Continuation of point clouds via persistence diagrams

Marcio Gameiro; Yasuaki Hiraoka; Ippei Obayashi

doi:https://doi.org/10.1016/j.physd.2015.11.011

Continuation of point clouds via persistence diagrams

Marcio Gameiro, Yasuaki Hiraoka, Ippei Obayashi

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P, its persistence diagram D, and a target persistence diagram D', we gradually move from D to D', by successively computing intermediate point clouds until we finally find a point cloud P' having D' as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.

Original language	American English
Pages (from-to)	118-132
Number of pages	15
Journal	Physica D: Nonlinear Phenomena
Volume	334
DOIs	https://doi.org/10.1016/j.physd.2015.11.011
State	Published - Nov 1 2016
Externally published	Yes

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

Keywords

Continuation
Persistence diagram
Persistent homology
Point cloud

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title = "Continuation of point clouds via persistence diagrams",

abstract = "In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P, its persistence diagram D, and a target persistence diagram D', we gradually move from D to D', by successively computing intermediate point clouds until we finally find a point cloud P' having D' as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.",

keywords = "Continuation, Persistence diagram, Persistent homology, Point cloud",

author = "Marcio Gameiro and Yasuaki Hiraoka and Ippei Obayashi",

note = "Funding Information: The authors would like to thank Shouhei Honda for useful discussions. M.G. was partially supported by FAPESP grants 2013/07460-7 and 2010/00875-9 , and by CNPq grant 306453/2009-6 , Brazil. Y.H. and I.O. were partially supported by JSPS Grant-in-Aid ( 24684007 , 26610042 ). Publisher Copyright: {\textcopyright} 2015 Elsevier B.V.",

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doi = "https://doi.org/10.1016/j.physd.2015.11.011",

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AU - Gameiro, Marcio

AU - Hiraoka, Yasuaki

AU - Obayashi, Ippei

N1 - Funding Information: The authors would like to thank Shouhei Honda for useful discussions. M.G. was partially supported by FAPESP grants 2013/07460-7 and 2010/00875-9 , and by CNPq grant 306453/2009-6 , Brazil. Y.H. and I.O. were partially supported by JSPS Grant-in-Aid ( 24684007 , 26610042 ). Publisher Copyright: © 2015 Elsevier B.V.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P, its persistence diagram D, and a target persistence diagram D', we gradually move from D to D', by successively computing intermediate point clouds until we finally find a point cloud P' having D' as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.

AB - In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P, its persistence diagram D, and a target persistence diagram D', we gradually move from D to D', by successively computing intermediate point clouds until we finally find a point cloud P' having D' as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.

KW - Continuation

KW - Persistence diagram

KW - Persistent homology

KW - Point cloud

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VL - 334

SP - 118

EP - 132

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -

Continuation of point clouds via persistence diagrams

Abstract

ASJC Scopus subject areas

Keywords

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