Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative Uniqueness of Kantorovich Potentials

William Ford

Abstract

This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal measure's support is rectifiably connected (path-connected by rectifiable paths), the optimal dual potentials are unique up to a constant. This represents the first uniqueness result applicable even when both marginal measures are concentrated on lower-dimensional subsets of the ambient space, and also applies in cases where optimal potentials are nowhere differentiable on the supports of the marginals. On the quantitative side, we control the diameter of the set of optimal dual potentials by the Hausdorff distance between the support of one of the marginal measures and a regular connected set. In this way, we quantify the extent to which optimisers are almost unique when the support of one marginal measure is almost connected. This is a consequence of a novel characterisation of the set of optimal dual potentials as the intersection of an explicit family of half-spaces.

Quantitative Uniqueness of Kantorovich Potentials

Abstract

This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal measure's support is rectifiably connected (path-connected by rectifiable paths), the optimal dual potentials are unique up to a constant. This represents the first uniqueness result applicable even when both marginal measures are concentrated on lower-dimensional subsets of the ambient space, and also applies in cases where optimal potentials are nowhere differentiable on the supports of the marginals. On the quantitative side, we control the diameter of the set of optimal dual potentials by the Hausdorff distance between the support of one of the marginal measures and a regular connected set. In this way, we quantify the extent to which optimisers are almost unique when the support of one marginal measure is almost connected. This is a consequence of a novel characterisation of the set of optimal dual potentials as the intersection of an explicit family of half-spaces.

Paper Structure

This paper contains 29 sections, 16 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions to qualitative uniqueness
  3. Contributions to quantitative uniqueness
  4. Characterisation of the set of Kantorovich potentials
  5. Related works
  6. Open problems
  7. Structure of the paper
  8. Kantorovich potentials
  9. Uniqueness in which space?
  10. $c$-concavity, choosing good representatives
  11. Duality, the compatibility condition
  12. Equivalence of pairwise and single-potential uniqueness
  13. Equivalence between quadratic and bilinear costs
  14. Qualitative uniqueness
  15. Proof of Theorem \ref{['thrm: uniqueness rect connected bounded']}: Qualitative uniqueness with $\mathcal{C}^1$ cost
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $\rho$ and $\mu$ be probability measures on $\mathbb{R}^d$ such that $\mathop{\mathrm{spt}}\nolimits \rho$ is rectifiably connected and $\mathop{\mathrm{spt}}\nolimits \mu$ is bounded. Suppose that $c \in \mathcal{C}^1(\mathbb{R}^d \times \mathbb{R}^d)$ and that there exist real-valued upper sem Then optimisers of the dual problem intro: c dual problem exist and are unique up to a constant $\r

Figures (3)

  • Figure 1: Interval of admissibility for $\phi(x_{i+1})- \phi(x_i)$.
  • Figure 2: The optimal transport plan between $\rho$ (middle) and $\mu$ (left and right).
  • Figure 3: The path \ref{['eq: path connecting c0 boundary']} connecting points on the chart of a $\mathcal{C}^0$ boundary.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.2
  • proof
  • Proposition 3.2
  • ...and 26 more