Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

  title={Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics},
  author={Adil Ahidar-Coutrix and Thibaut Le Gouic and Quentin Paris},
  journal={Probability Theory and Related Fields},
This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last… 
Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent
This work proves new geodesic convexity results on auxiliary functionals; this provides strong control of the Riemannian GD iterates, ultimately yielding a dimension-free convergence rate.
The exponentially weighted average forecaster in geodesic spaces of non-positive curvature
This paper addresses the problem of prediction with expert advice for outcomes in a geodesic space with non-positive curvature in the sense of Alexandrov by extending to this setting the definition and analysis of the classical exponentially weighted average forecaster.
Entropic-Wasserstein Barycenters: PDE Characterization, Regularity, and CLT
This paper characterizes Wasserstein barycenters in terms of a system of Monge–Ampère equations, and proves some global moment and Sobolev bounds as well as higher regularity properties.
An entropic generalization of Caffarelli's contraction theorem via covariance inequalities
The optimal transport map between the standard Gaussian measure and an α-strongly logconcave probability measure is α-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this
Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions
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
Randomized Wasserstein Barycenter Computation: Resampling with Statistical Guarantees
A hybrid resampling method to approximate finitely supported Wasserstein barycenters on large-scale datasets, which can be combined with any exact solver, and which is shown to be optimal and independent of the underlying dimension.
Online learning with exponential weights in metric spaces
  • Q. Paris
  • Mathematics, Computer Science
  • 2021
This paper extends the analysis of the exponentially weighted average forecaster, traditionally studied in a Euclidean settings, to a more abstract framework using the notion of barycenters, a suitable version of Jensen’s inequality and a synthetic notion of lower curvature bound in metric spaces known as the measure contraction property.
This work develops an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions that is shown to achieve near minimax optimality.
Wasserstein Distributionally Robust Optimization via Wasserstein Barycenters
The proposed formulation admits a tractable reformulation as a finite convex program, with powerful finite-sample and asymptotic guarantees, for Wasserstein distributionally robust optimization problems.


Ricci curvature for metric-measure spaces via optimal transport
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the
Probability Measures on Metric Spaces of Nonpositive Curvature
We present an introduction to metric spaces of nonpositive curvature (”NPC spaces”) and a discussion of barycenters of probability measures on such spaces. In our introduction to NPC spaces, we will
Barycenters in the Wasserstein Space
This paper provides existence, uniqueness, characterizations, and regularity of the barycenter and relates it to the multimarginal optimal transport problem considered by Gangbo and Świech in [Comm. Pure Appl. Math., 51 (1998), pp. 23–45].
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence
Existence and consistency of Wasserstein barycenters
It is proved the existence of Wasserstein barycenters of random probabilities defined on a geodesic space (E, d) and the consistency of this barycenter in a general setting, that includes taking baryCenters of empirical versions of the probability measures or of a growing set of probability measures.
Riemannian Lp center of mass: existence, uniqueness, and convexity
Let be a complete Riemannian manifold and a probability measure on . Assume . We derive a new bound (in terms of , the injectivity radius of and an upper bound on the sectional curvatures of ) on the
Barycenters in Alexandrov spaces of curvature bounded below
We investigate barycenters of probability measures on proper Alexandrov spaces of curvature bounded below, and show that they enjoy several properties relevant to or different from those in metric
Metric Spaces of Non-Positive Curvature
This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by
A geometric study of Wasserstein spaces: Euclidean spaces
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there
A rigidity theorem in Alexandrov spaces with lower curvature bound
Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm.