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Sharp local sparsity of regularized optimal transport

Albert González-Sanz, Rishabh S. Gvalani, Lukas Koch

Abstract

In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, $\mathrm{supp}(π_\varepsilon)$, shrinks to the support of the original optimal transport problem as $\varepsilon \to 0$. The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports $\mathrm{supp}(π_\varepsilon(\cdot \mid x))$ of the conditional measures, $π_\varepsilon(\cdot \mid x)$, behave like balls of radius $\varepsilon^\frac 1 {d(p-1)+2}$. This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (González-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.

Sharp local sparsity of regularized optimal transport

Abstract

In recent years, the use of entropy-regularized optimal transport with -type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, , shrinks to the support of the original optimal transport problem as . The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports of the conditional measures, , behave like balls of radius . This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (González-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.

Paper Structure

This paper contains 12 sections, 11 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Notation
  4. Main results
  5. Preliminaries
  6. Regularized optimal transport
  7. Convex analysis
  8. Proofs of results
  9. Proof of
  10. Proof of
  11. Explicit solution for self-transport with Lebesgue marginals
  12. Reynolds' transport theorem

Key Result

Theorem 3.1

For any smooth domain $K_0\Subset \Omega_0$, there exists $R_0=R_0(K_0)$ and $\varepsilon_0=\varepsilon_0(K_0)$ such that for every $x\in K_0$ and $\varepsilon\in (0,\varepsilon_0]$, If $\Omega_1$ is convex, the result holds for any $\varepsilon\in (0,1]$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (20)

  • Theorem 3.1: Interior sharp sparsity
  • Corollary 3.2: Interior strong convexity
  • Corollary 3.3: Rates of ROT map
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4: Interior regularity, GvalaniKoch2026
  • proof
  • ...and 10 more