Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform large-scale $\varepsilon$-regularity for entropic optimal transport

Rishabh S. Gvalani, Lukas Koch

TL;DR

The paper develops a large-scale ε-regularity theory for costs that perturb quadratic OT, framing minimisers as local quasi-minimisers and controlling long trajectories to obtain a $C^{2,α}$ Morrey-Campanato-type regularity up to the quasi-minimal scale. It then specializes the framework to entropic OT, establishing that the entropic cost $OT_ε(λ,μ)$ satisfies the required assumptions and yields a uniform ε-regularity bound that propagates a gradient-type $BMO$-like control down to the entropic scale. The main contributions are a general perturbative regularity theory for OT costs and its concrete realization for entropic OT, including a rigorous link between large-scale regularity and small-ε smoothing effects. This provides a rigorous understanding of how entropy regularization influences the fine structure of OT minimisers and offers a foundation for stable numerical and analytical treatment of entropic OT problems, including connections to the Schrödinger bridge problem.

Abstract

We study the regularity properties of the minimisers of entropic optimal transport providing a natural analogue of the $\varepsilon$-regularity theory of quadratic optimal transport in the entropic setting. More precisely, we show that if the minimiser of the entropic problem satisfies a gradient BMO-type estimate at some scale, the same estimate holds all the way down to the natural length-scale associated to the entropic regularisation. Our result follows from a more general $\varepsilon$-regularity theory for optimal transport costs which can be viewed as perturbations of quadratic optimal transport. We consider such a perturbed cost and require that, under a certain class of admissible affine rescalings, the minimiser remains a local quasi-minimiser of the quadratic problem (in an appropriate sense) and that the cost of "long trajectories" of minimisers (and their rescalings) is small. Under these assumptions, we show that the minimiser satisfies an appropriate $C^{2,α}$ Morrey$\unicode{x2013}$Campanato-type estimate which is valid up to the scale of quasi-minimality.

Uniform large-scale $\varepsilon$-regularity for entropic optimal transport

TL;DR

The paper develops a large-scale ε-regularity theory for costs that perturb quadratic OT, framing minimisers as local quasi-minimisers and controlling long trajectories to obtain a Morrey-Campanato-type regularity up to the quasi-minimal scale. It then specializes the framework to entropic OT, establishing that the entropic cost satisfies the required assumptions and yields a uniform ε-regularity bound that propagates a gradient-type -like control down to the entropic scale. The main contributions are a general perturbative regularity theory for OT costs and its concrete realization for entropic OT, including a rigorous link between large-scale regularity and small-ε smoothing effects. This provides a rigorous understanding of how entropy regularization influences the fine structure of OT minimisers and offers a foundation for stable numerical and analytical treatment of entropic OT problems, including connections to the Schrödinger bridge problem.

Abstract

We study the regularity properties of the minimisers of entropic optimal transport providing a natural analogue of the -regularity theory of quadratic optimal transport in the entropic setting. More precisely, we show that if the minimiser of the entropic problem satisfies a gradient BMO-type estimate at some scale, the same estimate holds all the way down to the natural length-scale associated to the entropic regularisation. Our result follows from a more general -regularity theory for optimal transport costs which can be viewed as perturbations of quadratic optimal transport. We consider such a perturbed cost and require that, under a certain class of admissible affine rescalings, the minimiser remains a local quasi-minimiser of the quadratic problem (in an appropriate sense) and that the cost of "long trajectories" of minimisers (and their rescalings) is small. Under these assumptions, we show that the minimiser satisfies an appropriate MorreyCampanato-type estimate which is valid up to the scale of quasi-minimality.
Paper Structure (6 sections, 7 theorems, 138 equations)

This paper contains 6 sections, 7 theorems, 138 equations.

Table of Contents

  1. Proof of Theorem \ref{['thm:main']}
  2. Controlling the mass of long trajectories
  3. One-step improvement
  4. Campanato iteration
  5. Application to entropic optimal transport and the proof of Theorem \ref{['thm:entropicmain']}
  6. Local quasiminimality of entropic optimal transport

Key Result

Theorem 1

Suppose $\lambda$ and $\mu$ have $C^{0,\alpha}$-densities for some $\alpha\in (0,1)$ and $\pi$ is a minimiser of the entropic optimal transport problem eq:eot. Define Then, there exists some $\varepsilon_1>0$ such that if for some $R_0>0$ the densities of $\lambda$ and $\mu$ are bounded away from zero on $B_{R_0}$ and then for any $r\leq R_0$ such that $\left(\frac{r}{R_0}\right)\gg \left(\frac{

Theorems & Definitions (13)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Lemma 4
  • Proposition 5
  • proof : Proof of Lemma \ref{['lem:soft']}
  • Theorem 6
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Proposition 7
  • ...and 3 more