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On the global convergence of Wasserstein gradient flow of the Coulomb discrepancy

Siwan Boufadène, François-Xavier Vialard

Abstract

In this work, we study the Wasserstein gradient flow of the Riesz energy defined on the space of probability measures. The Riesz kernels define a quadratic functional on the space of measure which is not in general geodesically convex in the Wasserstein geometry, therefore one cannot conclude to global convergence of the Wasserstein gradient flow using standard arguments. Our main result is the exponential convergence of the flow to the minimizer on a closed Riemannian manifold under the condition that the logarithm of the source and target measures are H{ö}lder continuous. To this goal, we first prove that the Polyak-Lojasiewicz inequality is satisfied for sufficiently regular solutions. The key regularity result is the global in-time existence of H{ö}lder solutions if the initial and target data are H{ö}lder continuous, proven either in Euclidean space or on a closed Riemannian manifold. For general measures, we prove using flow interchange techniques that there is no local minima other than the global one for the Coulomb kernel. In fact, we prove that a Lagrangian critical point of the functional for the Coulomb (or Energy distance) kernel is equal to the target everywhere except on singular sets with empty interior. In addition, singular enough measures cannot be critical points.

On the global convergence of Wasserstein gradient flow of the Coulomb discrepancy

Abstract

In this work, we study the Wasserstein gradient flow of the Riesz energy defined on the space of probability measures. The Riesz kernels define a quadratic functional on the space of measure which is not in general geodesically convex in the Wasserstein geometry, therefore one cannot conclude to global convergence of the Wasserstein gradient flow using standard arguments. Our main result is the exponential convergence of the flow to the minimizer on a closed Riemannian manifold under the condition that the logarithm of the source and target measures are H{ö}lder continuous. To this goal, we first prove that the Polyak-Lojasiewicz inequality is satisfied for sufficiently regular solutions. The key regularity result is the global in-time existence of H{ö}lder solutions if the initial and target data are H{ö}lder continuous, proven either in Euclidean space or on a closed Riemannian manifold. For general measures, we prove using flow interchange techniques that there is no local minima other than the global one for the Coulomb kernel. In fact, we prove that a Lagrangian critical point of the functional for the Coulomb (or Energy distance) kernel is equal to the target everywhere except on singular sets with empty interior. In addition, singular enough measures cannot be critical points.
Paper Structure (21 sections, 45 theorems, 134 equations)

This paper contains 21 sections, 45 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Polyak-Lojasiewicz Inequality and Exponential Convergence
  3. Regularity and Polyak-Lojasiewicz Inequality for the Energy Functional
  4. Exponential Convergence for Globally Regular Data
  5. Well-Posedness in Hölder Spaces
  6. On the Euclidean Space
  7. On a Compact Riemannian Manifold
  8. Critical Points for the Wasserstein Flow
  9. Critical Points and Lagrangian Critical Points for MMD Wasserstein Gradient Flows
  10. Characterization of Lagrangian Critical points for MMD Wasserstein Gradient Flows
  11. No Local Minima in the Wasserstein Geometry
  12. Heat Diffusion Perturbation
  13. Estimates of Mass Transfers in the Diffusion Process
  14. Proof of Theorem \ref{['NoLocMin']}
  15. Dimension of Measure and Critical Points
  16. ...and 6 more sections

Key Result

Proposition 2.2

Let $\mathcal{F}$ be a proper lower semicontinuous functional and $\mu_t$ be an absolutely continuous curve with tangent velocity $v_t$. We suppose $\mathcal{F} \circ \mu$ is approximately differentiable in time almost everywhere and that for all $t$ the set $\partial \mathcal{F}(\mu_t)$ of vector f

Theorems & Definitions (89)

  • Definition 2.1: Polyak-Lojasiewicz condition
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Proposition 2.7: Local Polyak-Lojasiewicz inequality
  • proof
  • Proposition 2.8: Stability of the Polyak-Lojasiewicz condition
  • proof : Sketch of proof
  • ...and 79 more