Intrinsic Sparsity of Kantorovich Solutions

TL;DR

This paper investigates the structure of solutions to the discrete Kantorovich optimal transport problem, revealing intrinsic sparsity properties related to the greatest common divisor of the number of points in the source and target sets.

Contribution

It establishes that when the source and target point sets differ in size, optimal solutions exhibit a sparsity pattern where each point connects to a limited number of points in the other set, depending on their sizes.

Findings

Existence of solutions with limited point connections proportional to the gcd of set sizes.

When m ≠ n, solutions are sparse with each point connecting to at most n/gcd(m,n) points.

The structure generalizes the bijective case when m = n, where solutions are fully matched.

Abstract

Let $X,Y$ be two finite sets of points having $\#X = m$ and $\#Y = n$ points with $\mu = (1/m) \sum_{i=1}^{m} \delta_{x_i}$ and $\nu = (1/n) \sum_{j=1}^{n} \delta_{y_j}$ being the associated uniform probability measures. A result of Birkhoff implies that if $m = n$, then the Kantorovich problem has a solution which also solves the Monge problem: optimal transport can be realized with a bijection $\pi: X \rightarrow Y$. This is impossible when $m \neq n$. We observe that when $m \neq n$, there exists a solution of the Kantorovich problem such that the mass of each point in $X$ is moved to at most $n/\gcd(m,n)$ different points in $Y$ and that, conversely, each point in $Y$ receives mass from at most $m/\gcd(m,n)$ points in $X$.