Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparsity and uniform regularity for regularised optimal transport

Rishabh S. Gvalani, Lukas Koch

TL;DR

The paper studies regularised quadratic optimal transport with subquadratic and entropic regularisation, proving ε-independent interior regularity for the transport-like map and potentials under bi-C^{1,α} regularity of the unregularised map. The authors develop a two-scale approach: large-scale ε-regularity via Campanato iteration and small-scale regularity ensuring Lipschitz and higher regularity of T_ε, verified through detailed local bounds on the support and energy comparisons. They establish convergence of T_ε and the potentials to their unregularised Kantorovich counterparts in local Hölder–Sobolev spaces and derive sharp, general bounds on the regularised transport support, yielding global bias bounds and improving previous results for quadratic regularisation. The framework unifies entropic and polynomial regularisations and provides ε-uniform estimates that enhance stability and convergence analyses for regularised OT problems with broad applicability in analysis and numerical methods.

Abstract

We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the potentials, under the assumption that the transport map solving the unregularised problem is bi-$C^{1,α}$-regular. For strictly subquadratic and entropic regularisation, the estimates improve to interior $C^1$ and $C^2$ estimates for the transport-like map and the potentials, respectively. Our estimates are uniform in the regularisation parameter. As a consequence of this, we obtain convergence of the transport-like map (resp. the potentials) to the unregularised transport map (resp. Kantorovich potentials) in $C^{0,1-}_{\mathrm{loc}}$ (resp. $C^{1,1-}_{\mathrm{loc}}$). Central to our approach are sharp local bounds on the size of the support for regularised optimal transport which we derive for a general convex, superlinear regularisation term. These bounds are of independent interest and imply global bias bounds for the regularised transport plans. Our global bounds, while not necessarily sharp, improve on the best known results in the literature for quadratic regularisation.

Sparsity and uniform regularity for regularised optimal transport

TL;DR

The paper studies regularised quadratic optimal transport with subquadratic and entropic regularisation, proving ε-independent interior regularity for the transport-like map and potentials under bi-C^{1,α} regularity of the unregularised map. The authors develop a two-scale approach: large-scale ε-regularity via Campanato iteration and small-scale regularity ensuring Lipschitz and higher regularity of T_ε, verified through detailed local bounds on the support and energy comparisons. They establish convergence of T_ε and the potentials to their unregularised Kantorovich counterparts in local Hölder–Sobolev spaces and derive sharp, general bounds on the regularised transport support, yielding global bias bounds and improving previous results for quadratic regularisation. The framework unifies entropic and polynomial regularisations and provides ε-uniform estimates that enhance stability and convergence analyses for regularised OT problems with broad applicability in analysis and numerical methods.

Abstract

We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the potentials, under the assumption that the transport map solving the unregularised problem is bi--regular. For strictly subquadratic and entropic regularisation, the estimates improve to interior and estimates for the transport-like map and the potentials, respectively. Our estimates are uniform in the regularisation parameter. As a consequence of this, we obtain convergence of the transport-like map (resp. the potentials) to the unregularised transport map (resp. Kantorovich potentials) in (resp. ). Central to our approach are sharp local bounds on the size of the support for regularised optimal transport which we derive for a general convex, superlinear regularisation term. These bounds are of independent interest and imply global bias bounds for the regularised transport plans. Our global bounds, while not necessarily sharp, improve on the best known results in the literature for quadratic regularisation.
Paper Structure (11 sections, 17 theorems, 152 equations)

This paper contains 11 sections, 17 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. A general monotonicity principle
  5. Qualitative theory of minimisers of regularised transport
  6. Transformation under affine changes of coordinates
  7. Local $L^\infty$-bounds
  8. Large-scale regularity
  9. Small-scale regularity
  10. Proof of \ref{['thm:main']} and \ref{['cor:convergence']}
  11. Regularised optimal transport

Key Result

Theorem 3

Let $p\in[1,2]$ and $h(z)=h_p(z)$. Fix $\alpha\in (0,1]$. Assume $\lambda,\mu$ are probability measures, bounded and bounded away from $0$ on their support, and with $C^{0,\alpha}$-densities. Denote $\Omega = \mathrm{supp}\;\lambda$. Let $T$ be the minimiser of $OT(\lambda,\mu)$ and $T_\varepsilon$ If $p<2$, $T_\varepsilon \in C^1(\omega)$ and the essential supremum may be replaced by a supremum.

Theorems & Definitions (33)

  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • Remark 7
  • Corollary 8
  • Proposition 9
  • proof
  • Lemma 10
  • proof
  • ...and 23 more