Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold

Martins Bruveris; Peter Michor

doi:10.1002/mana.201600523

Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold

Martins Bruveris (Corresponding author), Peter Michor

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of a form extending the Fisher-Rao metric, depending on two smooth functions of the total volume.
Here we determine the geodesics and the curvature of this metric and study geodesic and metric
completeness.

Original language	English
Pages (from-to)	511-523
Number of pages	13
Journal	Mathematische Nachrichten
Volume	292
Issue number	3
Early online date	12 Nov 2018
DOIs	https://doi.org/10.1002/mana.201600523
Publication status	Published - Mar 2019

Austrian Fields of Science 2012

101032 Functional analysis
101003 Applied geometry
101006 Differential geometry

Keywords

Fisher-Rao metric
information geometry
space of densities
surfaces of revolution
58D15
Fisher–Rao metric
Primary: 58B20

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10.1002/mana.201600523

Cite this

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abstract = "It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of a form extending the Fisher-Rao metric, depending on two smooth functions of the total volume.Here we determine the geodesics and the curvature of this metric and study geodesic and metriccompleteness.",

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AB - It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of a form extending the Fisher-Rao metric, depending on two smooth functions of the total volume.Here we determine the geodesics and the curvature of this metric and study geodesic and metriccompleteness.

KW - Fisher-Rao metric

KW - information geometry

KW - space of densities

KW - surfaces of revolution

KW - 58D15

KW - Fisher–Rao metric

KW - Primary: 58B20

UR - https://arxiv.org/pdf/1607.04550.pdf

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SN - 0025-584X

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JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

IS - 3

ER -

Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold

Abstract

Austrian Fields of Science 2012

Keywords

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