Vector-Valued Optimal Mass Transport
This work provides a theoretical foundation and computational framework for optimal transport on vector-valued data, benefiting applications in image processing, radar, and network problems.
The paper introduces a framework for optimal transport of vector-valued distributions, where mass can flow between vector entries as well as across space. The resulting Wasserstein-type metrics are computable via convex optimization and generalize classical optimal transport to vector-valued settings.
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.