The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.
Wasserstein GAN-Based Precipitation Downscaling with Optimal Transport for Enhancing Perceptual Realism
Graduate School of Science and Engineering, Chiba University（千叶大学科学技术研究生院）；Center for Environmental Remote Sensing, Chiba University（千叶大学环境遥感中心）；Institute for Advanced Academic Research, Chiba University（千叶大学高级学术研究所）；Research Institute of Disaster Medicine, Chiba University（千叶大学灾害医学研究所）
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）
Wasserstein Generative Adversarial Networks (WGANs) are the popular generative models built on the theory of Optimal Transport (OT) and the Kantorovich duality. Despite the success of WGANs, it is still unclear how well the underlying OT dual solvers approximate the OT cost (Wasserstein-1 distance, $\mathbb{W}_{1}$) and the OT gradient needed to update the generator. In this paper, we address these questions. We construct 1-Lipschitz functions and use them to build ray monotone transport plans. This strategy yields pairs of continuous benchmark distributions with the analytically known OT plan…
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）
Generative adversarial networks (GANs) are a widely used framework for learning generative models. Wasserstein GANs (WGANs), one of the most successful variants of GANs, require solving a minmax optimization problem to global optimality, but are in practice successfully trained using stochastic gradient descent-ascent. In this paper, we show that, when the generator is a one-layer network, stochastic gradient descent-ascent converges to a global solution with polynomial time and sample complexity.
LTCI（通信与信息处理实验室）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎理工学院）
Issued from Optimal Transport, the Wasserstein distance has gained importance in Machine Learning due to its appealing geometrical properties and the increasing availability of efficient approximations. In this work, we consider the problem of estimating the Wasserstein distance between two probability distributions when observations are polluted by outliers. To that end, we investigate how to leverage Medians of Means (MoM) estimators to robustify the estimation of Wasserstein distance. Exploiting the dual Kantorovitch formulation of Wasserstein distance, we introduce and discuss novel MoM-ba…
The combinatorial search space presents a significant challenge to learning causality from data. Recently, the problem has been formulated into a continuous optimization framework with an acyclicity constraint, allowing for the exploration of deep generative models to better capture data sample distributions and support the discovery of Directed Acyclic Graphs (DAGs) that faithfully represent the underlying data distribution. However, so far no study has investigated the use of Wasserstein distance for causal structure learning via generative models. This paper proposes a new model named DAG-W…
Wasserstein Generative Adversarial Networks (WGANs) provide a versatile class of models, which have attracted great attention in various applications. However, this framework has two main drawbacks: (i) Wasserstein-1 (or Earth-Mover) distance is restrictive such that WGANs cannot always fit data geometry well; (ii) It is difficult to achieve fast training of WGANs. In this paper, we propose a new class of \textit{Relaxed Wasserstein} (RW) distances by generalizing Wasserstein-1 distance with Bregman cost functions. We show that RW distances achieve nice statistical properties while not sacrifi…
Inspired by ideas from optimal transport theory we present Trust the Critics (TTC), a new algorithm for generative modelling. This algorithm eliminates the trainable generator from a Wasserstein GAN; instead, it iteratively modifies the source data using gradient descent on a sequence of trained critic networks. This is motivated in part by the misalignment which we observed between the optimal transport directions provided by the gradients of the critic and the directions in which data points actually move when parametrized by a trainable generator. Previous work has arrived at similar ideas …
Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance between a data distribution and sample distribution. Recent studies have proposed stabilizing the training process for the WGAN and implementing the Lipschitz constraint. In this study, we prove the local stability of optimizing the simple gradient penalty $μ$-WGAN(SGP $μ$-WGAN) under suitable assumptions regarding the equilibrium and penalty measure $μ$. The measure valued differentiation concept is employed to deal with the derivative of the penalty terms, which is helpful for handling abstract singular measures with lo…
Northeastern University（东北大学）；Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
Due to the widespread use of positron emission tomography (PET) in clinical practice, the potential risk of PET-associated radiation dose to patients needs to be minimized. However, with the reduction in the radiation dose, the resultant images may suffer from noise and artifacts that compromise diagnostic performance. In this paper, we propose a parameter-transferred Wasserstein generative adversarial network (PT-WGAN) for low-dose PET image denoising. The contributions of this paper are twofold: i) a PT-WGAN framework is designed to denoise low-dose PET images without compromising structural…
Generative adversarial networks (GANs) have enjoyed much success in learning high-dimensional distributions. Learning objectives approximately minimize an $f$-divergence ($f$-GANs) or an integral probability metric (Wasserstein GANs) between the model and the data distribution using a discriminator. Wasserstein GANs enjoy superior empirical performance, but in $f$-GANs the discriminator can be interpreted as a density ratio estimator which is necessary in some GAN applications. In this paper, we bridge the gap between $f$-GANs and Wasserstein GANs (WGANs). First, we list two constraints over v…
University of Copenhagen（哥本哈根大学）；IT University of Copenhagen（哥本哈根信息技术大学）
Generative Adversial Networks (GANs) have made a major impact in computer vision and machine learning as generative models. Wasserstein GANs (WGANs) brought Optimal Transport (OT) theory into GANs, by minimizing the $1$-Wasserstein distance between model and data distributions as their objective function. Since then, WGANs have gained considerable interest due to their stability and theoretical framework. We contribute to the WGAN literature by introducing the family of $(q,p)$-Wasserstein GANs, which allow the use of more general $p$-Wasserstein metrics for $p\geq 1$ in the GAN learning proce…
We provide statistical theory for conditional and unconditional Wasserstein generative adversarial networks (WGANs) in the framework of dependent observations. We prove upper bounds for the excess Bayes risk of the WGAN estimators with respect to a modified Wasserstein-type distance. Furthermore, we formalize and derive statements on the weak convergence of the estimators and use them to develop confidence intervals for new observations. The theory is applied to the special case of high-dimensional time series forecasting. We analyze the behavior of the estimators in simulations based on synth…
This paper explains the math behind a generative adversarial network (GAN) model and why it is hard to be trained. Wasserstein GAN is intended to improve GANs' training by adopting a smooth metric for measuring the distance between two probability distributions.
Sorbonne Université（索邦大学）；Criteo AI Lab（科睿唯安人工智能实验室）
Generative Adversarial Networks (GANs) have been successful in producing outstanding results in areas as diverse as image, video, and text generation. Building on these successes, a large number of empirical studies have validated the benefits of the cousin approach called Wasserstein GANs (WGANs), which brings stabilization in the training process. In the present paper, we add a new stone to the edifice by proposing some theoretical advances in the properties of WGANs. First, we properly define the architecture of WGANs in the context of integral probability metrics parameterized by neural ne…
University of Copenhagen（哥本哈根大学）；University of California, Los Angeles（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）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
Generative modelling is often cast as minimizing a similarity measure between a data distribution and a model distribution. Recently, a popular choice for the similarity measure has been the Wasserstein metric, which can be expressed in the Kantorovich duality formulation as the optimum difference of the expected values of a potential function under the real data distribution and the model hypothesis. In practice, the potential is approximated with a neural network and is called the discriminator. Duality constraints on the function class of the discriminator are enforced approximately, and th…
Wasserstein GANs with Gradient Penalty (WGAN-GP) are a very popular method for training generative models to produce high quality synthetic data. While WGAN-GP were initially developed to calculate the Wasserstein 1 distance between generated and real data, recent works (e.g. [23]) have provided empirical evidence that this does not occur, and have argued that WGAN-GP perform well not in spite of this issue, but because of it. In this paper we show for the first time that WGAN-GP compute the minimum of a different optimal transport problem, the so-called congested transport [7]. Congested tran…
Montreal Institute for Learning Algorithms（蒙特利尔学习算法研究所）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Google Brain（谷歌大脑）
Generative Adversarial Networks (GANs) are powerful generative models, but suffer from training instability. The recently proposed Wasserstein GAN (WGAN) makes progress toward stable training of GANs, but sometimes can still generate only low-quality samples or fail to converge. We find that these problems are often due to the use of weight clipping in WGAN to enforce a Lipschitz constraint on the critic, which can lead to undesired behavior. We propose an alternative to clipping weights: penalize the norm of gradient of the critic with respect to its input. Our proposed method performs better…
On optimal solutions of classical and sliced Wasserstein GANs with non-Gaussian data
Dept. of Applied Math., Univ. of Colorado, Boulder, CO 80309, USA（应用数学系，科罗拉多大学，伯尔德，科罗拉多州，80309，美国）；Department of EE and GICE, National Taiwan University, Taipei, Taiwan（电子工程系和信息科学与工程系，国立台湾大学，台北，台湾）；Institute of Communications Engineering, National Yang Ming Chiao Tung University, HsinChu, Taiwan（通讯工程研究所，国立阳明交通大学，新竹，台湾）；Bosch Center for AI, USA（博世人工智能中心，美国）
Zurich University of Applied Sciences（苏黎世应用科技大学）；University of Twente（特文特大学）
Modeling financial time series is challenging due to their high volatility and unexpected happenings on the market. Most financial models and algorithms trying to fill the lack of historical financial time series struggle to perform and are highly vulnerable to overfitting. As an alternative, we introduce in this paper a deep neural network called the WGAN-GP, a data-driven model that focuses on sample generation. The WGAN-GP consists of a generator and discriminator function which utilize an LSTM architecture. The WGAN-GP is supposed to learn the underlying structure of the input data, which …
Computer Vision Lab, ETH Zurich（苏黎世联邦理工学院计算机视觉实验室）；VISICS KU Leuven（比利时鲁汶大学VISICS）
Generative modeling over natural images is one of the most fundamental machine learning problems. However, few modern generative models, including Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued images that are frequently encountered in real-world applications. To fill the gap, this paper first formulates the problem of generating manifold-valued images and exploits three typical instances: hue-saturation-value (HSV) color image generation, chromaticity-brightness (CB) color image generation, and diffusion-tensor (DT) image generation. For the proposed generativ…
University of Central Florida（中佛罗里达大学）；Beijing Jiaotong University（北京交通大学）；Tencent AI Lab（腾讯人工智能实验室）
Despite being impactful on a variety of problems and applications, the generative adversarial nets (GANs) are remarkably difficult to train. This issue is formally analyzed by \cite{arjovsky2017towards}, who also propose an alternative direction to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, called Wasserstein GAN (WGAN), hinges on the 1-Lipschitz continuity of the discriminator. In this paper, we propose a novel approach to enforcing the Lipschitz continuity in the training procedure of WGANs. Our approach seamlessly connects WGAN with one of the …
University of Montreal（蒙特利尔大学）；Mila - Quebec AI Institute（Mila - 魁北克人工智能研究所）；Shanghai University of Finance and Economics（上海财经大学）；Shanghai Jiao Tong University（上海交通大学）；HEC Montreal（蒙特利尔高等商学院）；CIFAR AI Research Chair（CIFAR人工智能研究讲席）
Wasserstein GANs (WGANs), built upon the Kantorovich-Rubinstein (KR) duality of Wasserstein distance, is one of the most theoretically sound GAN models. However, in practice it does not always outperform other variants of GANs. This is mostly due to the imperfect implementation of the Lipschitz condition required by the KR duality. Extensive work has been done in the community with different implementations of the Lipschitz constraint, which, however, is still hard to satisfy the restriction perfectly in practice. In this paper, we argue that the strong Lipschitz constraint might be unnecessar…
Generative Adversarial Networks (GANs) have been impactful on many problems and applications but suffer from unstable training. The Wasserstein GAN (WGAN) leverages the Wasserstein distance to avoid the caveats in the minmax two-player training of GANs but has other defects such as mode collapse and lack of metric to detect the convergence. We introduce a novel inferential Wasserstein GAN (iWGAN) model, which is a principled framework to fuse auto-encoders and WGANs. The iWGAN model jointly learns an encoder network and a generator network motivated by the iterative primal dual optimization pr…
Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Facebook AI Research（Facebook 人工智能研究院）
We introduce a new algorithm named WGAN, an alternative to traditional GAN training. In this new model, we show that we can improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches. Furthermore, we show that the corresponding optimization problem is sound, and provide extensive theoretical work highlighting the deep connections to other distances between distributions.
The University of Texas at San Antonio（德克萨斯大学圣安东尼奥分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
Electroencephalography (EEG) data are difficult to obtain due to complex experimental setups and reduced comfort with prolonged wearing. This poses challenges to train powerful deep learning model with the limited EEG data. Being able to generate EEG data computationally could address this limitation. We propose a novel Wasserstein Generative Adversarial Network with gradient penalty (WGAN-GP) to synthesize EEG data. This network addresses several modeling challenges of simulating time-series EEG data including frequency artifacts and training instability. We further extended this network to a…
Cambridge Image Analysis Group, Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系剑桥图像分析组）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；University of Cambridge（剑桥大学）；University of Bath（巴斯大学）
Wasserstein GANs are based on the idea of minimising the Wasserstein distance between a real and a generated distribution. We provide an in-depth mathematical analysis of differences between the theoretical setup and the reality of training Wasserstein GANs. In this work, we gather both theoretical and empirical evidence that the WGAN loss is not a meaningful approximation of the Wasserstein distance. Moreover, we argue that the Wasserstein distance is not even a desirable loss function for deep generative models, and conclude that the success of Wasserstein GANs can in truth be attributed to …
Generative adversarial networks (GANs) have received a tremendous amount of attention in the past few years, and have inspired applications addressing a wide range of problems. Despite its great potential, GANs are difficult to train. Recently, a series of papers (Arjovsky & Bottou, 2017a; Arjovsky et al. 2017b; and Gulrajani et al. 2017) proposed using Wasserstein distance as the training objective and promised easy, stable GAN training across architectures with minimal hyperparameter tuning. In this paper, we compare the performance of Wasserstein distance with other training objectives on a…
Graz University of Technology（格拉茨工业大学）；Austrian Institute of Technology（奥地利技术研究所）
Generative Adversarial Networks (GANs) have been used to model the underlying probability distribution of sample based datasets. GANs are notoriuos for training difficulties and their dependence on arbitrary hyperparameters. One recent improvement in GAN literature is to use the Wasserstein distance as loss function leading to Wasserstein Generative Adversarial Networks (WGANs). Using this as a basis, we show various ways in which the Wasserstein distance is estimated for the task of generative modelling. Additionally, the secrets in training such models are shown and summarized at the end of …
Generative Adversarial Networks (GANs) have shown great promise in modeling high dimensional data. The learning objective of GANs usually minimizes some measure discrepancy, \textit{e.g.}, $f$-divergence~($f$-GANs) or Integral Probability Metric~(Wasserstein GANs). With $f$-divergence as the objective function, the discriminator essentially estimates the density ratio, and the estimated ratio proves useful in further improving the sample quality of the generator. However, how to leverage the information contained in the discriminator of Wasserstein GANs (WGAN) is less explored. In this paper, …
We study the ability of Wasserstein Generative Adversarial Network (WGAN) to generate missing audio content which is, in context, (statistically similar) to the sound and the neighboring borders. We deal with the challenge of audio inpainting long range gaps (500 ms) using WGAN models. We improved the quality of the inpainting part using a new proposed WGAN architecture that uses a short-range and a long-range neighboring borders compared to the classical WGAN model. The performance was compared with two different audio instruments (piano and guitar) and on virtuoso pianists together with a st…
Generative Adversarial Networks (GANs) have been promising in the field of image generation, however, they have been hard to train for language generation. GANs were originally designed to output differentiable values, so discrete language generation is challenging for them which causes high levels of instability in training GANs. Consequently, past work has resorted to pre-training with maximum-likelihood or training GANs without pre-training with a WGAN objective with a gradient penalty. In this study, we present a comparison of those approaches. Furthermore, we present the results of some e…
In this short note, we introduce the spectral-domain $\mathcal{W}_2$ Wasserstein distance for elliptical stochastic processes in terms of their power spectra. We also introduce the spectral-domain Gelbrich bound for processes that are not necessarily elliptical.
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under Wasserstein loss, using only metric properties, such as covering and packing numbers, of the sample space, and weak moment assumptions on the probability distributions.
University of Cambridge（剑桥大学）；Columbia University（哥伦比亚大学）；Google Brain（谷歌大脑）；Google Research（谷歌研究院）；The Alan Turing Institute（艾伦·图灵研究所）
Wasserstein distances are increasingly used in a wide variety of applications in machine learning. Sliced Wasserstein distances form an important subclass which may be estimated efficiently through one-dimensional sorting operations. In this paper, we propose a new variant of sliced Wasserstein distance, study the use of orthogonal coupling in Monte Carlo estimation of Wasserstein distances and draw connections with stratified sampling, and evaluate our approaches experimentally in a range of large-scale experiments in generative modelling and reinforcement learning.
To measure the similarity of documents, the Wasserstein distance is a powerful tool, but it requires a high computational cost. Recently, for fast computation of the Wasserstein distance, methods for approximating the Wasserstein distance using a tree metric have been proposed. These tree-based methods allow fast comparisons of a large number of documents; however, they are unsupervised and do not learn task-specific distances. In this work, we propose the Supervised Tree-Wasserstein (STW) distance, a fast, supervised metric learning method based on the tree metric. Specifically, we rewrite th…
HRL Laboratories, LLC.（HRL实验室有限责任公司）；Télécom ParisTech（巴黎高等电信学院）；University of Virginia（弗吉尼亚大学）
The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of dista…
Understanding proper distance measures between distributions is at the core of several learning tasks such as generative models, domain adaptation, clustering, etc. In this work, we focus on mixture distributions that arise naturally in several application domains where the data contains different sub-populations. For mixture distributions, established distance measures such as the Wasserstein distance do not take into account imbalanced mixture proportions. Thus, even if two mixture distributions have identical mixture components but different mixture proportions, the Wasserstein distance bet…
École Polytechnique（巴黎综合理工学院）；University of Texas, Austin（德克萨斯大学奥斯汀分校）；Hanoi University of Science and Technology（河内科技大学）
Generalized sliced Wasserstein distance is a variant of sliced Wasserstein distance that exploits the power of non-linear projection through a given defining function to better capture the complex structures of the probability distributions. Similar to sliced Wasserstein distance, generalized sliced Wasserstein is defined as an expectation over random projections which can be approximated by the Monte Carlo method. However, the complexity of that approximation can be expensive in high-dimensional settings. To that end, we propose to form deterministic and fast approximations of the generalized…
Wasserstein distance, which measures the discrepancy between distributions, shows efficacy in various types of natural language processing (NLP) and computer vision (CV) applications. One of the challenges in estimating Wasserstein distance is that it is computationally expensive and does not scale well for many distribution comparison tasks. In this paper, we aim to approximate the 1-Wasserstein distance by the tree-Wasserstein distance (TWD), where TWD is a 1-Wasserstein distance with tree-based embedding and can be computed in linear time with respect to the number of nodes on a tree. More …
