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...

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Veröffentlicht in:	Foundations of computational mathematics Jg. 18; H. 1; S. 1 - 44
Hauptverfasser:	Chizat, Lénaïc, Peyré, Gabriel, Schmitzer, Bernhard, Vialard, François-Xavier
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.02.2018 Springer Nature B.V
Schlagworte:	Applications of Mathematics Computer Science Continuity equation Economics Geodesy Interpolation Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Wasserstein Fisher–Rao metric Positive Radon measures metric Unbalanced optimal transport 49-XX
ISSN:	1615-3375, 1615-3383
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Abstract	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.
AbstractList	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.
Author	Chizat, Lénaïc Vialard, François-Xavier Peyré, Gabriel Schmitzer, Bernhard
Author_xml	– sequence: 1 givenname: Lénaïc surname: Chizat fullname: Chizat, Lénaïc organization: Project team Mokaplan, CEREMADE, CNRS, INRIA, Université Paris-Dauphine – sequence: 2 givenname: Gabriel surname: Peyré fullname: Peyré, Gabriel organization: Project team Mokaplan, CEREMADE, CNRS, INRIA, Université Paris-Dauphine – sequence: 3 givenname: Bernhard surname: Schmitzer fullname: Schmitzer, Bernhard organization: Project team Mokaplan, CEREMADE, CNRS, INRIA, Université Paris-Dauphine – sequence: 4 givenname: François-Xavier surname: Vialard fullname: Vialard, François-Xavier email: vialard@ceremade.dauphine.fr organization: Project team Mokaplan, CEREMADE, CNRS, INRIA, Université Paris-Dauphine
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References_xml	– reference: L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008. – reference: BegMFMillerMITrouvéAYounesLComputing large deformation metric mappings via geodesic flows of diffeomorphismsInternational journal of computer vision200561213915710.1023/B:VISI.0000043755.93987.aa – reference: BenamouJ-DNumerical resolution of an "unbalanced" mass transport problemESAIM. Mathematical Modelling and Numerical Analysis20033705851868202086710.1051/m2an:20030581037.65063 – reference: HaninLGKantorovich-Rubinstein norm and its application in the theory of Lipschitz spacesProceedings of the American Mathematical Society19921152345352109734410.1090/S0002-9939-1992-1097344-50768.46012 – reference: C. Villani. Topics in optimal transportation. Number 58. American Mathematical Soc., 2003. – reference: K. Guittet. Extended Kantorovich norms: a tool for optimization. Technical report, Tech. Rep. 4402, INRIA, 2002. – reference: RaoCInformation and accuracy attainable in the estimation of statistical parametersBulletin of the Calcutta Mathematical Society19453738191157480063.06420 – reference: AyNJostJLêHVSchwachhöferLInformation geometry and sufficient statisticsProbability Theory and Related Fields20151621327364335004710.1007/s00440-014-0574-81321.62008 – reference: KantorovichLOn the transfer of masses (in russian)Doklady Akademii Nauk1942372227229 – reference: HakerSZhuLTannenbaumAAngenentSOptimal mass transport for registration and warpingInternational Journal of computer vision200460322524010.1023/B:VISI.0000036836.66311.97 – reference: FigalliAThe optimal partial transport problemArchive for rational mechanics and analysis20101952533560259228710.1007/s00205-008-0212-71245.49059 – reference: TrouvéAYounesLMetamorphoses through lie group actionFoundations of Computational Mathematics200552173198214941510.1007/s10208-004-0128-z1099.68116 – reference: PapadakisNPeyréGOudetEOptimal transport with proximal splittingSIAM Journal on Imaging Sciences201471212238315878510.1137/1309200581295.90047 – reference: BauerMBruverisMMichorPWUniqueness of the Fisher-Rao metric on the space of smooth densitiesBull. 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Snippet	This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao...
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SubjectTerms	Applications of Mathematics Computer Science Continuity equation Economics Geodesy Interpolation Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis
Title	An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics
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