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Wasserstein dictionary learning is an unsupervised approach to learning a collection of probability distributions that generate observed distributions as Wasserstein barycentric combinations. Existing methods for Wasserstein dictionary…
Wasserstein distances are widely used in modern data analysis but pose significant computational and statistical challenges in high dimensions. The sliced Wasserstein distance alleviates these challenges by leveraging one-dimensional…
We consider a multimarginal optimal transport, which includes as a particular case the Wasserstein barycenter problem. In this problem one has to find an optimal coupling between $m$ probability measures, which amounts to finding a tensor…
We propose a variational approach to approximate measures with measures uniformly distributed over a 1 dimentional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of…
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…
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…
This paper deals with the estimation of a probability measure on the real line from data observed with an additive noise. We are interested in rates of convergence for the Wasserstein metric of order $p\geq 1$. The distribution of the…
We consider probabilistic models for sequential observations which exhibit gradual transitions among a finite number of states. We are particularly motivated by applications such as human activity analysis where observed accelerometer time…
It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the…
We study the problem of learning a real-valued function that satisfies the Demographic Parity constraint. It demands the distribution of the predicted output to be independent of the sensitive attribute. We consider the case that the…
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which…
We revisit the question of characterizing the convergence rate of plug-in estimators of optimal transport costs. It is well known that an empirical measure comprising independent samples from an absolutely continuous distribution on…
The sliced Wasserstein barycenter (SWB) is a widely acknowledged method for efficiently generalizing the averaging operation within probability measure spaces. However, achieving marginal fairness SWB, ensuring approximately equal distances…
We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges…
In this paper, we establish sharp upper and lower bounds on the convergence rate of the empirical measures of point processes under the Wasserstein distance. To this end, we first introduce a new metric on the space of counting measures…
As a natural approach to modeling system safety conditions, chance constraint (CC) seeks to satisfy a set of uncertain inequalities individually or jointly with high probability. Although a joint CC offers stronger reliability certificate,…
Motivated by the growing need for black-box optimization and data privacy, we introduce a collaborative Bayesian optimization (BO) framework that addresses both of these challenges. In this framework agents work collaboratively to optimize…
This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on…
In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Fr\'echet mean of some distribution $\mathbb{P}$ supported on a subspace of positive semi-definite Hermitian operators $\mathbb{H}_{+}(d)$.…
Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment…