Wasserstein barycenters over Riemannian manifolds
type: Item
instance of: scholarly article
title: Wasserstein barycenters over Riemannian manifolds
zbMATH Open document ID: 1373.60006
author: Young-Heon Kim
author: Brendan Pass
published in: Advances in Mathematics
publication date: 2017-01-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1412.7726
review text: Let \(M\) be a compact Riemannian manifold with the space of Borel probability measures on \(M\) denoted by \(P(M)\). The Wasserstein space associated to \(M\) is the metric space \(P(M)\) endowed with the Wasserstein metric. For any probability measure \(\Omega\) on that Wasserstein space, its Wasserstein barycenter is an element of \(P(M)\). The main results of the paper under review give sufficient conditions for the barycenter of \(\Omega\) to be absolutely continuous with respect to the volume measure of~\(M\). These conditions are more general than the ones from earlier work and as applications, the authors are able to obtain versions of the Jensen inequality and a generalized Brunn-Minkowski inequality.
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 58C35
zbMATH DE Number: 6670887
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Wasserstein barycenter
zbMATH Keywords: multi-marginal optimal transport
Wikidata QID: Q115362023
MaRDI profile type: Publication
OpenAlex ID: W2963587665
arXiv ID: 1412.7726
cites work: Barycenters in the Wasserstein Space
cites work: Q4954179
cites work: Q4663947
cites work: Existence and uniqueness of optimal maps on Alexandrov spaces
cites work: On the Euler-Lagrange equation for a variational problem
cites work: On the Euler-Lagrange equation for a variational problem: The general case. II
cites work: Q5524269
cites work: Q3797433
cites work: Q4871557
cites work: Matching for teams
cites work: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
cites work: Positively curved Riemannian locally symmetric spaces are positively squared distance curved
cites work: Locally nearly spherical surfaces are almost-positively \(c-\)curved
cites work: Q3979066
cites work: Absolute continuity of Wasserstein geodesics in the Heisenberg group
cites work: When is multidimensional screening a convex program?
cites work: Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2
cites work: The geometry of optimal transportation
cites work: Optimal maps for the multidimensional Monge-Kantorovich problem
cites work: Foundations of quantization for probability distributions
cites work: Counterexamples to Continuity of Optimal Transport Maps on Positively Curved Riemannian Manifolds
cites work: Continuity, curvature, and the general covariance of optimal transportation
cites work: Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)
cites work: Multi-marginal optimal transport on Riemannian manifolds
cites work: On a generalization of cyclic monotonicity and distances among random vectors
cites work: Jensen's inequality on convex spaces
cites work: On the regularity of solutions of optimal transportation problems
cites work: Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna
cites work: Ricci curvature for metric-measure spaces via optimal transport
cites work: Regularity of potential functions of the optimal transportation problem
cites work: A convexity principle for interacting gases
cites work: Polar factorization of maps on Riemannian manifolds
cites work: Hexagonal Economic Regions Solve the Location Problem
cites work: Maximum Submatrix Traces for Positive Definite Matrices
cites work: Optimal transportation with infinitely many marginals
cites work: Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions
cites work: Multi-marginal optimal transport: Theory and applications
cites work: Q4395406
cites work: Absolute continuity and summability of transport densities: simpler proofs and new estimates
cites work: Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling
cites work: Q4453431
cites work: On the geometry of metric measure spaces. I
cites work: On the geometry of metric measure spaces. II
cites work: Q5484522
cites work: On the second boundary value problem for Monge-Ampère type equations and optimal transportation
cites work: Q4805362
cites work: Optimal Transport
cites work: The Brunn-Minkowski inequality for random sets
DOI: 10.1016/J.AIM.2016.11.026
description: scientific article