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Quantitative Stability in Discrete Optimal Transport

William Ford

TL;DR

This work tackles quantitative stability in discrete optimal transport by analyzing stability of transport plans and dual potentials under perturbations. It develops a dual- and graph-theoretic framework for the fully discrete problem, and employs Kantorovich functionals with glueing arguments to extend strong convexity-based stability to broader classes of measures, including John domains. It provides a detailed study of perturbations of support positions, showing plan stability via glue-compositions along trajectories that preserve uniqueness, and it establishes new dual uniqueness criteria for continuous OT when supports are Lipschitz-path connected and may be lower-dimensional. The results bridge linear programming and OT theory, yielding insights relevant to numerical discretizations and data-driven perturbations, and outline open problems for stability and uniqueness in more general settings.

Abstract

This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative stability of optimal transport plans with respect to Wasserstein distance on the product space. Chapter 2: Extention of strong convexity inequalities for the Kantorovich functional to a larger class of source measures, using glueing arguments recently used for the quantitative stability of optimal transport maps. Chapters 3/4: A qualitative description of the behaviour of the fully discrete transport problem under perturbation of the support positions, as well as quantitative stability under uniqueness assumptions. Chapter 5: Extention of known uniqueness criteria for the dual transport problem. We show that when one marginal measure has Lipschitz-path connected support and the other has bounded support, the values of dual optimisers are unique up to a constant for a large family of costs, including $p$-costs for all $p>1$.

Quantitative Stability in Discrete Optimal Transport

TL;DR

This work tackles quantitative stability in discrete optimal transport by analyzing stability of transport plans and dual potentials under perturbations. It develops a dual- and graph-theoretic framework for the fully discrete problem, and employs Kantorovich functionals with glueing arguments to extend strong convexity-based stability to broader classes of measures, including John domains. It provides a detailed study of perturbations of support positions, showing plan stability via glue-compositions along trajectories that preserve uniqueness, and it establishes new dual uniqueness criteria for continuous OT when supports are Lipschitz-path connected and may be lower-dimensional. The results bridge linear programming and OT theory, yielding insights relevant to numerical discretizations and data-driven perturbations, and outline open problems for stability and uniqueness in more general settings.

Abstract

This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative stability of optimal transport plans with respect to Wasserstein distance on the product space. Chapter 2: Extention of strong convexity inequalities for the Kantorovich functional to a larger class of source measures, using glueing arguments recently used for the quantitative stability of optimal transport maps. Chapters 3/4: A qualitative description of the behaviour of the fully discrete transport problem under perturbation of the support positions, as well as quantitative stability under uniqueness assumptions. Chapter 5: Extention of known uniqueness criteria for the dual transport problem. We show that when one marginal measure has Lipschitz-path connected support and the other has bounded support, the values of dual optimisers are unique up to a constant for a large family of costs, including -costs for all .
Paper Structure (19 sections, 32 theorems, 189 equations, 10 figures)

This paper contains 19 sections, 32 theorems, 189 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Background on optimal transport
  4. Motivation for quantitative stability
  5. Stability of transport plans
  6. Stability with one measure fixed absolutely continuous
  7. The Kantorovich functional
  8. Glueing strong convexity inequalities
  9. The discrete transport problem
  10. Primal and dual polyhedra
  11. The sets of optimal plans and potentials
  12. Perturbation of support positions
  13. Discussion on how to use linear programming connections
  14. Structure of primal optimisers under deformations
  15. Uniqueness of continuous Kantorovich potentials
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}$ and $\mathcal{Y}$ be compact metric spaces, let $\rho_n \in \mathcal{P}(\mathcal{X})$ and $\mu_n \in \mathcal{P}(\mathcal{Y})$ be sequences of probability measures converging weakly (in duality with continuous bounded functions) to limits $\rho_n \rightharpoonup \rho$ and $\mu_n \r

Figures (10)

  • Figure 1: The optimal map $T_{\rho \to \mu_\theta}$. Taken from letrouit2025lectures.
  • Figure 2: Uniqueness vs non-uniqueness of the minimisers of $\langle \hat{\gamma} |C \rangle$ depending on the direction of $-C$.
  • Figure 3: Laguerre cell decompositions of the source (left) and target (middle) domains induced by an optimal dual vector, and corresponding graph $G_\Gamma$ (right).
  • Figure 4: Optimal plans between $\rho_\varepsilon$ and $\mu$.
  • Figure 5: Measures $\rho_X$, $\mu_{Y_0}$, $\mu_{Y(t)}$ and $\mu_{Y_1}$ and couplings between them (left); and two potential glue-compositions of $\gamma_0$ and $\pi_t$ (middle and right).
  • ...and 5 more figures

Theorems & Definitions (81)

  • Theorem 1.1
  • Example 1.2
  • Proposition 1.3
  • proof
  • Lemma 1.4
  • proof
  • Proposition 1.5: Optimal plans cannot be Lipschitz
  • proof
  • Remark 1.6
  • Proposition 2.1: Characterisation of the Subdifferential
  • ...and 71 more