Vector-Valued Optimal Mass Transport

TL;DR

This paper introduces a novel framework for vector-valued optimal mass transport, enabling mass transfer between vector components and across space, with applications in image processing, radar, and network resource management.

Contribution

It develops a new theory of optimal transport for vector-valued distributions on graphs and continuous spaces, generalizing Wasserstein metrics for these settings.

Findings

Distance computation via convex optimization

Generalization of Wasserstein metrics to vector-valued distributions

Applicable to multi-color images, radar, and network resource problems

Abstract

We introduce the problem of transporting vector-valued distributions. In this, a salient feature is that mass may flow between vectorial entries as well as across space (discrete or continuous). The theory relies on a first step taken to define an appropriate notion of optimal transport on a graph. The corresponding distance between distributions is readily computable via convex optimization and provides a suitable generalization of Wasserstein-type metrics. Building on this, we define Wasserstein-type metrics on vector-valued distributions supported on continuous spaces as well as graphs. Motivation for developing vector-valued mass transport is provided by applications such as multi-color image processing, polarimetric radar, as well as network problems where resources may be vectorial.