Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Metric conditions that guarantee existence and uniqueness of Optimal Transport maps

Shucheng Li, Mattia Magnabosco, Timo Schultz

Abstract

We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the Riemannian twist condition that we call local metric twist condition, showing, under this assumption on the cost function, existence and uniqueness of optimal transport maps. Secondly, we prove the same result for cost equal to $d^2$ in a metric space $(X, d)$ satisfying a quantitative non-branching assumption, that we call locally-uniformly non-branching.

Metric conditions that guarantee existence and uniqueness of Optimal Transport maps

Abstract

We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the Riemannian twist condition that we call local metric twist condition, showing, under this assumption on the cost function, existence and uniqueness of optimal transport maps. Secondly, we prove the same result for cost equal to in a metric space satisfying a quantitative non-branching assumption, that we call locally-uniformly non-branching.

Paper Structure

This paper contains 3 sections, 12 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main1']}
  3. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

Let $(\mathsf{X},\mathsf d, \mathfrak m)$ be a proper metric measure space with $\mathfrak m$ doubling on $\mathsf{X}$ and let $c:\mathsf{X}\times \mathsf{X}\to [0,\infty]$ be any lower-semi-continuous cost function that satisfies the local metric twist condition. Then, for every pair of measures $\

Figures (3)

  • Figure 1: Definition of [LMTC] (Local Metric Twist Condition): as $x' \in F_{s|r_2}(\bar{x} | y,z)$, the sum of the distances between points connected by a red line is smaller than the sum of the distances between points connected by a blue line, for every choice of $y' \in B_{r_2}(y)$ and $z' \in B_{r_2}(z)$.
  • Figure 2: Definition of $C_{\gamma,k}$
  • Figure 3: Picture of the strategy to prove Theorem \ref{['thm:NonBranch+ DenselyScattered -> !OTM']}.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1: see Figure \ref{['img: def:LMTC']}
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 21 more