Related papers: The Wasserstein distance for Ricci shrinkers

In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure $\varrho$. The system's response $f$ pushes forward $\varrho$ to a new measure $f\circ \varrho$ which we…

Wasserstein distances provide a metric on a space of probability measures. We consider the space $\Omega$ of all probability measures on the finite set $\chi = \{1, \dots ,n\}$ where $n$ is a positive integer. 1-Wasserstein distance,…

Let $M$ be a connected compact Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $\mu(\d x):=\e^{V(x)}\d x$ is a probability measure, where $\d x$ is the volume measure, and let $L=\Delta+\nabla V$. The exact…

Let $X_t$ be the (reflecting) diffusion process generated by $L:=\Delta+\nabla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^1(M)$ such that $\mu(d x):= e^{V(x)}d x$ is a probability…

The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the…

Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit interval. For fixed $\mathbb{P}$ the Wasserstein distance between $\hat{\mathbb{P}}_N$ and…

Let $\mu_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $\mu$ on $\mathbb{R}^d$. We are interested in the rate of convergence of $\mu_N$ to $\mu$, when measured in the Wasserstein distance of…

We obtain an estimate for the expected subspace robust Wasserstein distance between any probability measure on the unit ball of a separable Hilbert space, and its empirical distribution from $n$ i.i.d. samples.

We study the weighted total variation distance between probability measures. Using Fourier-analytic tools, we present estimates in terms of Wasserstein distances between the respective probabilities, under appropriate smoothness and moment…

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…

We investigate the stability of the Wasserstein distance, a metric structure on the space of probability measures arising from the theory of optimal transport, under metric ultralimits. We first show that if $(X_{i},d_{i})_{i\in\mathbb{N}}$…

The sliced Wasserstein distance as well as its variants have been widely considered in comparing probability measures defined on $\mathbb R^d$. Here we derive the notion of sliced Wasserstein distance for measures on an infinite dimensional…

We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $\mu$ and the corresponding empirical measure $\mu_N$, whenever $\mu$ has finite $q$-th moment for some…

Let $\pi\in \Pi(\mu,\nu)$ be a coupling between two probability measures $\mu$ and $\nu$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $\mu$ and $\nu$, which we call…

The question of optimally approximating an arbitrary probability measure in the Wasserstein distance by a discrete one with uniform weights is considered. Estimates are obtained for the optimal approximation distance, with an explicit rate…

The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on…

For a complete connected Riemannian manifold $M$ let $V\in C^2(M)$ be such that $\mu(d x)={\rm e}^{-V(x)} \mbox{vol}(d x)$ is a probability measure on $M$. Taking $\mu$ as reference measure, we derive inequalities for probability measures…

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 provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization…

We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal…