Related papers: Wasserstein barycenters on metric graphs

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 barycenter problem is to compute the average of $m$ given probability measures, which has been widely studied in many different areas; however, real-world data sets are often noisy and huge, which impedes its applications in…

We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of…

We propose a unifying framework for generalising the Wasserstein-1 metric to a discrepancy measure between nonnegative measures of different mass. This generalization inherits the convexity and computational efficiency from the…

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…

Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are…

We suggest that the tools of contraction analysis for deterministic systems can be applied towards studying the convergence behavior of stochastic dynamical systems in the Wasserstein metric. In particular, we consider the case of Ito…

We obtain $q$-Wasserstein convergence rates in the invariance principle for nonuniformly hyperbolic flows, where $q\ge1$ depends on the degree of nonuniformity. Utilizing a martingale-coboundary decomposition for nonuniformly expanding…

We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein distance of general M-estimators, with an almost sharp (up to a logarithmic term) behavior in the number of observations. We focus on situations where the…

In this brief note, it is shown that the Bures-Wasserstein (BW) metric on the space positive definite matrices lends itself to convex optimization. In other words, the computation of the BW metric can be posed as a convex optimization…

We produce an upper bound for the Hausdorff dimension of the graph of a Weierstrass-type function. Whilst strictly weaker than existing results, it has the advantage of being directly computable from the theory of hyperbolic iterated…

In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{Cut}(p, \mathfrak{g})$ of a point $p$ in a compact Riemannian manifold $(M, \mathfrak{g})$ under $C^2$ perturbation of the metric.…

We study the complexity of approximating Wassertein barycenter of $m$ discrete measures, or histograms of size $n$ by contrasting two alternative approaches, both using entropic regularization. The first approach is based on the Iterative…

We consider metrics which are preserved under a $p$-Wasserstein transport map, up to a possible contraction. In the case $p=1$ this corresponds to a metric which is uniformly curved in the sense of coarse Ricci curvature. We investigate the…

We give a survey, known and new results on the beingness of fixed points of the maximal operator in the more general settings of metric measure space. In particular, we prove that the fixed points of the uncentered one must be the constant…

We show that the variance of a probability measure $\mu$ on a compact subset $X$ of a complete metric space $M$ is bounded by the square of the circumradius $R$ of the canonical embedding of $X$ into the space $P(M)$ of probability measures…

In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and…

Persistence diagrams do not admit an inner product structure compatible with any Wasserstein metric. Hence, when applying kernel methods to persistence diagrams, the underlying feature map necessarily causes distortion. We prove persistence…

The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with…

We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $\beta$, and then give convergence theorems for martingales of…