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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Vydáno v:Foundations of computational mathematics Ročník 18; číslo 1; s. 1 - 44
Hlavní autoři: Chizat, Lénaïc, Peyré, Gabriel, Schmitzer, Bernhard, Vialard, François-Xavier
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.02.2018
Springer Nature B.V
Témata:
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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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-016-9331-y