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Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

Guillaume Carlier, Katharina Eichinger, Alexey Kroshnin

TL;DR

After characterizing these barycenters in terms of a system of Monge-Amp\`ere equations, it is proved some global moment and Sobolev bounds as well as higher regularity properties and a central limit theorem is established for entropic-Wasserstein baryCenters.

Abstract

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a regularization of Wasserstein barycenters first presented by Agueh and Carlier (2011). After characterizing these barycenters in terms of a system of Monge-Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.

Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

TL;DR

After characterizing these barycenters in terms of a system of Monge-Amp\`ere equations, it is proved some global moment and Sobolev bounds as well as higher regularity properties and a central limit theorem is established for entropic-Wasserstein baryCenters.

Abstract

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a regularization of Wasserstein barycenters first presented by Agueh and Carlier (2011). After characterizing these barycenters in terms of a system of Monge-Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.

Paper Structure

This paper contains 15 sections, 22 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Setting, assumptions and preliminaries
  3. Characterization
  4. Properties of the entropic barycenter
  5. Global bounds
  6. Stability
  7. A maximum principle
  8. Higher regularity
  9. The bounded case
  10. The case of log-concave measures on Rd
  11. Statistical properties
  12. Stochastic setting and law of large numbers
  13. Central limit theorem
  14. Linearization of Monge--Ampère equations
  15. Auxiliary probability results

Key Result

Proposition 2.2

Assuming finitemomentP, then barycentpbm admits a unique solution.

Theorems & Definitions (60)

  • Example 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 50 more