Stratified Monge-Kantorovich optimal transport problems

TL;DR

This paper explores generalized Monge-Kantorovich optimal transport problems with relaxed absolute continuity, analyzing solutions concentrated on multiple maps and measures supported on different dimensional subsets, extending classical results.

Contribution

It introduces new results on the uniqueness and structure of solutions for multi-layer and singular measure optimal transport problems, broadening the classical theory.

Findings

Solutions for K≥2 are unique and concentrated on multiple maps.

The paper characterizes solutions supported on multi-dimensional subsets.

Extends classical optimal transport to measures with mixed dimensional support.

Abstract

In this paper, we investigate Monge-Kantorovich problems for which the absolute continuity of marginals is relaxed. For $X,Y\subseteq\mathbb{R}^{n+1}$ let $(X,\mathcal{B}_X,\mu)$ and $(Y,\mathcal{B}_Y,\nu)$ be two Borel probability spaces, $c:X\times Y\to\mathbb{R}$ be a cost function, and consider the problem \begin{align*}\tag{MKP}\label{MKPEQ} \inf\left\{\int_{X\times Y} c(x,y)\,d\lambda\ :\ \lambda \in\Pi(\mu,\nu) \right\}. \end{align*} Inspired by the seminal paper \cite{GANGBOMCCANN2} with applications in shape recognition problem, we first consider \eqref{MKPEQ} for the cost $c(x,y)=h(x-y)$ with $h$ strictly convex defined on the multi-layers target space \begin{align*} X=\overline{X}\times\{\overline{x}\},\quad\text{and}\quad Y=\bigcup_{k=1}^K \left(\overline{Y}_{k}\times \{\overline{y}_k\}\right), \end{align*} where $\overline{X}, \overline{Y}_{k}\subseteq \mathbb{R}^{n}$ for $k\in \{1,\ldots,K\},$ $\overline{x}\in \mathbb{R}$, and $\{\overline{y}_1,..., \overline{y}_K\}\subseteq \mathbb{R}$. Here, we assume that $\mu|_\overline{X}\ll\mathcal{L}^n$ (the Lebesgue measure on $\mathbb{R}^n$), but $\mu$ is singular w.r.t. $\mathcal{L}^{n+1}$. When $K=1$, this translates to the standard \eqref{MKPEQ} for which the unique solution is concentrated on a map. We show that for $K\geq 2,$ the solution is still unique but it concentrates on the graph of several maps. Next, we study \eqref{MKPEQ} for a closed subset $X\subseteq \mathbb{R}^{n+1}$ and its $n$-dimensional submanifold $X_0$ with the first marginal of the form \begin{align*} \int_X f(x)\,d\mu(x)=\int_X f(x)\alpha(x)\,d\mathcal{L}^{n+1}(x)+\int_{X_0} f(x_0)\,d S(x_0),\ \ \forall f\in C_b(X). \end{align*} Here, $S$ is a measure on $X_0$ such that $S\ll \mathcal{L}^{n}$ on each coordinate chart of $X_0$. This can be seen as a two-layers problem as the measure $\mu$ charges both $n$- and $n+1$-dimensional subsets.