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Displacement smoothness of entropic optimal transport

Guillaume Carlier, Lénaïc Chizat, Maxime Laborde

TL;DR

It is proved that when the cost function is C k +1 with k ∈ N ∗ then this map is Lipschitz continuous from the L 2 -Wasserstein space to the space of C k functions, covering the multi-marginal case.

Abstract

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is $\mathcal{C}^{k+1}$ with $k\in \mathbb{N}^*$ then this map is Lipschitz continuous from the $L^2$-Wasserstein space to the space of $\mathcal{C}^k$ functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.

Displacement smoothness of entropic optimal transport

TL;DR

It is proved that when the cost function is C k +1 with k ∈ N ∗ then this map is Lipschitz continuous from the L 2 -Wasserstein space to the space of C k functions, covering the multi-marginal case.

Abstract

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is with then this map is Lipschitz continuous from the -Wasserstein space to the space of functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.
Paper Structure (19 sections, 17 theorems, 109 equations)

This paper contains 19 sections, 17 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Schrödinger Map
  3. Main result in the two-marginals case
  4. Discussion of prior work
  5. The multi-marginal case
  6. Notation and assumptions on the domain
  7. Multi-marginal Schrödinger System
  8. Existence and weak continuity of the Schrödinger map
  9. Main result : regularity of the Schrödinger map
  10. Proofs
  11. Invertibility of the differential of $T$
  12. Differentiability of $G$
  13. Proof of Thm. \ref{['thm:main']}
  14. Proof of Proposition \ref{['cor:lipschitz-sobolev-multi']}
  15. Smoothness of entropic optimal transport and Wasserstein gradient flows
  16. ...and 4 more sections

Key Result

Theorem 1.1

If $c\in \mathcal{C}^{k+1}(\mathcal{X}_1\times \mathcal{X}_2)$ for $k\in \mathbb{N}^*$, then there exists $C>0$ that only depends on $\Vert c\Vert_{\mathcal{C}^{k+1}}$ such that for all $\mathop{\mathrm{\boldsymbol{\mu}}}\nolimits,\mathop{\mathrm{\boldsymbol{\mu}}}\nolimits' \in \mathcal{P}(\mathcal

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1: Schrödinger system/potentials/map
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Lemma 3.1
  • ...and 19 more