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An Interpolating Distance between Optimal Transport and Fisher-Rao

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.
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Contributor : Gabriel Peyré <>
Submitted on : Tuesday, February 9, 2016 - 10:09:36 PM
Last modification on : Wednesday, February 19, 2020 - 8:59:43 AM

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  • HAL Id : hal-01271984, version 1
  • ARXIV : 1506.06430



Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, François-Xavier Vialard. An Interpolating Distance between Optimal Transport and Fisher-Rao. Foundations of Computational Mathematics, Springer Verlag, 2010. ⟨hal-01271984⟩



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