, we adopt the deep information bottleneck model, identify its shortcomings and propose a model that circumvents them. To this end, we apply a copula transformation which, by restoring the invariance properties of the information bottleneck method, leads to disentanglement of the features in the latent space. Building on that, we show how this transformation translates to sparsity of the latent space in the new model. We evaluate our method on artificial and real data. In recent years, deep latent variable models BID13 BID22 BID7 have become a popular toolbox in the machine learning community for a wide range of applications BID14 BID19 BID11. At the same time, the compact representation, sparsity and interpretability of the latent feature space have been identified as crucial elements of such models. In this context, multiple contributions have been made in the field of relevant feature extraction BID3 BID0 and learning of disentangled representations of the latent space BID5 BID2 BID9.In this paper, we consider latent space representation learning. We focus on disentangling features with the copula transformation and, building on that, on forcing a compact low-dimensional representation with a sparsity-inducing model formulation. To this end, we adopt the deep information bottleneck (DIB) model BID0 which combines the information bottleneck and variational autoencoder methods. The information bottleneck (IB) principle BID26 identifies relevant features with respect to a target variable. It takes two random vectors x and y and searches for a third random vector t which, while compressing x, preserves information contained in y. A variational autoencoder (VAE) BID13 BID22 ) is a generative model which learns a latent representation t of x by using the variational approach. Although DIB produces good in terms of image classification and adversarial attacks, it suffers from two major shortcomings. First, the IB solution only depends on the copula of x and y and is thus invariant to strictly monotone transformations of the marginal distributions. DIB does not preserve this invariance, which means that it is unnecessarily complex by also implicitly modelling the marginal distributions. We elaborate on the fundamental issues arising from this lack of invariance in Section 3. Second, the latent space of the IB is not sparse which in the fact that a compact feature representation is not feasible. Our contribution is two-fold: In the first step, we restore the invariance properties of the information bottleneck solution in the DIB. We achieve this by applying a transformation of x and y which makes the latent space only depend on the copula. This is a way to fully represent all the desirable features inherent to the IB formulation. The model is also simplified by ensuring robust and fully non-parametric treatment of the marginal distributions. In addition, the problems arising from the lack of invariance to monotone transformations of the marginals are solved. In the second step, once the invariance properties are restored, we exploit the sparse structure of the latent space of DIB. This is possible thanks to the copula transformation in conjunction with using the sparse parametrisation of the information bottleneck, proposed by BID21. It translates to a more compact latent space that in a better interpretability of the model. The remainder of this paper is structured as follows: In Section 2, we review publications on related models. Subsequently, in Section 3, we describe the proposed copula transformation and show how it fixes the shortcomings of DIB, as well as elaborate on the sparsity induced in the latent space. In Section 4, we present of both synthetic and real data experiments. We conclude our paper in Section 5. The IB principle was introduced by BID26. The idea is to compress the random vector x while retaining the information of the random vector y. This is achieved by solving the following variational problem: min p(t|x) I(x; t)−λI(t; y), with the assumption that y is conditionally independent of t given x, and where I stands for mutual information. In recent years, copula models were combined with the IB principle in BID20 and extended to the sparse meta-Gaussian IB BID21 to become invariant against strictly monotone transformations. Moreover, the IB method has been applied to the analysis of deep neural networks in BID25, by quantifying mutual information between the network layers and deriving an information theoretic limit on DNN efficiency. The variational bound and reparametrisation trick for autoencoders were introduced in BID13 BID22. The variational autoencoder aims to learn the posterior distribution of the latent space p(t|x) and the decoder p(x|t). The general idea of combining the two approaches is to identify the solution t of the information bottleneck with the latent space t of the variational autoencoder. Consequently, the terms I(x; t) and I(t; y) in the IB problem can be expressed in terms of the parametrised conditionals p(t|x), p(y|t).Variational lower bounds on the information bottleneck optimisation problem have been considered in BID3 and BID0. Both approaches, however, treat the differential entropy of the marginal distribution as a positive constant, which is not always justified (see Section 3). A related model is introduced in BID18, where a penalty on the entropy of output distributions of neural networks is imposed. These approaches do not introduce the invariance against strictly monotone transformations and thus do not address the issues we identify in Section 3.A sizeable amount of work on modelling the latent space of deep neural networks has been done. The authors of BID1 propose the use of a group sparsity regulariser. Other techniques, e.g. in BID17 ) are based on removing neurons which have a limited impact on the output layer, but they frequently do not scale well with the overall network size. More recent approaches include training neural networks of smaller size to mimic a deep network BID10 BID23. In addition, multiple contributions have been proposed in the area of latent space disentanglement BID5 BID2 BID9 BID6. None of the approaches consider the influence of the copula on the modelled latent space. Copula models have been proposed in the context of Bayesian variational methods in BID24, BID27 and BID8. The former approaches focus on treating the latent space variables as indicators of local approximations of the original space. None of the three approaches relate to the information bottleneck framework. In order to specify our model, we start with a parametric formulation of the information bottleneck: max DISPLAYFORM0 where I stands for mutual information with its parameters in the subscript. A parametric form of the conditionals p φ (t|x) and p θ (y|t) as well as the information bottleneck Markov chain t − x − y are assumed. A graphical illustration of the proposed model is depicted in Figure 1. DISPLAYFORM1 Figure 1: Deep information bottleneck with the copula augmentation. Green circles describe random variables and orange rectangles denote deep nets parametrising the random variables. The blue circle stands for latent random variables whereas the red circle denotes the copula transformed random variables. The two terms in Eq. have the following forms: DISPLAYFORM2 and DISPLAYFORM3 because of the Markov assumption in the information bottleneck model p φ (t|x, y) = p φ (t|x). We denote with h(y) = −E p(y) [log p(y)] the entropy for discrete y and the differential entropy for continuous y. We then assume a conditional independence copula and Gaussian margins: DISPLAYFORM4 where t j is the jth marginal of t = (t 1, . . ., t d), c t|x is the copula density of t|x, u(t|x):= F t|x (t|x) is the uniform density indexed by t|x, and the functions µ j (x), σ 2 j (x) are implemented by deep networks. We make the same assumption about p θ (y|t). As we stated in Section 1, the deep information bottleneck model derived in Section 3.1 is not invariant to strictly increasing transformations of the marginal distributions. The IB method is formulated in terms of mutual information I(x, y), which depends only on the copula and therefore does not depend on monotone transformations of the marginals: I(x, y) = M I(x, y) − M I(x) − M I(y), where M I(x), for x = (x 1, . . ., x d), denotes the multi-information, which is equal to the negative copula entropy, as shown by BID16: DISPLAYFORM0 Issues with lack of invariance to marginal transformations.1. On the encoder side (Eq.), the optimisation is performed over the parametric conditional DISPLAYFORM1. When a monotone transformation x j → x j is applied, the required invariance property can only be guaranteed if the model for φ (in our case a deep network) is flexible enough to compensate for this transformation, which can be a severe problem in practice (see example in Section 4.1).2. On the decoder side, assuming Gaussian margins in p θ (y j |t) might be inappropriate for modelling y if the domain of y is not equal to the real numbers, e.g. when y is defined only on a bounded interval. If used in a generative way, the model might produce samples outside the domain of y. Even if other distributions than Gaussian are considered, such as truncated Gaussian, one still needs to make assumptions concerning the marginals. According to the IB formulation, such assumptions are unnecessary.3. Also on the decoder side, we have: I φ (t; y) = E p(x,y) E p φ (t|x) log p θ (y|t) + h(y). The authors of BID0 argue that since h(y) is constant, it can be ignored in computing I φ (t; y). This is true for a fixed or for a discrete y, but not for the class of monotone transformations of y, which should be the case for a model specified with mutual informations only. Since the left hand side of this equation (I φ (t; y)) is invariant against monotone transformations, and h(y) in general depends on monotone transformations, the first term on the right hand side (E p(x,y) E p φ (t|x) log p θ (y|t)) cannot be invariant to monotone transformations. In fact, under such transformations, the differential entropy h(y) can take any value from −∞ to +∞, which can be seen easily by decomposing the entropy into the copula entropy and the sum of marginal entropies (here, j stands for the jth dimension): DISPLAYFORM2 The first term (i.e. the copula entropy which is equal to the negative multi-information, as in Eq. FORMULA5) is a non-positive number. The marginal entropies h(y j) can take any value when using strictly increasing transformations (for instance, the marginal entropy of a uniform distribution on [a, b] is log(b − a)). As a consequence, the entropy term h(y) in Eq. FORMULA3 can be treated as a constant only either for one specific y or for discrete y, but not for all elements of the equivalence class containing all monotone transformations of y. Moreover, every such transformation would lead to different (I(x, t), I(y, t)) pairs in the information curve, which basically makes this curve arbitrary. Thus, h(y) being constant is a property that needs to be restored. The issues described in Section 3.2 can be fixed by using transformed variables (for a d dimensional x = (x 1, . . ., x d), x j stands for the jth dimension): DISPLAYFORM0 where Φ is the Gaussian cdf andF is the empirical cdf. The same transformation is applied to y. In the copula literature, these transformed variables are sometimes called normal scores. Note that the mapping is (approximately) invertible: x j =F −1 (Φ(x j)), withF −1 being the empirical quantiles treated as a function (e.g. by linear interpolation). This transformation fixes the invariance problem on the encoding side (issue 1), as well as the problems on the decoding side: problem 2 disappeared because the transformed variablesx j are standard normal distributed, and problem 3 disappeared because the decoder part (Eq. FORMULA3) now has the form: DISPLAYFORM1 where c inv (u(ỹ)) is indeed constant for all strictly increasing transformations applied to y. Having solved the IB problem in the transformed space, we can go back to the original space by using the inverse transformation according to Eq. DISPLAYFORM2 The ing model is thus a variational autoencoder with x replaced byx in the first term and y replaced byỹ in the second term. Technical details. We assume a simple prior p(t) = N (t; 0, I). Therefore, the KL divergence D KL (p φ (t|x) p(t)) is a divergence between two Gaussian distributions and admits an analytical form. We then estimate DISPLAYFORM3 and all the gradients on (mini-)batches. For the decoder side, E p(x,ỹ) E p φ (t|x) log p θ (ỹ|t) is needed. We train our model using the backpropagation algorithm. However, this algorithm can only handle deterministic nodes. In order to overcome this problem, we make use of the reparametrisation trick BID13 BID22: DISPLAYFORM4 withỹ j = Φ −1 (F (y j)). In this section we explain how the sparsity constraint on the information bottleneck along with the copula transformation in sparsity of the latent space t. We first introduce the Sparse Gaussian Information Bottleneck and subsequently show how augmenting it with the copula transformation leads to the sparse t. Sparse Gaussian Information Bottleneck. Recall that the information bottleneck compresses x to a new variable t by minimising I(x; t) − λI(t; y). This ensures that some amount of information with respect to a second "relevance" variable y is preserved in the compression. The assumption that x and y are jointly Gaussian-distributed leads to the Gaussian Information Bottleneck BID4 where the solution t can be proved to also be Gaussian distributed. In particular, if we denote the marginal distribution of x: x ∼ N (0, Σ x), the optimal t is a noisy projection of x of the following form: DISPLAYFORM0 The mutual information between x and t is then equal to: I(x; t) = 1 2 log |AΣ x A + I|. In the sparse Gaussian Information Bottleneck, we additionally assume that A is diagonal, so that the compressed t is a sparse version of x. Intuitively, sparsity follows from the observation that for a pair of random variables x, x, any full-rank projection Ax of x would lead to the same mutual information since I(x, x) = I(x; Ax), and a reduction in mutual information can only be achieved by a rank-deficient matrix A. For diagonal projections, this immediately implies sparsity of A.Sparse latent space of the Deep Information Bottleneck. We now proceed to explain the sparsity induced in the latent space of the copula version of the DIB introduced in Section 3.3. We will assume a possibly general, abstract pre-transformation of x, f β, which accounts for the encoder network along with the copula transformation of x. Then we will show how allowing for this abstract pre-transformation, in connection with the imposed sparsity constraint of the sparse information bottleneck described above, translates to the sparsity of the latent space of the copula DIB. By sparsity we understand the number of active neurons in the last layer of the encoder. To this end, we use the Sparse Gaussian Information Bottleneck model described above. We analyse the encoder part of the DIB, described with I(x, t). Consider the general Gaussian Information Bottleneck (with x and y jointly Gaussian and a full matrix A) and the deterministic pre-transformation, f β (x), performed on x. The pre-transformation is parametrised by a set of parameters β, which might be weights of neurons should f β be implemented as a neural network. Denote by M a n × p matrix which contains n i.i.d. samples of Af β (x), i.e. M = AZ with Z = (f β (x 1),..., f β (x n)). The optimisation of mutual information I(x, t) in min I(x; t) − λI(t; y) is then performed over M and β. Given f β and the above notation, the estimator of I(x; t) = 1 2 log |AΣ x A + I| becomes: DISPLAYFORM1 which would further simplify toÎ(x; t) = 1 2 i log(D ii + 1), if the pre-transformation f β were indeed such that D:= 1 n M M were diagonal. This is equivalent to the Sparse Gaussian Information Bottleneck model described above. Note that this means that the sparsity constraint in the Sparse Gaussian IB does not cause any loss of generality of the IB solution as long as the abstract 1 n M M in Eq.. We can, however, approximate this case by forcing this diagonalisation in Eq., i.e. by only considering the diagonal part of the matrix: I (x; t) = 1 2 log diag(DISPLAYFORM2 We now explain why this approximation (replacingÎ(x; t) with I (x; t)) is justified and how it leads to f β finding a low-dimensional representation of the latent space. Note that for any positive definite matrix B, the determinant |B| is always upper bounded by i B ii = |diag(B)|, which is a consequence of Hadamard's inequality. Thus, instead of minimisingÎ(x; t), we minimise an upper bound I (x; t) ≥Î(x; t) in the Information Bottleneck cost function. Equality is obtained if the transformation f β, which we assume to be part of an "end-to-end" optimisation procedure, indeed successfully diagonalised D = 1 n M M. Note that equality in the Hadamard's inequality is equivalent to D + I being orthogonal, thus f β is forced to find the "most orthogonal" representation of the inputs in the latent space. Using a highly flexible f β (for instance, modelled by a deep neural network), we might approximate this situation reasonably well. This explains how the copula transformation translates to a low-dimensional representation of the latent space. We indeed see disentanglement and sparse structure of the latent space learned by the copula DIB model by comparing it to the plain DIB without the copula transformation. We demonstrate it in Section 4. We now proceed to experimentally verify the contributions of the copula Deep Information Bottleneck. The goal of the experiments is to test the impact of the copula transformation. To this end, we perform a series of pair-wise experiments, where DIB without and with (cDIB) the copula transformation are tested in the same set-up. We use two datasets (artificial and real-world) and devise multiple experimental set-ups. First, we construct an artificial dataset such that a high-dimensional latent space is needed for its reconstruction (the dataset is reconstructed when samples from the latent space spatially coincide with it in its high-dimensional space). We perform monotone transformations on this dataset and test the difference between DIB and cDIB on reconstruction capabilities as well as classification predictive score. Dataset and set-up. The model used to generate the data consists of two input vectors x 1 and x 2 drawn form a uniform distribution defined on and vectors k 1 and k 2 drawn uniformly from. Additional inputs are x i=3...10 = a i * k 1 +(1−a i) * k 2 +0.3 * b i with a i, b i drawn from a uniform distribution defined on. All input vectors x 1...10 form the input matrix X. Latent variables z 1 = x 2 1 + x 2 2 and z 2 = z 1 + x 4 are defined and then normalised by dividing through their maximum value. Finally, random noise is added. Two target variables y 1 = z 2 * cos(1.75 * π * z 1) and y 2 = z 2 * sin(1.75 * π * z 1) are then calculated. y 1 and y 2 form a spiral if plotted in two dimensions. The angle and the radius of the spiral are highly correlated, which leads to the fact that a one-dimensional latent space can only reconstruct the backbone of the spiral. In order to reconstruct the details of the radial function, one has to use a latent space of at least two dimensions. We generate 200k samples from X and y. X is further transformed to beta densities using strictly increasing transformations. We split the samples into test (20k samples) and training (180k samples) sets. The generated samples are then transformed with the copula transformation (Eq.) toX and y and split in the same way into test and training sets. This gives us the four input sets X train, X test, X train,X test and the four target sets y train, y test,ỹ train,ỹ test.We use a latent layer with ten nodes that model the means of the ten-dimensional latent space t. The variance of the latent space is set to 1 for simplicity. The encoder as well as the decoder consist of a neural network with two fully-connected hidden layers with 50 nodes each. We use the softplus function as the activation function. Our model is trained using mini batches (size = 500) with the Adam optimiser BID12 for 70000 iterations using a learning rate of 0.0006. Experiment 1. In the first experiment, we compare the information curves produced by the DIB and its copula augmentation (Figure 2(a) ). To this end, we use the sets (X train, y train) and (X train,ỹ train) and record the values of I(x; t) and I(y; t) while multiplying the λ parameter every 500 iterations by 1.06 during training. One can observe an increase in the mutual information from approximately 6 in the DIB to approximately 11 in the copula DIB. At the same time, only two dimensions are used in the latent space t by the copula DIB. The version without copula does not provide competitive despite using 10 out of 18 dimensions of the latent space t. In Appendix B, we extend this experiment to comparison of information curves for other pre-processing techniques as well as to subjecting the training data to monotonic transformations other than the beta transformation. Experiment 2. Building on Experiment 1, we use the trained models for assessing their predictive quality on test data (X test, y test) and (X test,ỹ test). We compute predictive scores of the latent space t with respect to the generated y in the form of mutual information I(t; y) for all values of the parameter λ. The ing information curve shows an increased predictive capability of cDIB in Figure 2 (b) and exhibits no difference to the information curve produced in Experiment 1. Thus, the increased mutual information reported in Experiment 1 cannot only be attributed to overfitting. Experiment 3. In the third experiment, we qualitatively assess the reconstruction capability of cDIB compared to plain DIB (Figure 3). We choose the value of λ such that in both models two dimensions are active in the latent space. Figure 3 (b) shows a detailed reconstruction of y. The reconstruction quality of plain DIB on test data in a tight backbone which is not capable of reconstructing y (Figure 3(a) ). Figure 2: Information curves for the artificial experiment. The red curve describes the information curve with copula transformation whereas the orange one illustrates the plain information curve. The numbers represent the dimensions in the latent space t which are needed to reconstruct the output y. Experiment 4. We further inspect the information curves of DIB and cDIB by testing how the copula transformation adds resilience of the model against outliers and adversarial attacks in the training phase. To simulate an adversarial attack, we randomly choose 5% of all entries in the datasets X train andX train and replace them with outliers by adding uniformly sampled noise within the range. We again compute information curves for the training procedure and compare normal training with training with data subject to an attack for the copula and non-copula models. The (Figure 4(a) ) show that the copula model is more robust against outlier data than the plain one. We attribute this behaviour directly to the copula transformation, as ranks are less sensitive to outliers than raw data. Experiment 5. In this experiment, we investigate how the copula transformation affects convergence of the neural networks making up the DIB. We focus on the encoder and track the values of the loss function. Figure 4 (b) shows a sample comparison of convergence of DIB and cDIB for λ = 100. One can see that the cDIB starts to converge around iteration no. 1000, whereas the plain DIB takes longer. This can be explained by the fact that in the copula model the marginals are normalised to the same range of normal quantiles by the copula transformation. This translates to higher convergence rates. We continue analysing the impact of the copula transformation on the latent space of the DIB with a real-world dataset. We first report information curves analogous to Experiment 1 (Section 4.1) and proceed to inspect the latent spaces of both models along with sensitivity analysis with respect to λ. Dataset and Set-up. We consider the unnormalised Communities and Crime dataset BID15 from the UCI repository 1. The dataset consisted of 125 predictive, 4 non-predictive and 18 target variables with 2215 samples in total. In a preprocessing step, we removed all missing values from the dataset. In the end, we used 1901 observations with 102 predictive and 18 target variables in our analysis. 0.7 2.1 3.5 4.9 6.2 7.6 9.0 10.4 11.8 13.2 14.5 15. We use a latent layer with 18 nodes that models the means of the 18-dimensional latent space t. Again, the variance of the latent and the output space is set to 1. The stochastic encoder as well as the stochastic decoder consist of a neural network with two fully-connected hidden layers with 100 nodes each. Softplus is employed as the activation function. The decoder uses a Gaussian likelihood. Our model is trained for 150000 iterations using mini batches with a size of 1255. As before, we use Adam BID12 ) with a learning rate of 0.0005.Experiment 6. Analogously to Experiment 1 (Section 4.1), information curves stemming from the DIB and cDIB models have been computed. We record the values of I(x; t) and I(y; t) while multiplying the λ parameter every 500 iterations by 1.01 during training. Again, the information curve for the copula model yields larger values of mutual information, which we attribute to the increased flexibility of the model, as we pointed out in Section 3.3. In addition, the application of the copula transformation leads to a much lower number of used dimensions in the latent space. For example, copula DIB uses only four dimensions in the latent space for the highest λ values. DIB, on the other hand, needs eight dimensions in the latent space and nonetheless in lower mutual information scores. In order to show that our information curves are significantly different, we perform a Kruskal-Wallis rank test (p-value of 1.6 * 10 −16).Experiment 7. This experiment illustrates the difference in the disentanglement of the latent spaces of the DIB model with and without the copula transformation. We select two variables which yielded highest correlation with the target variable arsons and plot them along with their densities. In order to obtain the corresponding class labels (rainbow colours in Figure 6), we separate the values of arsons in eight equally-sized bins. A sample comparison of latent spaces of DIB and cDIB for λ = 21.55 is depicted in Figure 6. A more in-depth analysis of sensitivity of the learned latent space to λ is presented in Appendix A. The latent space t of DIB appears consistently less structured than that of cDIB, which is also reflected in the densities of the two plotted variables. In contrast, we can identify a much clearer structure in the latent space with respect to our previously calculated class labels. Figure 6: Latent space t consisting of two dimensions along with marginal densities without (a) and with (b) the copula transformation. The copula transformation leads to a disentangled latent space, which is reflected in non-overlapping modes of marginal distributions. We have presented a novel approach to compact representation learning of deep latent variable models. To this end, we showed that restoring invariance properties of the Deep Information Bottleneck with a copula transformation leads to disentanglement of the features in the latent space. Subsequently, we analysed how the copula transformation translates to sparsity in the latent space of the considered model. The proposed model allows for a simplified and fully non-parametric treatment of marginal distributions which has the advantage that it can be applied to distributions with arbitrary marginals. We evaluated our method on both artificial and real data. We showed that in practice the copula transformation leads to latent spaces that are disentangled, have an increased prediction capability and are resilient to adversarial attacks. All these properties are not sensitive to the only hyperparameter of the model, λ. In Section 3.2, we motivated the copula transformation for the Deep Information Bottleneck with the lack of invariance properties present in the original Information Bottleneck model, making the copula augmentation particularly suited for the DIB. The relevance of the copula transformation, however, reaches beyond the variational autoencoder, as evidenced by e.g. resilience to adversarial attacks or the positive influence on convergence rates presented in Section 4. These advantages of our model that do not simply follow from restoring the Information Bottleneck properties to the DIB, but are additional benefits of the copula. The copula transformation thus promises to be a simple but powerful addition to the general deep learning toolbox. We augment Experiment 7 from Section 4 with sensitivity analysis of the latent space with respect to the chosen value of the only hyperparameter, λ. To this end, we recompute Experiment 7 for different values of λ ranging between 1.79 and 1897.15 (which corresponds to the reported information curves). The are reported in FIG6. As can be seen, the latent space of the copula DIB is consistently better structured then that of the plain DIB. Building on Experiment 1 from Section 4, we again compare the information curves produced by the DIB and its copula augmentation. We compare the copula transformation with data normalisation (transformation to mean 0 and variance 1) in Figure 9 (a). We also replace the beta transformation with gamma in the experimental set-up described in Section 4 and report the in Figure 9 (b). As in Experiment 1, one con see that the information curve for the copula version of DIB lies above the plain one. The latent space uses fewer dimensions as well. Figure 9: Different extensions of Experiment 1: (a) comparison of the copula transformation to normalising the input data (to zero mean and unit variance), (b) Experiment 1 with a gamma instead of a beta transformation. All curves are computed over the same range of the hyperparameter λ. The copula pre-transformation yields higher information curves and uses fewer dimensions in the latent space.

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We apply the copula transformation to the Deep Information Bottleneck which leads to restored invariance properties and a disentangled latent space with superior predictive capabilities.

This paper 1) characterizes functions representable by ReLU DNNs, 2) formally studies the benefit of depth in such architectures,  3) gives an algorithm to implement empirical risk minimization to global optimality for two layer ReLU nets.