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Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…
The Wasserstein barycenter (WB) is an important tool for summarizing sets of probability measures. It finds applications in applied probability, clustering, image processing, etc. When the measures' supports are finite, computing a…
The Wasserstein barycenter is a geometric construct which captures the notion of centrality among probability distributions, and which has found many applications in machine learning. However, most algorithms for finding even an approximate…
In this paper, we focus on the analysis of the regularized Wasserstein barycenter problem. We provide uniqueness and a characterization of the barycenter for two important classes of probability measures: (i) Gaussian distributions and (ii)…
This paper studies the statistical estimation of exact Wasserstein barycenters. Existing non-asymptotic results for empirical barycenters exhibit a severe curse of dimensionality. Motivated by the semi-dual formulation of the barycenter…
Optimal transport maps between two probability distributions $\mu$ and $\nu$ on $\mathbb{R}^d$ have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled…
We study first order methods to compute the barycenter of a probability distribution $P$ over the space of probability measures with finite second moment. We develop a framework to derive global rates of convergence for both gradient…
We revisit the problem of recovering a low-rank positive semidefinite matrix from rank-one projections using tools from optimal transport. More specifically, we show that a variational formulation of this problem is equivalent to computing…
In this paper, we introduce a generalization of the Wasserstein barycenter, to a case where the initial probability measures live on different subspaces of R^d. We study the existence and uniqueness of this barycenter, we show how it is…
A data-driven formulation of the optimal transport problem is presented and solved using adaptively refined meshes to decompose the problem into a sequence of finite linear programming problems. Both the marginal distributions and their…
This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the…
This paper discusses the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximate solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution…
We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in…
We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability…
Collecting and aggregating information from several probability measures or histograms is a fundamental task in machine learning. One of the popular solution methods for this task is to compute the barycenter of the probability measures…
We study barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). We prove that -- $p$-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely…
The primary choice to summarize a finite collection of random objects is by using measures of central tendency, such as mean and median. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fr\'{e}chet or…
Inspired by recent advances in distributed algorithms for approximating Wasserstein barycenters, we propose a novel distributed algorithm for this problem. The main novelty is that we consider time-varying computational networks, which are…
We propose a novel approach to the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our…
Consider a multi-agent system whereby each agent has an initial probability measure. In this paper, we propose a distributed algorithm based upon stochastic, asynchronous and pairwise exchange of information and displacement interpolation…