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 embedded distributions. We examine empirically the representational capacity of our learned Wasserstein embeddings, showing that they can embed a wide variety of metric structures with smaller distortion than an equivalent Euclidean embedding. We also investigate an application to word embedding, demonstrating a unique advantage of Wasserstein embeddings: We can visualize the high-dimensional embedding directly, since it is a probability distribution on a low-dimensional space. This obviates the need for dimensionality reduction techniques like t-SNE for visualization.
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…
As interest in graph data has grown in recent years, the computation of various geometric tools has become essential. In some area such as mesh processing, they often rely on the computation of geodesics and shortest paths in discretized manifolds. A recent example of such a tool is the computation of Wasserstein barycenters (WB), a very general notion of barycenters derived from the theory of Optimal Transport, and their entropic-regularized variant. In this paper, we examine how WBs on discretized meshes relate to the geometry of the underlying manifold. We first provide a generic stability …
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…
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）
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…
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…
Structure-Preserving Multi-View Embedding Using Gromov-Wasserstein Optimal Transport
Instituto Federal de Educação, Ciência e Tecnologia do Ceará（塞阿拉联邦教育、科学与技术学院）；Sigma Nova；Department of Teleinformatics Engineering, Federal University of Ceará（塞阿拉联邦大学远程信息工程系）
Infinia ML, Inc.（Infinia ML公司）；Duke University（杜克大学）；Georgia Institute of Technology（佐治亚理工学院）
A novel Gromov-Wasserstein learning framework is proposed to jointly match (align) graphs and learn embedding vectors for the associated graph nodes. Using Gromov-Wasserstein discrepancy, we measure the dissimilarity between two graphs and find their correspondence, according to the learned optimal transport. The node embeddings associated with the two graphs are learned under the guidance of the optimal transport, the distance of which not only reflects the topological structure of each graph but also yields the correspondence across the graphs. These two learning steps are mutually-beneficia…
Infinia ML, Inc.（Infinia ML公司）；Duke University（杜克大学）；Tencent AI Lab（腾讯人工智能实验室）
We propose a novel Wasserstein method with a distillation mechanism, yielding joint learning of word embeddings and topics. The proposed method is based on the fact that the Euclidean distance between word embeddings may be employed as the underlying distance in the Wasserstein topic model. The word distributions of topics, their optimal transports to the word distributions of documents, and the embeddings of words are learned in a unified framework. When learning the topic model, we leverage a distilled underlying distance matrix to update the topic distributions and smoothly calculate the co…
Université de Technologie de Compiègne（贡比涅技术大学）；INSA-Rouen（鲁昂国立应用科学学院）；Université de Rouen Normandie（鲁昂诺曼底大学）；Criteo AI Lab（Criteo人工智能实验室）
Optimal Transport (OT) metrics allow for defining discrepancies between two probability measures. Wasserstein distance is for longer the celebrated OT-distance frequently-used in the literature, which seeks probability distributions to be supported on the $\textit{same}$ metric space. Because of its high computational complexity, several approximate Wasserstein distances have been proposed based on entropy regularization or on slicing, and one-dimensional Wassserstein computation. In this paper, we propose a novel extension of Wasserstein distance to compare two incomparable distributions, tha…
Measuring similarities between different tasks is critical in a broad spectrum of machine learning problems, including transfer, multi-task, continual, and meta-learning. Most current approaches to measuring task similarities are architecture-dependent: 1) relying on pre-trained models, or 2) training networks on tasks and using forward transfer as a proxy for task similarity. In this paper, we leverage the optimal transport theory and define a novel task embedding for supervised classification that is model-agnostic, training-free, and capable of handling (partially) disjoint label sets. In s…
Classical optimal transport problem seeks a transportation map that preserves the total mass betwenn two probability distributions, requiring their mass to be the same. This may be too restrictive in certain applications such as color or shape matching, since the distributions may have arbitrary masses and/or that only a fraction of the total mass has to be transported. Several algorithms have been devised for computing partial Wasserstein metrics that rely on an entropic regularization, but when it comes with exact solutions, almost no partial formulation of neither Wasserstein nor Gromov-Was…
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge. Recent work has advocated a two-step approach to improve robustness and facilitate the computation of optimal transport, using for instance projections on random real lines, or a preliminary quantization of the measures to reduce the size of their support. We propose in this work a "max-min" robust variant of the Wasserstein distance by considering the maximal possible distance that can be realized between two measures, assuming they can be projected orthogonally on a lower $k$-dim…
IBM Research（IBM研究院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
In this paper we propose to perform model ensembling in a multiclass or a multilabel learning setting using Wasserstein (W.) barycenters. Optimal transport metrics, such as the Wasserstein distance, allow incorporating semantic side information such as word embeddings. Using W. barycenters to find the consensus between models allows us to balance confidence and semantics in finding the agreement between the models. We show applications of Wasserstein ensembling in attribute-based classification, multilabel learning and image captioning generation. These results show that the W. ensembling is a…
The ability to align points across two related yet incomparable point clouds (e.g. living in different spaces) plays an important role in machine learning. The Gromov-Wasserstein (GW) framework provides an increasingly popular answer to such problems, by seeking a low-distortion, geometry-preserving assignment between these points. As a non-convex, quadratic generalization of optimal transport (OT), GW is NP-hard. While practitioners often resort to solving GW approximately as a nested sequence of entropy-regularized OT problems, the cubic complexity (in the number $n$ of samples) of that appr…
University of California, San Diego（加州大学圣地亚哥分校）；Halicioğlu Data Science Institute（哈利西奥卢数据科学研究所）
Discriminating between distributions is an important problem in a number of scientific fields. This motivated the introduction of Linear Optimal Transportation (LOT), which embeds the space of distributions into an $L^2$-space. The transform is defined by computing the optimal transport of each distribution to a fixed reference distribution, and has a number of benefits when it comes to speed of computation and to determining classification boundaries. In this paper, we characterize a number of settings in which LOT embeds families of distributions into a space in which they are linearly separ…
Wasserstein Wormhole: Scalable Optimal Transport Distance with Transformers
Sloan Kettering Institute（斯隆凯特琳研究所）；Memorial Sloan Kettering Cancer Center（纪念斯隆凯特琳癌症中心）；Tri-Institutional Training Program in Computational Biology and Medicine（三机构计算生物学与医学培训项目）；Weill Cornell Medicine（威尔康奈尔医学院）；Columbia University（哥伦比亚大学）；Yale University（耶鲁大学）；Howard Hughes Medical Institute（霍华德·休斯医学研究所）
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
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 Sink…
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…
Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets
Department of Electrical and Computer Engineering, Yale University（耶鲁大学电气与计算机工程系）；Department of Biostatistics, Harvard University（哈佛大学生物统计学系）；Program in Applied Mathematics, Yale University（耶鲁大学应用数学项目）；Interdepartmental Program in Computational Biology and Bioinformatics, Yale University（耶鲁大学计算生物学与生物信息学跨学科项目）；Department of Pathology, Yale University School of Medicine（耶鲁大学医学院病理学系）
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Univ. Paris-Sud（巴黎第十一大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；Inria Saclay - Ile-de-France（法国国家信息与自动化研究所萨克雷-法兰西岛分部）
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)Hölder continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent Hölder-stability results for optimal transport maps.
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…
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…
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, …
Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences. Such problems can be used to form an examplar measure out of various probability measures, as in the Wasserstein barycenter problem, or to carry out parametric inference and density fitting, where the loss is measured in terms of an optimal transport cost to the measure of observations. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein dist…
Computationally solving multi-marginal optimal transport (MOT) with squared Euclidean costs for $N$ discrete probability measures has recently attracted considerable attention, in part because of the correspondence of its solutions with Wasserstein-$2$ barycenters, which have many applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. While entropic regularization has been successfully applied to approximate Wasserstein barycenters, this loses the sparsity of the optimal solution, making it difficult to solve the MOT problem directly …
Linearized Wasserstein dimensionality reduction with approximation guarantees
University of California, San Diego（加州大学圣地亚哥分校）；Halicioğlu Data Science Institute（哈利乔奥卢数据科学研究所）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow these measures with the 2-Wasserstein metric, with two important benefits: (i) For such measures, the squared 2-Wasserstein metric has a closed form, equal to a weighted sum of the squared Euclidean distance between means and the squared Bures metric between covariance matrices. The …
Leibniz Institute of Agricultural Engineering and Bioeconomy e.V.（莱布尼茨农业工程与生物经济研究所）；University of Potsdam（波茨坦大学）
Deep metric learning employs deep neural networks to embed instances into a metric space such that distances between instances of the same class are small and distances between instances from different classes are large. In most existing deep metric learning techniques, the embedding of an instance is given by a feature vector produced by a deep neural network and Euclidean distance or cosine similarity defines distances between these vectors. In this paper, we study deep distributional embeddings of sequences, where the embedding of a sequence is given by the distribution of learned deep feat…
HRL Laboratories, LLC.（HRL有限责任公司）；University of Virginia（弗吉尼亚大学）
We present Wasserstein Embedding for Graph Learning (WEGL), a novel and fast framework for embedding entire graphs in a vector space, in which various machine learning models are applicable for graph-level prediction tasks. We leverage new insights on defining similarity between graphs as a function of the similarity between their node embedding distributions. Specifically, we use the Wasserstein distance to measure the dissimilarity between node embeddings of different graphs. Unlike prior work, we avoid pairwise calculation of distances between graphs and reduce the computational complexity …
Université de Bretagne Sud（南布列塔尼大学）；IRISA, UMR 6074, CNRS（法国国家信息与自动化研究所）；OCA（蔚蓝海岸天文台）；Lagrange, UMR 7293, CNRS（拉格朗日实验室）；I3S, UMR 7271, CNRS（计算机、信号与系统实验室）
The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain adaptation, dimensionality reduction or generative models. However, its use is still limited by a heavy computational cost. Our goal is to alleviate this problem by providing an approximation mechanism that allows to break its inherent complexity. It relies on the search of an embedding where the Euclidean distance mimics the Wasserstein distance. We show that…
Ecole Normale Supérieure（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；Université PSL（巴黎文理研究大学）；INSERM（法国国家健康与医学研究院）；Institut de Biologie de l’Ecole Normale Supérieure（巴黎高等师范学院生物学研究所）
Defining meaningful distances between samples in a dataset is a fundamental problem in machine learning. Optimal Transport (OT) lifts a distance between features (the "ground metric") to a geometrically meaningful distance between samples. However, there is usually no straightforward choice of ground metric. Supervised ground metric learning approaches exist but require labeled data. In absence of labels, only ad-hoc ground metrics remain. Unsupervised ground metric learning is thus a fundamental problem to enable data-driven applications of OT. In this paper, we propose for the first time a c…
Interaction between pharmacological agents can trigger unexpected adverse events. Capturing richer and more comprehensive information about drug-drug interactions (DDI) is one of the key tasks in public health and drug development. Recently, several knowledge graph embedding approaches have received increasing attention in the DDI domain due to their capability of projecting drugs and interactions into a low-dimensional feature space for predicting links and classifying triplets. However, existing methods only apply a uniformly random mode to construct negative samples. As a consequence, these…
Predictive states for stochastic processes are a nonparametric and interpretable construct with relevance across a multitude of modeling paradigms. Recent progress on the self-supervised reconstruction of predictive states from time-series data focused on the use of reproducing kernel Hilbert spaces. Here, we examine how Wasserstein distances may be used to detect predictive equivalences in symbolic data. We compute Wasserstein distances between distributions over sequences ("predictions"), using a finite-dimensional embedding of sequences based on the Cantor for the underlying geometry. We sh…
Transport-based metrics and related embeddings (transforms) have recently been used to model signal classes where nonlinear structures or variations are present. In this paper, we study the geodesic properties of time series data with a generalized Wasserstein metric and the geometry related to their signed cumulative distribution transforms in the embedding space. Moreover, we show how understanding such geometric characteristics can provide added interpretability to certain time series classifiers, and be an inspiration for more robust classifiers.
