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Quantitative stability in optimal transport for general power costs

Octave Mischler, Dario Trevisan

TL;DR

The paper advances the quantitative stability theory for optimal transport with a broad class of power costs $\mathcal{W}_p$, $p>1$, by deriving explicit $L^2(\rho)$-type bounds that quantify how Kantorovich potentials and OT maps react to perturbations in the target measure. It develops a unified approach that handles quadratic costs and generalizes to $p\ge 2$ and $1<p<2$ via new convexity and interpolation tools, notably a log-concavity-based Prékopa–Leindler extension and fractional Sobolev techniques. The results require a log-concave source $\rho$ with bounded convex support and compact target supports (or moment bounds), with explicit exponents depending on $p$, providing practical insights for numerical stability and semi-discrete analyses. Overall, the work broadens the stability framework beyond quadratic costs and offers precise, implementable rates that can inform algorithm design and error estimates in OT-based applications.

Abstract

We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant for both theoretical and practical applications. Our results apply to a wide range of costs, including all Wasserstein distances with power cost exponent strictly larger than $1$ and leverage mostly assumptions on the source measure, such as log-concavity and bounded support. Our work provides a significant step forward in the understanding of stability of optimal transport problems, as previous results where mostly limited to the case of the quadratic cost.

Quantitative stability in optimal transport for general power costs

TL;DR

The paper advances the quantitative stability theory for optimal transport with a broad class of power costs , , by deriving explicit -type bounds that quantify how Kantorovich potentials and OT maps react to perturbations in the target measure. It develops a unified approach that handles quadratic costs and generalizes to and via new convexity and interpolation tools, notably a log-concavity-based Prékopa–Leindler extension and fractional Sobolev techniques. The results require a log-concave source with bounded convex support and compact target supports (or moment bounds), with explicit exponents depending on , providing practical insights for numerical stability and semi-discrete analyses. Overall, the work broadens the stability framework beyond quadratic costs and offers precise, implementable rates that can inform algorithm design and error estimates in OT-based applications.

Abstract

We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant for both theoretical and practical applications. Our results apply to a wide range of costs, including all Wasserstein distances with power cost exponent strictly larger than and leverage mostly assumptions on the source measure, such as log-concavity and bounded support. Our work provides a significant step forward in the understanding of stability of optimal transport problems, as previous results where mostly limited to the case of the quadratic cost.
Paper Structure (20 sections, 22 theorems, 165 equations)

This paper contains 20 sections, 22 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related literature
  4. Comments on the proof technique
  5. Extensions and open problems
  6. Structure of the paper
  7. Notation and basic facts
  8. General notation
  9. Optimal transport and its entropic regularization
  10. Dual formulations and Kantorovich functional
  11. Semi-discrete problem
  12. Fractional Sobolev spaces
  13. Quantitative stability of potentials
  14. The quadratic case
  15. The case $p\geq2$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d \ge1$, $\lambda$ be a $\log$-concave probability measure on ${\mathbb R}^d$ with bounded support, $\rho$ be a probability measure on ${\mathbb R}^d$ absolutely continuous with respect to $\lambda$, with density bounded from above and below by strictly positive constants, let $\mathcal{Y} \sub with and where $(\phi_\mu, \psi_\mu)$ denotes the Kantorovich potentials for $\mathcal{W}_p(\rho,

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4: Ambrosio and Gigli
  • Theorem 1.5: Mérigot, Delalande and Chazal
  • Lemma 2.1
  • Lemma 2.2
  • Example 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 37 more