Related papers: On Combinatorial Properties of Greedy Wasserstein …
Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is…
Given a graph $G=(V,E)$, suppose we are interested in selecting a sequence of vertices $(x_j)_{j=1}^n$ such that $\left\{x_1, \dots, x_k\right\}$ is `well-distributed' uniformly in $k$. We describe a greedy algorithm motivated by potential…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein…
Optimal transport distances are powerful tools to compare probability distributions and have found many applications in machine learning. Yet their algorithmic complexity prevents their direct use on large scale datasets. To overcome this…
Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one,…
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…
We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the function $f:\mathbb{T} \rightarrow \mathbb{R}$ satisfy $\hat{f}(k) \geq c|k|^{-2}$ and define a…
We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight…
The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for…
Optimal transportation theory and the related $p$-Wasserstein distance ($W_p$, $p\geq 1$) are widely-applied in statistics and machine learning. In spite of their popularity, inference based on these tools has some issues. For instance, it…
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…
Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the…
Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is…
Random measures provide flexible parameters for Bayesian nonparametric models. Given two different priors for a random measure, we develop a natural framework to investigate the rate at which the corresponding posteriors merge, as the…
We discuss the classical problem of measuring the regularity of distribution of sets of $N$ points in $\mathbb{T}^d$. A recent line of investigation is to study the cost ($=$ mass $\times$ distance) necessary to move Dirac measures placed…
The Wasserstein distances $W_p$ ($p\geq 1$), defined in terms of solution to the Monge-Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou-Brenier formula characterizes the…
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…
We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $\Lambda$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with…
Optimal transport is widely used to learn distributions, enforce distributional constraints, and model uncertainty. In applications, transport losses are often computed from samples through tractable representations, such as one-dimensional…