Related papers: From optimal transportation to optimal teleportati…
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…
Given two n-dimensional measures $\mu$ and $\nu$ on Polish spaces, we propose an optimal transportation's formulation, inspired by classical Kan-torovitch's formulation in the scalar case. In particular, we established a strong duality…
Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability…
The Wasserstein metric or earth mover's distance (EMD) is a useful tool in statistics, machine learning and computer science with many applications to biological or medical imaging, among others. Especially in the light of increasingly…
Optimal transport is widely used in pure and applied mathematics to find probabilistic solutions to hard combinatorial matching problems. We extend the Wasserstein metric and other elements of optimal transport from the matching of sets to…
Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from…
Measuring dependence between random variables is a fundamental problem in Statistics, with applications across diverse fields. While classical measures such as Pearson's correlation have been widely used for over a century, they have…
This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative…
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…
Let $(X_t)_{t \geq 0}$ be a diffusion process defined on a compact Riemannian manifold, and for $\alpha > 0$, let $$ \mu_t^{(\alpha)} = \frac{\alpha}{t^\alpha} \int_{0}^{t} \delta_{X_s} \, s^{\alpha - 1} \mathrm{d} s $$ be the associated…
We construct a probability measure $\mu$ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu}\in L_{p}(\mathbb{R}^d)$.
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
Combining p-values from multiple independent tests is a fundamental task in statistical inference, but presents unique challenges when the p-values are discrete. We extend a recent optimal transport-based framework for combining discrete…
Computational implementation of optimal transport barycenters for a set of target probability measures requires a form of approximation, a widespread solution being empirical approximation of measures. We provide an $O(\sqrt{N/n})$…
We study the average $p-$Wasserstein distance between a finite sample of an infinite hyperuniform point process on $\mathbb{R}^2$ and its mean for any $p\geq 1$. The average Wasserstein transport cost is shown to be bounded from above and…
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with compact support, and let $\mu \boxplus \nu$ denote their additive free convolution. We show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of $\mu$ and…
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in…
A common way to discretize a probability measure is to use an empirical measure as a discrete approximation. But how far from being optimal is this approximation in the p-Wasserstein distance? In this paper, we study this question in two…
Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability…
We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal…