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Unstable optimal transport maps

Cyril Letrouit

TL;DR

This work shows that optimal transport maps can be highly unstable with respect to perturbations of the marginals, even when the source density is well-behaved or uniform on bounded domains. It identifies two instability mechanisms: (i) an unbounded source density that blows up near two boundary points, which destroys any Hölder-type stability between maps and perturbed targets, and (ii) proximity to non-uniqueness in the optimal plans where instability occurs just before uniqueness is lost. The authors construct explicit densities and multi-scale configurations to prove that the map $\mu \mapsto T_\mu$ can fail to be Hölder with any exponent in the unbounded-density setting and can be limited to at most $\frac{1}{3}$-Hölder in certain uniform-density contexts, highlighting fundamental limits to quantitative stability results in optimal transport and informing numerical and statistical approaches that rely on map stability.

Abstract

The stability of optimal transport maps with respect to perturbations of the marginals is a question of interest for several reasons, ranging from the justification of the linearized optimal transport framework to numerical analysis and statistics. Under various assumptions on the source measure, it is known that optimal transport maps are stable with respect to variations of the target measure. In this note, we focus on the mechanisms that can, on the contrary, lead to instability. We identify two of them, which we illustrate through examples of absolutely continuous source measures $ρ$ in $\mathbb{R}^d$ for which optimal transport maps are less stable, or even very unstable. We first show that instability may arise from the unboundedness of the density: we exhibit a source density on the unit ball of $\mathbb{R}^d$ which blows up superpolynomially at two points of the boundary and for which optimal transport maps are highly unstable. Then we prove that even for uniform densities on bounded open sets, optimal transport maps can be rather unstable close enough to configurations where uniqueness of optimal plans is lost.

Unstable optimal transport maps

TL;DR

This work shows that optimal transport maps can be highly unstable with respect to perturbations of the marginals, even when the source density is well-behaved or uniform on bounded domains. It identifies two instability mechanisms: (i) an unbounded source density that blows up near two boundary points, which destroys any Hölder-type stability between maps and perturbed targets, and (ii) proximity to non-uniqueness in the optimal plans where instability occurs just before uniqueness is lost. The authors construct explicit densities and multi-scale configurations to prove that the map can fail to be Hölder with any exponent in the unbounded-density setting and can be limited to at most -Hölder in certain uniform-density contexts, highlighting fundamental limits to quantitative stability results in optimal transport and informing numerical and statistical approaches that rely on map stability.

Abstract

The stability of optimal transport maps with respect to perturbations of the marginals is a question of interest for several reasons, ranging from the justification of the linearized optimal transport framework to numerical analysis and statistics. Under various assumptions on the source measure, it is known that optimal transport maps are stable with respect to variations of the target measure. In this note, we focus on the mechanisms that can, on the contrary, lead to instability. We identify two of them, which we illustrate through examples of absolutely continuous source measures in for which optimal transport maps are less stable, or even very unstable. We first show that instability may arise from the unboundedness of the density: we exhibit a source density on the unit ball of which blows up superpolynomially at two points of the boundary and for which optimal transport maps are highly unstable. Then we prove that even for uniform densities on bounded open sets, optimal transport maps can be rather unstable close enough to configurations where uniqueness of optimal plans is lost.
Paper Structure (10 sections, 3 theorems, 38 equations, 4 figures)

This paper contains 10 sections, 3 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results.
  3. Previous works.
  4. Acknowledgments.
  5. Proof of Theorem \ref{['t:highlyunstable']}
  6. Proof of Theorem \ref{['t:unstableW1']}
  7. Idea of the construction.
  8. The construction
  9. Preliminary computations.
  10. End of the proof of Theorem \ref{['t:unstableW1']}

Key Result

Theorem 1.1

Let $d\geq 2$. There exists an absolutely continuous $\rho\in\mathcal{P}(B_{\mathbb{R}^d}(0,1))$ with density bounded below, such that for any ball $\mathcal{Y}=B_{\mathbb{R}^d}(0,R)$ with $R>0$, any $C,\alpha>0$ and $p\geq 1$, the inequality fails.

Figures (4)

  • Figure 1: Illustration of \ref{['e:cone']}: in purple the ball $B(A',\theta/4)$, in orange the cone in \ref{['e:cone']}, in grey dashed lines the set \ref{['e:cone']}, and in blue the boundary of the support of $\rho$.
  • Figure 2: Any transport plan between $\frac{1}{2} (\delta_A+\delta_{A'})$ and $\frac{1}{2}(\delta_B+\delta_{B'})$ is optimal
  • Figure 3: Part of the support of $\rho$, projected on the $(x_1,x_2)$-plane
  • Figure 4: The support of the measures $\rho$, $\mu$ and $\nu_i$

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1