Fréchet Means for Distributions of Persistence Diagrams

Katharine Turner; Yuriy Mileyko; Sayan Mukherjee; John Harer

doi:10.1007/s00454-014-9604-7

Fréchet Means for Distributions of Persistence Diagrams

Katharine Turner^*, Yuriy Mileyko, Sayan Mukherjee, John Harer

^*Corresponding author for this work

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

148 Citations (Scopus)

Abstract

Given a distribution ρ on persistence diagrams and observations (Formula presented.) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X₁,...,X_n. If the underlying measure ρ is a combination of Dirac masses (Formula presented.) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

Original language	English
Pages (from-to)	44-70
Number of pages	27
Journal	Discrete and Computational Geometry
Volume	52
Issue number	1
DOIs	https://doi.org/10.1007/s00454-014-9604-7
Publication status	Published - Jul 2014
Externally published	Yes

Access to Document

10.1007/s00454-014-9604-7

Cite this

@article{43009d0f15e0452e9b8909c0df15da25,

title = "Fr{\'e}chet Means for Distributions of Persistence Diagrams",

abstract = "Given a distribution ρ on persistence diagrams and observations (Formula presented.) we introduce an algorithm in this paper that estimates a Fr{\'e}chet mean from the set of diagrams X1,...,Xn. If the underlying measure ρ is a combination of Dirac masses (Formula presented.) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fr{\'e}chet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.",

keywords = "Alexandrov space, Fr{\'e}chet mean, Persistence diagram, Persistent homology, Topological data analysis",

author = "Katharine Turner and Yuriy Mileyko and Sayan Mukherjee and John Harer",

year = "2014",

month = jul,

doi = "10.1007/s00454-014-9604-7",

language = "English",

volume = "52",

pages = "44--70",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer",

number = "1",

}

TY - JOUR

T1 - Fréchet Means for Distributions of Persistence Diagrams

AU - Turner, Katharine

AU - Mileyko, Yuriy

AU - Mukherjee, Sayan

AU - Harer, John

PY - 2014/7

Y1 - 2014/7

N2 - Given a distribution ρ on persistence diagrams and observations (Formula presented.) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,...,Xn. If the underlying measure ρ is a combination of Dirac masses (Formula presented.) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

AB - Given a distribution ρ on persistence diagrams and observations (Formula presented.) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,...,Xn. If the underlying measure ρ is a combination of Dirac masses (Formula presented.) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

KW - Alexandrov space

KW - Fréchet mean

KW - Persistence diagram

KW - Persistent homology

KW - Topological data analysis

UR - http://www.scopus.com/inward/record.url?scp=84904427571&partnerID=8YFLogxK

U2 - 10.1007/s00454-014-9604-7

DO - 10.1007/s00454-014-9604-7

M3 - Article

AN - SCOPUS:84904427571

SN - 0179-5376

VL - 52

SP - 44

EP - 70

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -

Fréchet Means for Distributions of Persistence Diagrams

Abstract

Access to Document

Other files and links

Fingerprint

Cite this