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A regularized transportation cost stemming from entropic approximation

Camilla Brizzi, Luigi De Pascale, Anna Kausamo

TL;DR

The paper develops a comprehensive theory for entropic regularization of optimal transport with lower semicontinuous costs under natural summability bounds. It proves uniform bounds and regularity for the entropic transforms, establishes convergence of the Sinkhorn algorithm, and analyzes the limit as the regularization parameter $\\varepsilon$ vanishes, showing convergence to a Kantorovich-type dual pair for a regularized cost $\\tilde c$ and to OT minimizers with this cost in the primal. The results extend to the multi-marginal setting and encompass costs like the Coulomb interaction, providing a robust framework for understanding variational limits beyond convergence to the original unregularized problem. The findings offer a principled way to interpret entropic regularizations as approximations that remain well-posed under broad conditions and reveal the precise form of the limiting problem.

Abstract

We study the entropic regularizations of optimal transport problems under suitable summability assumptions on the point-wise transport cost. These summability assumptions already appear in the literature. However, we show that the weakest compactness conditions that can be derived are already enough to obtain the convergence of the regularized functionals. This approach allows us to characterize the variational limit of the regularization even when it does not converge to the original problem. The results apply also to problems with more than two marginals.

A regularized transportation cost stemming from entropic approximation

TL;DR

The paper develops a comprehensive theory for entropic regularization of optimal transport with lower semicontinuous costs under natural summability bounds. It proves uniform bounds and regularity for the entropic transforms, establishes convergence of the Sinkhorn algorithm, and analyzes the limit as the regularization parameter vanishes, showing convergence to a Kantorovich-type dual pair for a regularized cost and to OT minimizers with this cost in the primal. The results extend to the multi-marginal setting and encompass costs like the Coulomb interaction, providing a robust framework for understanding variational limits beyond convergence to the original unregularized problem. The findings offer a principled way to interpret entropic regularizations as approximations that remain well-posed under broad conditions and reveal the precise form of the limiting problem.

Abstract

We study the entropic regularizations of optimal transport problems under suitable summability assumptions on the point-wise transport cost. These summability assumptions already appear in the literature. However, we show that the weakest compactness conditions that can be derived are already enough to obtain the convergence of the regularized functionals. This approach allows us to characterize the variational limit of the regularization even when it does not converge to the original problem. The results apply also to problems with more than two marginals.
Paper Structure (13 sections, 37 theorems, 143 equations)

This paper contains 13 sections, 37 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. uniform bounds on the $(c,\varepsilon)$-transforms and the existence of maximizers
  3. Uniform continuity of the $(c,\varepsilon)$-transform and convergence of the Sinkhorn's algorithm
  4. Regularity of the $(c,\varepsilon)$-transform
  5. Sinkhorn algorithm
  6. The limit as $\varepsilon \to 0$
  7. The dual problem
  8. The primal problem
  9. The multi-marginal case
  10. Properties of the dual functional $\mathcal{D}_\varepsilon$.
  11. A semicontinuity theorem
  12. Existence of a Vitali covering for separable metric spaces
  13. $\lambda$-essential relaxation

Key Result

Proposition 2.1

For every $u\in L^{exp,\varepsilon}_\mu$ and for every $v\in L^{exp,\varepsilon}_\nu$, $\mathcal{D}_\varepsilon(u,\cdot)$ and $\mathcal{D}_\varepsilon(\cdot, v)$ are uniquely maximized respectively by and by which are called the entropic transforms or $(c,\varepsilon)$-transforms of $u$ and $v$.

Theorems & Definitions (77)

  • Example 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 67 more