Takatsu - Geometry of sliced-disintegrated Monge--Kantorovich metrics

Abstract

The Monge—Kantorovich transport problem is a variational problem on the space of probability measures over a complete separable metric space. This provides the so-called Monge—Kantorovich metric on the space of probability measures and the metric is applied in the fields of geometric analysism, PDE’s and applied mathematics. Since the Monge—Kantorovich metric is expensive to compute, there is great interest in alternative metrics on spaces of probability measures. In this talk, I explain two different two-parameter families of metrics derived from a slice-wise/disintegrated optimal transport problem. One family contains sliced and max-sliced Wasserstein metrics, which are used in applied mathematics. This talk is based on joint work with Jun KITAGAWA (Michigan State University).

Notation

$P(X):=\{\text{prob meas. on }X\}$

Theorem: $1\le p<\infty ,1\le q,\le \text{inft}$. Define a metric $M{K}_{p,q}$ on $P_p({\mathbb{R}}^n):=\{\mu \in P({\mathbb{R}}^{n:}{\int }_{{\mathbb{R}}^n}\}$ such that:

1. complete seperable metric
2. geodesic iff. eiterh n=1 or p=1

Next, define $M{K}_{p,q}^{\sigma }$ as a metric on $P_{p,q}^{\sigma }(\mathbb{R}\times {\mathbb{S}}^{n-1}))$ s.t.

1. complete
2. separable if $q<\infty$
3. geodesic

Optimal transport problem

This is the problem asking what the best way is to transport a finite mass of material from one plae to another