Related papers: Learning Probability Measures with respect to Opti…
We consider the problem of learning parametric distributions from their quantized samples in a network. Specifically, $n$ agents or sensors observe independent samples of an unknown parametric distribution; and each of them uses $k$ bits to…
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels,…
The goal of regression and classification methods in supervised learning is to minimize the empirical risk, that is, the expectation of some loss function quantifying the prediction error under the empirical distribution. When facing scarce…
Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
The aim of this article is to introduce and address the problem to smoothly interpolate (empirical) probability measures. To this end, we lift the concept of a spline curve from the setting of points in a Euclidean space that that of…
We give a new probabilistic construction of solutions to real Monge-Amp\`ere equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal…
We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for…
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 study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference…
This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality, in particular, we provide examples…
Recently, Optimal Transport has been proposed as a probabilistic framework in Machine Learning for comparing and manipulating probability distributions. This is rooted in its rich history and theory, and has offered new solutions to…
In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability…
We consider an optimal transport problem between laws of random probability measures: given a base cost function, we build the associated OT cost between probability measures that in turn we use to define the OT cost between probability…
We show that in any complete metric space the probability measures $\mu$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $\nu$ living on the…
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…
The objective in statistical Optimal Transport (OT) is to consistently estimate the optimal transport plan/map solely using samples from the given source and target marginal distributions. This work takes the novel approach of posing…
The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite dimensional Hilbert space. It allows us, for example, to define a distance…
Learning conditional distributions $\pi^*(\cdot|x)$ is a central problem in machine learning, which is typically approached via supervised methods with paired data $(x,y) \sim \pi^*$. However, acquiring paired data samples is often…
In recent years, Bayesian inference in large-scale inverse problems found in science, engineering and machine learning has gained significant attention. This paper examines the robustness of the Bayesian approach by analyzing the stability…