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On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals

Pedram Emami, Brendan Pass

Abstract

A natural and important question in multi-marginal optimal transport is whether the \emph{Monge ansatz} is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is $m$-empirical (that is, uniformly supported on $m$ points). By direct computation, we provide an example showing that the ansatz \emph{can fail} when the underlying dimension $d$ is $2$, number of marginals $N$ to be matched is $3$ and the size $m$ of their supports is $3$. As a consequence, the set of $m$-empirical measures is not barycentrically convex when $N \geq 3$, $d \geq 2$ and $m \geq3$. It is a well known consequence of the Birkhoff-von Neumann Theorem that the Monge ansatz holds for $N=2$, standard techniques show it holds when $d=1$, and we provide a simple proof here that \emph{it holds whenever $m=2$}. Therefore, the $N$, $d$ and $m$ in our counterexample are as small as possible.

On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals

Abstract

A natural and important question in multi-marginal optimal transport is whether the \emph{Monge ansatz} is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is -empirical (that is, uniformly supported on points). By direct computation, we provide an example showing that the ansatz \emph{can fail} when the underlying dimension is , number of marginals to be matched is and the size of their supports is . As a consequence, the set of -empirical measures is not barycentrically convex when , and . It is a well known consequence of the Birkhoff-von Neumann Theorem that the Monge ansatz holds for , standard techniques show it holds when , and we provide a simple proof here that \emph{it holds whenever }. Therefore, the , and in our counterexample are as small as possible.
Paper Structure (4 sections, 2 theorems, 7 equations, 1 figure)

This paper contains 4 sections, 2 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A counterexample to Monge solutions for marginals with three-point support in $\mathbb{R}^2$
  3. Existence of Monge solutions for uniform empirical measures with two-point supports in $\mathbb{R}^d$
  4. Discussion

Key Result

Theorem 1

Let $\mu_i=\frac{1}{2}$δ_x_i^1+δ_x_i^2$\in E_2$ be $2$-empirical measures on $\mathbb{R}^d$ for $i=1,...,N$. Then problem eqn: OT problem with cost eqn: multi-marginal quadratic cost has a Monge solution.

Figures (1)

  • Figure 1: The distribution of the errors of minimal Monge costs for $500$ occurrences of multi-marginal optimal transport problems with $3$-empirical marginals in $\mathbb{R}^3$ which do not have Monge solutions.

Theorems & Definitions (5)

  • Example 1
  • Theorem 1
  • Proof
  • Corollary 1
  • Proof