Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of optimal transport maps on Riemannian manifolds

Jun Kitagawa, Cyril Letrouit, Quentin Mérigot

TL;DR

The paper delivers quantitative stability bounds for optimal transport maps and Kantorovich potentials on smooth Riemannian manifolds with the quadratic cost $c(x,y)=\tfrac{1}{2}\mathrm{dist}(x,y)^2$, under perturbations of the target measure and with a fixed source $\rho$ supported on a John domain. It combines entropy-regularized OT with gluing techniques (via a Boman chain condition) and a Crofton-type integral-geometric framework to extend stability results beyond Euclidean spaces. The main results show $L^2(\rho)$-stability of Kantorovich potentials: $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)} \le C\,W_1(\mu,\nu)^{1/2}$, and, when the boundary of the domain has finite $(d-1)$-dimensional Hausdorff measure, a bound on the mean-squared transport map distance: $\int_M \mathrm{dist}(T_\mu(x),T_\nu(x))^2 \,d\rho(x) \le C\,W_1(\mu,\nu)^{1/6}$. These results extend prior Euclidean stability analyses to Riemannian manifolds and even suggest applicability to broader metric spaces, highlighting the power of entropic regularization and geometric-gluing methods in non-Euclidean settings.

Abstract

We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $ρ$ under variations of the target measure $μ$, when the cost function is the squared Riemannian distance on a Riemannian manifold. Previous works were restricted to subsets of Euclidean spaces, or made specific assumptions either on the manifold, or on the regularity of the transport maps. Our proof techniques combine entropy-regularized optimal transport with spectral and integral-geometric techniques. As some of the arguments do not rely on the Riemannian structure, our work also paves the way towards understanding stability of optimal transport in more general geometric spaces.

Stability of optimal transport maps on Riemannian manifolds

TL;DR

The paper delivers quantitative stability bounds for optimal transport maps and Kantorovich potentials on smooth Riemannian manifolds with the quadratic cost , under perturbations of the target measure and with a fixed source supported on a John domain. It combines entropy-regularized OT with gluing techniques (via a Boman chain condition) and a Crofton-type integral-geometric framework to extend stability results beyond Euclidean spaces. The main results show -stability of Kantorovich potentials: , and, when the boundary of the domain has finite -dimensional Hausdorff measure, a bound on the mean-squared transport map distance: . These results extend prior Euclidean stability analyses to Riemannian manifolds and even suggest applicability to broader metric spaces, highlighting the power of entropic regularization and geometric-gluing methods in non-Euclidean settings.

Abstract

We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure under variations of the target measure , when the cost function is the squared Riemannian distance on a Riemannian manifold. Previous works were restricted to subsets of Euclidean spaces, or made specific assumptions either on the manifold, or on the regularity of the transport maps. Our proof techniques combine entropy-regularized optimal transport with spectral and integral-geometric techniques. As some of the arguments do not rely on the Riemannian structure, our work also paves the way towards understanding stability of optimal transport in more general geometric spaces.

Paper Structure

This paper contains 21 sections, 16 theorems, 131 equations.

Table of Contents

  1. Introduction and main results
  2. Motivation and setting
  3. Optimal transport
  4. The optimal transport map in Riemannian manifolds
  5. Main result
  6. Related works
  7. Proof strategy and organization of the paper.
  8. Notation.
  9. Acknowledgments.
  10. Strong concavity of the Kantorovich regularized functionals
  11. The Kantorovich regularized functional and its first two derivatives
  12. The functional $\mathcal{I}_\rho^\varepsilon$
  13. Strong concavity of Kantorovich's regularized functional
  14. Stability of Kantorovich potentials
  15. Existence and uniqueness of Kantorovich potentials and optimal transport maps
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $M$ be a smooth and connected Riemannian manifold, endowed with the quadratic cost $c(x,y)=\frac{1}{2} {\rm dist}(x,y)^2$ where ${\rm dist}$ denotes the Riemannian distance. Let $\mathcal{X}\subset M$ be a John domain and $\mathcal{Y}\subset M$ be compact, and let $\rho$ be a probability measure where $\phi_\mu$ denotes the (unique) Kantorovich potential from $\rho$ to $\mu$, and similarly $\p

Theorems & Definitions (36)

  • Definition 1.1: John domains in metric spaces
  • Theorem 1.2: Main result
  • Corollary 1.3: Closed manifolds
  • Remark 2.1: Limit as $\varepsilon \to 0$
  • Definition 2.2: Gradient and Hessian
  • Lemma 2.3: Gradient of $\mathcal{K}_\rho^\varepsilon$
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 26 more