Related papers: $p$-Wasserstein barycenters
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 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…
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…
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 construct a positive measure on the space of positively oriented $2$-vectors in $\mathbb{R}^4$, whose barycenter is a simple $2$-vector, yet which cannot be approximated by weighted Gaussian images of Lipschitz $Q$-graphs for any fixed…
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…
Given a collection of probability measures, a practitioner sometimes needs to find an "average" distribution which adequately aggregates reference distributions. A theoretically appealing notion of such an average is the Wasserstein…
We develop a notion of projections between sets of probability measures using the geometric properties of the 2-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and…
The two-sample homogeneity testing problem is fundamental in statistics and becomes particularly challenging in high dimensions, where classical tests can suffer substantial power loss. We develop a learning-assisted procedure based on the…
We derive the limiting distribution of the barycenter $b_n$ of an i.i.d. sample of $n$ random points on a planar cone with angular spread larger than $2\pi$. There are three mutually exclusive possibilities: (i) (fully sticky case) after a…
For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every…
The so-called \emph{simplified} Wasserstein barycenter problem, also known as the cheapest hub problem, consists in selecting one point from each of $k$ given sets, each set consisting of $n$ points, with the aim of minimizing the sum of…
We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not…
The consensus problem -- achieving agreement among a network of agents -- is a central theme in both theory and applications. Recently, this problem has been extended from Euclidean spaces to the space of probability measures, where the…
The computation of exact barycenters for a set of discrete measures is of interest in applications where sparse solutions are desired, and to assess the quality of solutions returned by approximate algorithms and heuristics. The task is…
We provide new bounds for the rate of convergence of the multivariate Central Limit Theorem in Wasserstein distances of order $p \geq 2$. In particular, we obtain what we conjecture to be the asymptotically optimal rate whenever the density…
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 $(X_{i},d_{i})_{i\in\mathbb{N}}$…
Increasingly complex data analysis tasks motivate the study of the dependency of distributions of multivariate continuous random variables on scalar or vector predictors. Statistical regression models for distributional responses so far…
The Hellinger-Kantorovich (HK) distance is an unbalanced extension of the Wasserstein-2 distance. It was shown recently that the HK barycenter exhibits a much more complex behaviour than the Wasserstein barycenter. Motivated by this…
Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric $\bar{d}_1$ that combines…