This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sinkhorn divergence between two Gaussian measures on a Hilbert space, along with the fixed point equations for the barycenter of a set of Gaussian measures. Our formulations fully exploit the regularization aspect of the entropic formulation and are valid both in singular and nonsingular settings. In the infinite-dimensional setting, both the entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable, in contrast to the exact 2-Wasserstein distance, which is not differentiable. Our Sinkhorn barycenter equation is new and always has a unique solution. In contrast, the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. In the setting of reproducing kernel Hilbert spaces (RKHS), our distance formulas are given explicitly in terms of the corresponding kernel Gram matrices, providing an interpolation between the kernel Maximum Mean Discrepancy (MMD) and the kernel 2-Wasserstein distance.
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能研究中心）
This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that these distances/divergences are fully represented by reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators associated with the corresponding covariance functions. Using this representation, we show that the Sinkhorn divergence between two centered Gaussian processes can be consistently and efficiently estimated from the divergence bet…
Stanford University（斯坦福大学）；University of Delaware（特拉华大学）
Generative Adversarial Networks are a popular method for learning distributions from data by modeling the target distribution as a function of a known distribution. The function, often referred to as the generator, is optimized to minimize a chosen distance measure between the generated and target distributions. One commonly used measure for this purpose is the Wasserstein distance. However, Wasserstein distance is hard to compute and optimize, and in practice entropic regularization techniques are used to improve numerical convergence. The influence of regularization on the learned solution, …
This report has several purposes. First, our report is written to investigate the reproducibility of the submitted paper On the regularization of Wasserstein GANs (2018). Second, among the experiments performed in the submitted paper, five aspects were emphasized and reproduced: learning speed, stability, robustness against hyperparameter, estimating the Wasserstein distance, and various sampling method. Finally, we identify which parts of the contribution can be reproduced, and at what cost in terms of resources. All source code for reproduction is open to the public.
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
The variational autoencoder is a well defined deep generative model that utilizes an encoder-decoder framework where an encoding neural network outputs a non-deterministic code for reconstructing an input. The encoder achieves this by sampling from a distribution for every input, instead of outputting a deterministic code per input. The great advantage of this process is that it allows the use of the network as a generative model for sampling from the data distribution beyond provided samples for training. We show in this work that utilizing batch normalization as a source for non-determinism …
Generative models have the ability to synthesize data points drawn from the data distribution, however, not all generated samples are high quality. In this paper, we propose using a combination of coresets selection methods and ``entropic regularization'' to select the highest fidelity samples. We leverage an Energy-Based Model which resembles a variational auto-encoder with an inference and generator model for which the latent prior is complexified by an energy-based model. In a semi-supervised learning scenario, we show that augmenting the labeled data-set, by adding our selected subset of s…
Wasserstein-GANs have been introduced to address the deficiencies of generative adversarial networks (GANs) regarding the problems of vanishing gradients and mode collapse during the training, leading to improved convergence behaviour and improved image quality. However, Wasserstein-GANs require the discriminator to be Lipschitz continuous. In current state-of-the-art Wasserstein-GANs this constraint is enforced via gradient norm regularization. In this paper, we demonstrate that this regularization does not encourage a broad distribution of spectral-values in the discriminator weights, hence …
Generative adversarial networks (GANs) are one of the most popular approaches when it comes to training generative models, among which variants of Wasserstein GANs are considered superior to the standard GAN formulation in terms of learning stability and sample quality. However, Wasserstein GANs require the critic to be 1-Lipschitz, which is often enforced implicitly by penalizing the norm of its gradient, or by globally restricting its Lipschitz constant via weight normalization techniques. Training with a regularization term penalizing the violation of the Lipschitz constraint explicitly, in…
The Australian National University（澳大利亚国立大学）；Data61
Robustness to adversarial attacks is an important concern due to the fragility of deep neural networks to small perturbations and has received an abundance of attention in recent years. Distributionally Robust Optimization (DRO), a particularly promising way of addressing this challenge, studies robustness via divergence-based uncertainty sets and has provided valuable insights into robustification strategies such as regularization. In the context of machine learning, the majority of existing results have chosen $f$-divergences, Wasserstein distances and more recently, the Maximum Mean Discrep…
Duke University（杜克大学）；Infinia ML Inc.（Infinia ML公司）
A new algorithmic framework is proposed for learning autoencoders of data distributions. We minimize the discrepancy between the model and target distributions, with a \emph{relational regularization} on the learnable latent prior. This regularization penalizes the fused Gromov-Wasserstein (FGW) distance between the latent prior and its corresponding posterior, allowing one to flexibly learn a structured prior distribution associated with the generative model. Moreover, it helps co-training of multiple autoencoders even if they have heterogeneous architectures and incomparable latent spaces. W…
VinAI Research（VinAI研究院）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
Relational regularized autoencoder (RAE) is a framework to learn the distribution of data by minimizing a reconstruction loss together with a relational regularization on the latent space. A recent attempt to reduce the inner discrepancy between the prior and aggregated posterior distributions is to incorporate sliced fused Gromov-Wasserstein (SFG) between these distributions. That approach has a weakness since it treats every slicing direction similarly, meanwhile several directions are not useful for the discriminative task. To improve the discrepancy and consequently the relational regulari…
Approximating distributions over complicated manifolds, such as natural images, are conceptually attractive. The deep latent variable model, trained using variational autoencoders and generative adversarial networks, is now a key technique for representation learning. However, it is difficult to unify these two models for exact latent-variable inference and parallelize both reconstruction and sampling, partly due to the regularization under the latent variables, to match a simple explicit prior distribution. These approaches are prone to be oversimplified, and can only characterize a few modes…
Cohomological Obstructions to Global Counterfactuals: A Sheaf-Theoretic Foundation for Generative Causal Models
School of Management, University of Science and Technology of China（管理学院，中国科学技术大学）；School of Computer Science and Engineering, University of Science and Technology of China（计算机科学与工程学院，中国科学技术大学）
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）
Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete probability distributions in a Wasserstein space, endowed with an optimal transport metric. Wasserstein spaces are much larger and more flexible than Euclidean spaces, in that they can successfully embed a wider variety of metric structures. We exploit this flexibility by learning an embedding that captures semantic information in the Wasserstein distance between emb…
We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions with different dimensions. When the metric is the inner product, which we refer to as inner product Gromov-Wasserstein (IGW), we demonstrate that the optimal transportation plans of entropic IGW and its unbalanced variant are (unbalanced) Gaussian distributions. Via an application of von Neumann's trace inequality, we obtain closed-form expressions for the entropic IGW between these Gaussian distributions. Finally, we consider an entropic inner product Gromov-Wasserstein barycenter o…
Université Paris-Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Université Lyon（里昂大学）；CNRS（法国国家科学研究中心）；ENS Lyon（里昂高等师范学院）；UCB Lyon 1（里昂第一大学克洛德·贝尔纳大学）；Ecole Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；Université Paris Nanterre（巴黎南泰尔大学）
Dimension reduction (DR) methods provide systematic approaches for analyzing high-dimensional data. A key requirement for DR is to incorporate global dependencies among original and embedded samples while preserving clusters in the embedding space. To achieve this, we combine the principles of optimal transport (OT) and principal component analysis (PCA). Our method seeks the best linear subspace that minimizes reconstruction error using entropic OT, which naturally encodes the neighborhood information of the samples. From an algorithmic standpoint, we propose an efficient block-majorization-m…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is {\it strictly weaker} than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert-Schmidt norm. This is in contrast to the previous known result that a sequence of c…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；New York University（纽约大学）；Georgia Institute of Technology（佐治亚理工学院）
Wasserstein distributionally robust optimization (DRO) has recently achieved empirical success for various applications in operations research and machine learning, owing partly to its regularization effect. Although connection between Wasserstein DRO and regularization has been established in several settings, existing results often require restrictive assumptions, such as smoothness or convexity, that are not satisfied for many problems. In this paper, we develop a general theory on the variation regularization effect of the Wasserstein DRO - a new form of regularization that generalizes tot…
In this paper we propose a new family of algorithms, ATENT, for training adversarially robust deep neural networks. We formulate a new loss function that is equipped with an additional entropic regularization. Our loss function considers the contribution of adversarial samples that are drawn from a specially designed distribution in the data space that assigns high probability to points with high loss and in the immediate neighborhood of training samples. Our proposed algorithms optimize this loss to seek adversarially robust valleys of the loss landscape. Our approach achieves competitive (or…
UCLA（加州大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
We propose regularization strategies for learning discriminative models that are robust to in-class variations of the input data. We use the Wasserstein-2 geometry to capture semantically meaningful neighborhoods in the space of images, and define a corresponding input-dependent additive noise data augmentation model. Expanding and integrating the augmented loss yields an effective Tikhonov-type Wasserstein diffusion smoothness regularizer. This approach allows us to apply high levels of regularization and train functions that have low variability within classes but remain flexible across clas…
University Bretagne Sud（布列塔尼南部大学）；CNRS（法国国家科学研究中心）；INRIA（法国国家信息与自动化研究所）；IRISA（法国信息与自动化系统研究所）；InterDigital R&D（InterDigital研发中心）；Université de Paris（巴黎大学）；Wageningen University（瓦赫宁根大学）；École Polytechnique（巴黎综合理工学院）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Noisy labels often occur in vision datasets, especially when they are obtained from crowdsourcing or Web scraping. We propose a new regularization method, which enables learning robust classifiers in presence of noisy data. To achieve this goal, we propose a new adversarial regularization scheme based on the Wasserstein distance. Using this distance allows taking into account specific relations between classes by leveraging the geometric properties of the labels space. Our Wasserstein Adversarial Regularization (WAR) encodes a selective regularization, which promotes smoothness of the classifi…
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；University of Washington（华盛顿大学）
We describe a graph-based semi-supervised learning framework in the context of deep neural networks that uses a graph-based entropic regularizer to favor smooth solutions over a graph induced by the data. The main contribution of this work is a computationally efficient, stochastic graph-regularization technique that uses mini-batches that are consistent with the graph structure, but also provides enough stochasticity (in terms of mini-batch data diversity) for convergence of stochastic gradient descent methods to good solutions. For this work, we focus on results of frame-level phone classifi…
Artificial and Natural Intelligence Toulouse Institute (ANITI)（图卢兹人工智能与自然智能研究所）；Institut de Recherche Technologique (IRT) Saint Exupéry（圣埃克苏佩里技术研究所）
The increasingly common use of neural network classifiers in industrial and social applications of image analysis has allowed impressive progress these last years. Such methods are however sensitive to algorithmic bias, i.e. to an under- or an over-representation of positive predictions or to higher prediction errors in specific subgroups of images. We then introduce in this paper a new method to temper the algorithmic bias in Neural-Network based classifiers. Our method is Neural-Network architecture agnostic and scales well to massive training sets of images. It indeed only overloads the los…
Georgia Institute of Technology（佐治亚理工学院）；East China Normal University（华东师范大学）
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regularized variations with large regularization parameter will degradate the performance in several important machine learning applications, and small regularization parameter will fail due to numerical stability issues with existing algorithms. We address this challenge by developi…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Massachusetts Institute of Technology（麻省理工学院）
Wasserstein barycenters provide a geometric notion of the weighted average of probability measures based on optimal transport. In this paper, we present a scalable algorithm to compute Wasserstein-2 barycenters given sample access to the input measures, which are not restricted to being discrete. While past approaches rely on entropic or quadratic regularization, we employ input convex neural networks and cycle-consistency regularization to avoid introducing bias. As a result, our approach does not resort to minimax optimization. We provide theoretical analysis on error bounds as well as empir…
Regularization and normalization have become indispensable components in training deep neural networks, resulting in faster training and improved generalization performance. We propose the projected error function regularization loss (PER) that encourages activations to follow the standard normal distribution. PER randomly projects activations onto one-dimensional space and computes the regularization loss in the projected space. PER is similar to the Pseudo-Huber loss in the projected space, thus taking advantage of both $L^1$ and $L^2$ regularization losses. Besides, PER can capture the inte…
Extending Mean-Field Variational Inference via Entropic Regularization: Theory and Computation
Department of Statistics Columbia University（统计系 哥伦比亚大学）；Department of Computer Science and Department of Statistics Columbia University（计算机科学系和统计系 哥伦比亚大学）
We derive generalization and excess risk bounds for neural nets using a family of complexity measures based on a multilevel relative entropy. The bounds are obtained by introducing the notion of generated hierarchical coverings of neural nets and by using the technique of chaining mutual information introduced in Asadi et al. NeurIPS'18. The resulting bounds are algorithm-dependent and exploit the multilevel structure of neural nets. This, in turn, leads to an empirical risk minimization problem with a multilevel entropic regularization. The minimization problem is resolved by introducing a mu…
Technische Universität Darmstadt（达姆施塔特工业大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
An optimal feedback controller for a given Markov decision process (MDP) can in principle be synthesized by value or policy iteration. However, if the system dynamics and the reward function are unknown, a learning agent must discover an optimal controller via direct interaction with the environment. Such interactive data gathering commonly leads to divergence towards dangerous or uninformative regions of the state space unless additional regularization measures are taken. Prior works proposed bounding the information loss measured by the Kullback-Leibler (KL) divergence at every policy improv…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Artificial Intelligence Research Institute（人工智能研究院）；Massachusetts Institute of Technology（麻省理工学院）
Wasserstein barycenters have become popular due to their ability to represent the average of probability measures in a geometrically meaningful way. In this paper, we present an algorithm to approximate the Wasserstein-2 barycenters of continuous measures via a generative model. Previous approaches rely on regularization (entropic/quadratic) which introduces bias or on input convex neural networks which are not expressive enough for large-scale tasks. In contrast, our algorithm does not introduce bias and allows using arbitrary neural networks. In addition, based on the celebrity faces dataset…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）
We propose a novel end-to-end non-minimax algorithm for training optimal transport mappings for the quadratic cost (Wasserstein-2 distance). The algorithm uses input convex neural networks and a cycle-consistency regularization to approximate Wasserstein-2 distance. In contrast to popular entropic and quadratic regularizers, cycle-consistency does not introduce bias and scales well to high dimensions. From the theoretical side, we estimate the properties of the generative mapping fitted by our algorithm. From the practical side, we evaluate our algorithm on a wide range of tasks: image-to-imag…
Decentralized Entropic Optimal Transport for Distributed Distribution Comparison
School of Computer Science and Technology, East China Normal University（华东师范大学计算机科学与技术学院）；Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）
Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
We consider prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. This semi-adversarial setting includes (at the extremes) the classical i.i.d. setting, when the unknown constraint set is restricted to be a singleton, and the unconstrained adversarial setting, when the constraint set is the set of all distributions. The Hedge algorithm -- long known to be minimax (rate) optimal in the adversarial regime -- was recently shown to be simultaneously minimax optimal for i.i.d. data. In this work, we propose to relax the i.i.d.…
Entropy regularization in optimal transport (OT) has been the driver of many recent interests for Wasserstein metrics and barycenters in machine learning. It allows to keep the appealing geometrical properties of the unregularized Wasserstein distance while having a significantly lower complexity thanks to Sinkhorn's algorithm. However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters. This side effect has prompted an increasing temptation in the community to settle for a slower algorithm such as log-domain stabilized Sinkhorn which breaks the parallel …
University College London（伦敦大学学院）；INRIA（法国国家信息与自动化研究所）；École Normale Supérieure（巴黎高等师范学院）；Istituto Italiano di Tecnologia（意大利技术研究院）
Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn distance, proving that it enjoys the same smoothness as its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and ap…
CSIRO’s Data61（澳大利亚联邦科学与工业研究组织数据61中心）；University of Melbourne（墨尔本大学）
We use distributionally-robust optimization for machine learning to mitigate the effect of data poisoning attacks. We provide performance guarantees for the trained model on the original data (not including the poison records) by training the model for the worst-case distribution on a neighbourhood around the empirical distribution (extracted from the training dataset corrupted by a poisoning attack) defined using the Wasserstein distance. We relax the distributionally-robust machine learning problem by finding an upper bound for the worst-case fitness based on the empirical sampled-averaged f…
This paper introduces a new nonlinear dictionary learning method for histograms in the probability simplex. The method leverages optimal transport theory, in the sense that our aim is to reconstruct histograms using so-called displacement interpolations (a.k.a. Wasserstein barycenters) between dictionary atoms; such atoms are themselves synthetic histograms in the probability simplex. Our method simultaneously estimates such atoms, and, for each datapoint, the vector of weights that can optimally reconstruct it as an optimal transport barycenter of such atoms. Our method is computationally tra…
We propose a family of relaxations of the optimal transport problem which regularize the problem by introducing an additional minimization step over a small region around one of the underlying transporting measures. The type of regularization that we obtain is related to smoothing techniques studied in the optimization literature. When using our approach to estimate optimal transport costs based on empirical measures, we obtain statistical learning bounds which are useful to guide the amount of regularization, while maintaining good generalization properties. To illustrate the computational ad…
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to $ε$-accuracy by adding an entropic regularization of order $ε$ and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample comple…
We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cutur…
