We propose a new algorithm that uses an auxiliary neural network to express the potential of the optimal transport map between two data distributions. In the sequel, we use the aforementioned map to train generative networks. Unlike WGANs, where the Euclidean distance is ${\it implicitly}$ used, this new method allows to ${\it explicitly}$ use ${\it any}$ transportation cost function that can be chosen to match the problem at hand. For example, it allows to use the squared distance as a transportation cost function, giving rise to the Wasserstein-2 metric for probability distributions, which results in fast and stable gradient descends. It also allows to use image centered distances, like the structure similarity index, with notable differences in the results.
We investigate the use of entropy-regularized optimal transport (EOT) cost in developing generative models to learn implicit distributions. Two generative models are proposed. One uses EOT cost directly in an one-shot optimization problem and the other uses EOT cost iteratively in an adversarial game. The proposed generative models show improved performance over contemporary models for image generation on MNSIT.
Generative adversarial networks (GANs) are the state of the art in generative modeling. Unfortunately, most GAN methods are susceptible to mode collapse, meaning that they tend to capture only a subset of the modes of the true distribution. A possible way of dealing with this problem is to use an ensemble of GANs, where (ideally) each network models a single mode. In this paper, we introduce a principled method for training an ensemble of GANs using semi-discrete optimal transport theory. In our approach, each generative network models the transportation map between a point mass (Dirac measure…
We propose a potential flow generator with $L_2$ optimal transport regularity, which can be easily integrated into a wide range of generative models including different versions of GANs and flow-based models. We show the correctness and robustness of the potential flow generator in several 2D problems, and illustrate the concept of "proximity" due to the $L_2$ optimal transport regularity. Subsequently, we demonstrate the effectiveness of the potential flow generator in image translation tasks with unpaired training data from the MNIST dataset and the CelebA dataset.
Generative modelling is a key tool in unsupervised machine learning which has achieved stellar success in recent years. Despite this huge success, even the best generative models such as Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) come with their own shortcomings, mode collapse and mode mixture being the two most prominent problems. In this paper we develop a new generative model capable of generating samples which resemble the observed data, and is free from mode collapse and mode mixture. Our model is inspired by the recently proposed Autoencoder-Optimal Transp…
Inria（法国国家信息与自动化研究所）；LTCI, Télécom ParisTech, Université Paris-Saclay（巴黎-萨克雷大学巴黎高等电信学院信息处理与通信实验室）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）
By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of complicated datasets and sampling from them. The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes. We formulate the problem as a gradient flow in the space of probability measures. The connection…
Element AI（Element AI公司）；MILA, Université de Montréal（蒙特利尔大学MILA研究所）；Stanford University（斯坦福大学）
Computing optimal transport maps between high-dimensional and continuous distributions is a challenging problem in optimal transport (OT). Generative adversarial networks (GANs) are powerful generative models which have been successfully applied to learn maps across high-dimensional domains. However, little is known about the nature of the map learned with a GAN objective. To address this problem, we propose a generative adversarial model in which the discriminator's objective is the $2$-Wasserstein metric. We show that during training, our generator follows the $W_2$-geodesic between the init…
Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎理工学院）；ENS Cachan（卡尚高等师范学校）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；University of Oxford（牛津大学）
Minimum expected distance estimation (MEDE) algorithms have been widely used for probabilistic models with intractable likelihood functions and they have become increasingly popular due to their use in implicit generative modeling (e.g. Wasserstein generative adversarial networks, Wasserstein autoencoders). Emerging from computational optimal transport, the Sliced-Wasserstein (SW) distance has become a popular choice in MEDE thanks to its simplicity and computational benefits. While several studies have reported empirical success on generative modeling with SW, the theoretical properties of su…
ETH Zurich（苏黎世联邦理工学院）；Merantix GmbH（梅兰蒂克斯有限责任公司）；KU Leuven（鲁汶大学）
Generative models such as Variational Auto Encoders (VAEs) and Generative Adversarial Networks (GANs) are typically trained for a fixed prior distribution in the latent space, such as uniform or Gaussian. After a trained model is obtained, one can sample the Generator in various forms for exploration and understanding, such as interpolating between two samples, sampling in the vicinity of a sample or exploring differences between a pair of samples applied to a third sample. In this paper, we show that the latent space operations used in the literature so far induce a distribution mismatch betw…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Columbia University（哥伦比亚大学）
This paper provides a simple procedure to fit generative networks to target distributions, with the goal of a small Wasserstein distance (or other optimal transport costs). The approach is based on two principles: (a) if the source randomness of the network is a continuous distribution (the "semi-discrete" setting), then the Wasserstein distance is realized by a deterministic optimal transport mapping; (b) given an optimal transport mapping between a generator network and a target distribution, the Wasserstein distance may be decreased via a regression between the generated data and the mapped…
Sony Computer Science Laboratories Inc（索尼计算机科学实验室股份有限公司）；CSIRO’s Data61（英联邦科学与工业研究组织数据61研究所）
We define a novel class of distances between statistical multivariate distributions by modeling an optimal transport problem on their marginals with respect to a ground distance defined on their conditionals. These new distances are metrics whenever the ground distance between the marginals is a metric, generalize both the Wasserstein distances between discrete measures and a recently introduced metric distance between statistical mixtures, and provide an upper bound for jointly convex distances between statistical mixtures. By entropic regularization of the optimal transport, we obtain a fast…
Stony Brook University（石溪大学）；Dalian University of Technology（大连理工大学）；Harvard University（哈佛大学）
Variational Auto-Encoders enforce their learned intermediate latent-space data distribution to be a simple distribution, such as an isotropic Gaussian. However, this causes the posterior collapse problem and loses manifold structure which can be important for datasets such as facial images. A GAN can transform a simple distribution to a latent-space data distribution and thus preserve the manifold structure, but optimizing a GAN involves solving a Min-Max optimization problem, which is difficult and not well understood so far. Therefore, we propose a GAN-like method to transform a simple distr…
The field of deep generative modeling is dominated by generative adversarial networks (GANs). However, the training of GANs often lacks stability, fails to converge, and suffers from model collapse. It takes an assortment of tricks to solve these problems, which may be difficult to understand for those seeking to apply generative modeling. Instead, we propose two novel generative autoencoders, AE-OTtrans and AE-OTgen, which rely on optimal transport instead of adversarial training. AE-OTtrans and AEOTgen, unlike VAE and WAE, preserve the manifold of the data; they do not force the latent distr…
A generative flow for conditional sampling via optimal transport
University of the Philippines Diliman（菲律宾大学迪里曼分校）；California Institute of Technology（加州理工学院）；Columbia University（哥伦比亚大学）；Baruch College（巴鲁克学院）；University of Edinburgh（爱丁堡大学）；Georgia Institute of Technology（佐治亚理工学院）；Grinnell College（格林内尔学院）；University of Michigan（密歇根大学）；Yale University（耶鲁大学）
The framework of optimal transport has been leveraged to extend the notion of rank to the multivariate setting while preserving desirable properties of the resulting goodness-of-fit (GoF) statistics. In particular, the rank energy (RE) and rank maximum mean discrepancy (RMMD) are distribution-free under the null, exhibit high power in statistical testing, and are robust to outliers. In this paper, we point to and alleviate some of the practical shortcomings of these proposed GoF statistics, namely their high computational cost, high statistical sample complexity, and lack of differentiability …
Space Applications Centre（空间应用中心）；Indian Space Research Organisation（印度空间研究组织）；Skolkovo Institute of Science and Technology（斯科尔科沃科技学院）；Artificial Intelligence Research Institute (AIRI)（人工智能研究院（AIRI））
With the discovery of Wasserstein GANs, Optimal Transport (OT) has become a powerful tool for large-scale generative modeling tasks. In these tasks, OT cost is typically used as the loss for training GANs. In contrast to this approach, we show that the OT map itself can be used as a generative model, providing comparable performance. Previous analogous approaches consider OT maps as generative models only in the latent spaces due to their poor performance in the original high-dimensional ambient space. In contrast, we apply OT maps directly in the ambient space, e.g., a space of high-dimension…
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…
Princeton University（普林斯顿大学）；Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Chinese Academy of Sciences（中国科学院）
We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Ampère flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative…
Xi’an Jiaotong University（西安交通大学）；University of Iowa（爱荷华大学）；Wuhan University（武汉大学）；Duke-NUS Medical School（杜克-新加坡国立大学医学院）
We propose an Euler particle transport (EPT) approach for generative learning. The proposed approach is motivated by the problem of finding an optimal transport map from a reference distribution to a target distribution characterized by the Monge-Ampere equation. Interpreting the infinitesimal linearization of the Monge-Ampere equation from the perspective of gradient flows in measure spaces leads to a stochastic McKean-Vlasov equation. We use the forward Euler method to solve this equation. The resulting forward Euler map pushes forward a reference distribution to the target. This map is the …
University of Maryland（马里兰大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Optimal Transport (OT) distances such as Wasserstein have been used in several areas such as GANs and domain adaptation. OT, however, is very sensitive to outliers (samples with large noise) in the data since in its objective function, every sample, including outliers, is weighed similarly due to the marginal constraints. To remedy this issue, robust formulations of OT with unbalanced marginal constraints have previously been proposed. However, employing these methods in deep learning problems such as GANs and domain adaptation is challenging due to the instability of their dual optimization s…
Optimal transport distances for directed, weighted graphs: a case study with cell-cell communication networks
Institute for Computational Genomics, RWTH Aachen Medical Faculty（亚琛工业大学医学院计算基因组学研究所）；Department of Computer Science, RWTH Aachen University（亚琛工业大学计算机科学系）
Generative adversarial networks (GANs) are an expressive class of neural generative models with tremendous success in modeling high-dimensional continuous measures. In this paper, we present a scalable method for unbalanced optimal transport (OT) based on the generative-adversarial framework. We formulate unbalanced OT as a problem of simultaneously learning a transport map and a scaling factor that push a source measure to a target measure in a cost-optimal manner. In addition, we propose an algorithm for solving this problem based on stochastic alternating gradient updates, similar in practi…
Dalian University of Technology（大连理工大学）；Wuhan University（武汉大学）；Beijing Normal University（北京师范大学）；Harvard University（哈佛大学）；Stony Brook University（石溪大学）
In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models. By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can giv…
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 …
The Fréchet Inception Distance (FID) has been used to evaluate hundreds of generative models. We introduce FastFID, which can efficiently train generative models with FID as a loss function. Using FID as an additional loss for Generative Adversarial Networks improves their FID.
Generative Adversarial Networks have been shown to be powerful in generating content. To this end, they have been studied intensively in the last few years. Nonetheless, training these networks requires solving a saddle point problem that is difficult to solve and slowly converging. Motivated from techniques in the registration of point clouds and by the fluid flow formulation of mass transport, we investigate a new formulation that is based on strict minimization, without the need for the maximization. The formulation views the problem as a matching problem rather than an adversarial one and …
Great computational effort is invested in generating equilibrium states for molecular systems using, for example, Markov chain Monte Carlo. We present a probabilistic model that generates statistically independent samples for molecules from their graph representations. Our model learns a low-dimensional manifold that preserves the geometry of local atomic neighborhoods through a principled learning representation that is based on Euclidean distance geometry. In a new benchmark for molecular conformation generation, we show experimentally that our generative model achieves state-of-the-art accu…
A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science
Kyushu University（九州大学）；University of Tübingen（蒂宾根大学）；Tübingen AI Center（蒂宾根人工智能中心）；University Hospital Tübingen（蒂宾根大学医院）；M3 Research Center, University Hospital Tübingen（蒂宾根大学医院M3研究中心）；University Medical Center of Bonn（波恩大学医学中心）；Technical University of Munich（慕尼黑工业大学）
Automatic evaluation of the goodness of Generative Adversarial Networks (GANs) has been a challenge for the field of machine learning. In this work, we propose a distance complementary to existing measures: Topology Distance (TD), the main idea behind which is to compare the geometric and topological features of the latent manifold of real data with those of generated data. More specifically, we build Vietoris-Rips complex on image features, and define TD based on the differences in persistent-homology groups of the two manifolds. We compare TD with the most commonly used and relevant measures…
University of Science and Technology Beijing（北京科技大学）；Technical University of Denmark（丹麦技术大学）
Deep generative models are tremendously successful in learning low-dimensional latent representations that well-describe the data. These representations, however, tend to much distort relationships between points, i.e. pairwise distances tend to not reflect semantic similarities well. This renders unsupervised tasks, such as clustering, difficult when working with the latent representations. We demonstrate that taking the geometry of the generative model into account is sufficient to make simple clustering algorithms work well over latent representations. Leaning on the recent finding that dee…
