An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics
This paper defines a new transport metric over the space of nonnegative 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 the new 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 Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. 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.
KeywordsUnbalanced optimal transport Wasserstein \(L^2\) metric Fisher–Rao metric Positive Radon measures
Mathematics Subject Classification49-XX
The work of Bernhard Schmitzer has been supported by the Fondation Sciences Mathématiques de Paris. The work of Gabriel Peyré has been supported by the European Research Council (ERC project SIGMA-Vision). We would like to thank Yann Brenier and Jean-David Benamou for stimulating discussions.
- 1.L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008.Google Scholar
- 11.P. Combettes and J.-C. Pesquet. Proximal splitting methods in signal processing. In Fixed-point algorithms for inverse problems in science and engineering, pages 185–212. Springer, 2011.Google Scholar
- 16.K. Guittet. Extended Kantorovich norms: a tool for optimization. Technical report, Tech. Rep. 4402, INRIA, 2002.Google Scholar
- 20.L. Kantorovich. On the transfer of masses (in russian). Doklady Akademii Nauk, 37(2):227–229, 1942.Google Scholar
- 25.B. Piccoli and F. Rossi. On properties of the Generalized Wasserstein distance. arXiv:1304.7014, 2013.
- 31.C. Villani. Topics in optimal transportation. Number 58. American Mathematical Soc., 2003.Google Scholar