Fréchet Means for Distributions of Persistence Diagrams
Given a distribution ρ on persistence diagrams and observations X 1 , … , X n ∼ i i d ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X 1 , … , X n . If the underlying measure ρ is a combination of Dirac masses ρ = 1 m ∑ i = 1 m δ Z i then we prove th...
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| Published in | Discrete & computational geometry Vol. 52; no. 1; pp. 44 - 70 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.07.2014
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0179-5376 1432-0444 |
| DOI | 10.1007/s00454-014-9604-7 |
Cover
| Abstract | Given a distribution
ρ
on persistence diagrams and observations
X
1
,
…
,
X
n
∼
i
i
d
ρ
we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams
X
1
,
…
,
X
n
. If the underlying measure
ρ
is a combination of Dirac masses
ρ
=
1
m
∑
i
=
1
m
δ
Z
i
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. |
|---|---|
| AbstractList | Given a distribution
ρ
on persistence diagrams and observations
X
1
,
…
,
X
n
∼
i
i
d
ρ
we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams
X
1
,
…
,
X
n
. If the underlying measure
ρ
is a combination of Dirac masses
ρ
=
1
m
∑
i
=
1
m
δ
Z
i
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. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Given a distribution ... on persistence diagrams and observations ... we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams ... If the underlying measure ... is a combination of Dirac masses ... 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 ... We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.[PUBLICATION ABSTRACT] (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).Given a distribution ... on persistence diagrams and observations ... we introduce an algorithm in this paper that estimates a Frechet mean from the set of diagrams ... If the underlying measure ... is a combination of Dirac masses ... then we prove the algorithm converges to a local minimum and a law of large numbers result for a Frechet mean computed by the algorithm given observations drawn iid from ... We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields. |
| Author | Mukherjee, Sayan Harer, John Mileyko, Yuriy Turner, Katharine |
| Author_xml | – sequence: 1 givenname: Katharine surname: Turner fullname: Turner, Katharine email: kate@math.uchicago.edu organization: Department of Mathematics, University of Chicago – sequence: 2 givenname: Yuriy surname: Mileyko fullname: Mileyko, Yuriy organization: Department of Mathematics, University of Hawaii at Manoa – sequence: 3 givenname: Sayan surname: Mukherjee fullname: Mukherjee, Sayan organization: Departments of Statistical Science, Computer Science, and Mathematics, Institute for Genome Sciences & Policy, Duke University – sequence: 4 givenname: John surname: Harer fullname: Harer, John organization: Departments of Mathematics, Computer Science and Electrical and Computer Engineering, Center for Systems Biology, Duke University |
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| Cites_doi | 10.1214/10-IMSCOLL609 10.1093/acprof:oso/9780198506263.001.0001 10.1007/978-3-662-12494-9 10.4007/annals.2009.169.903 10.1007/s00454-008-9053-2 10.4310/SDG.2006.v11.n1.a6 10.1090/conm/516/10167 10.1090/S1061-0022-06-00916-2 10.1016/j.disc.2008.02.037 10.1137/0105003 10.1088/0266-5611/27/12/124007 10.1137/1.9781611973099.107 10.1070/RM1992v047n02ABEH000877 10.1007/s00454-009-9144-8 10.1007/s10208-010-9060-6 10.1214/aoap/1015345393 |
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| Keywords | Persistence diagram Persistent homology Fréchet mean Topological data analysis Alexandrov space |
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| References_xml | – reference: NiyogiPSmaleSWeinbergerSFinding the homology of submanifolds with high confidence from random samplesDiscrete Comput. Geom.20083941944110.1007/s00454-008-9053-21148.680482383768 – reference: Lunagómez, S., Mukherjee, S., Wolpert, R.L.: Geometric representations of hypergraphs for prior specification and posterior sampling (2009). http://arxiv.org/abs/0912.3648 – reference: Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised a topological view of unsupervised learning from noisy data. Manuscript (2008) – reference: Kahle, M.: Random geometric complexes (2011). http://arxiv.org/abs/0910.1649 – reference: PenroseMDRandom Geometric Graphs2003New YorkOxford University Press10.1093/acprof:oso/9780198506263.001.00011029.60007 – reference: Adler, R.J., Bobrowski, O., Borman, M.S., Subag, E., Weinberger, S.: Persistent homology for random fields and complexes. In: Berger, J.O., Tony Cai, T., Johnstone, I.M. (eds.) Borrowing Strength: Theory Powering Applications—A Festschrift for Lawrence D. Brown, vol. 6. Institute of Mathematical Statistics, Beachwood (2010) – reference: GromovMGerstenSMHyperbolic groupsEssays in Group Theory. Mathematical Sciences Research Institute Publications1987New YorkSpringer75263 – reference: LottJVillaniCRicci curvature for metric-measure spaces via optimal transportAnn. Math.200916990399110.4007/annals.2009.169.9031178.530382480619 – reference: BuragoYGromovMPerel’manGA.D. Alexandrov spaces with curvature bounded belowRuss. Math. Surv.199247215810.1070/RM1992v047n02ABEH0008770802.530181185284 – reference: Kahle, M., Meckes, E.: Limit theorems for Betti numbers of random simplicial complexes (2010). http://arxiv.org/abs/1009.4130v3 – reference: PetruninASemiconcave functions in Alexandrov’s geometrySurv. Differ. Geom.20071113720110.4310/SDG.2006.v11.n1.a62408266 – reference: PenroseMDYukichJECentral limit theorems for some graphs in computational geometryAnn. Appl. Probab.2001114100510411044.600161878288 – reference: SturmK-TAuscherPCoulhonTGrigoryanAProbability measures on metric spaces of nonpositive curvatureHeat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces2002ProvidenceAmerican Mathematical Society – reference: Cohen-SteinerDEdelsbrunnerHHarerJMileykoYLipschitz functions have Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}_p$$\end{document}-stable persistenceFound. Comput. Math.20101012713910.1007/s10208-010-9060-61192.550072594441 – reference: KahleMTopology of random clique complexesDiscrete Math.200930961658167110.1016/j.disc.2008.02.0371215.051632510573 – reference: EdelsbrunnerHHarerJComputational Topology: An Introduction2010ProvidenceAmerican Mathematical Society – reference: MolchanovITheory of Random Sets2005LondonSpringer1109.60001 – reference: LytchakAOpen map theorem for metric spacesSt. Petersbg. Math. J.200617347749110.1090/S1061-0022-06-00916-21152.530332167848 – reference: ChazalFCohen-SteinerDLieutierAA sampling theory for compact sets in Euclidean spaceDiscrete Comput. Geom.20094146147910.1007/s00454-009-9144-81165.680612486371 – reference: MileykoYMukherjeeSHarerJProbability measures on the space of persistence diagramsInverse Probab.2012271212400710.1088/0266-5611/27/12/1240072854323 – reference: Bendich, P., Mukherjee, S., Wang B.: Local homology transfer and stratification learning. In: ACM-SIAM Symposium on Discrete Algorithms (2012) – reference: MunkresJAlgorithms for the assignment and transportation problemsJ. Soc. Ind. Appl. Math.195751323810.1137/01050030083.1530293429 – reference: OhtaSBarycenters in Alexandrov spaces with curvature bounded belowAdv. Geom.2012125715871276.530733005101 – reference: BirdsonMRHaefligerAMetric Spaces of Non-positive Curvature1999BerlinSpringer-Verlag10.1007/978-3-662-12494-9 – reference: BubenikPCarlssonGKimPTLuoZ-MVianaMAGWynnHPStatistical topology via Morse theory, persistence, and nonparametric estimationAlgebraic Methods in Statistics and Probability II. 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| Snippet | Given a distribution
ρ
on persistence diagrams and observations
X
1
,
…
,
X
n
∼
i
i
d
ρ
we introduce an algorithm in this paper that estimates a Fréchet mean... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Given a distribution ... on persistence diagrams and observations ... we introduce... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).Given a distribution ... on persistence diagrams and observations ... we introduce... |
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| SubjectTerms | Algorithms Combinatorics Computation Computational Mathematics and Numerical Analysis Computer simulation Convergence Diagrams Estimates Gaussian Geometry Law Mathematics Mathematics and Statistics Normal distribution Texts |
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| Title | Fréchet Means for Distributions of Persistence Diagrams |
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