An Interpolating Distance between Optimal Transport and Fisher-Rao

TL;DR

This paper introduces a new metric that interpolates between optimal transport and Fisher-Rao, enabling the comparison of measures with different masses, with theoretical proofs and an application to image interpolation.

Contribution

It defines a new convex metric generalizing optimal transport and Fisher-Rao, proves existence of geodesics, and demonstrates practical application in image interpolation.

Findings

The metric interpolates between Wasserstein and Fisher-Rao.

Existence of geodesics in the new metric space is proven.

Application to image interpolation shows practical utility.

Abstract

This paper defines a new transport metric over the space of non-negative measures. This metric interpolates between the quadratic Wasserstein and the Fisher-Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher-Rao metric, and is averaged with the transportation term. This gives rise to a convex variational problem defining our metric. Our first contribution is a proof of the existence of geodesics (i.e. solutions to this variational problem). We then show that (generalized) optimal transport and Fisher-Rao metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Diracs are made of translating mixtures of Diracs. Lastly, we propose a numerical scheme making use of first order proximal splitting methods and we show an application of this new distance to image interpolation.