  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.
