A statistical approach to persistent homology

By Peter Bubenik, Peter T. Kim. 2006

122 citations recorded by the source corpus.

Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying distribution.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6a703287aa12f944336831d5c051dfaa5fd76c66

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS (2007, background): “de Silva and Carlsson (2004) and Bubenik and Kim (2007), or statistical inference in shape analysis, cf. e.g. Munk et al. (2007), which often requires a generalization of the underlying space to a quotient of a manifold, cf. e.g. Kendall et al. (1999) or even more general structures such as semimetr”
The topology of probability distributions on manifolds (2013, background): “icial complexes generated from stochastic processes [1,2,5,13,28,29, 30,32,46,47] and non-asymptotic bounds on the convergence or consistency of topological summaries as the number of points increase [11,17,18,20,23,44,45]. The central idea in these papers has been to study statistical properties of”
Topological complexity of smooth random functions (2011, background): “A concise, yet very clear introduction to the topics that concern us can be found in [17, 41], while [48, 84] are good examples of a thorough coverage of homology theory.”
Maximally Persistent Cycles in Random Geometric Complexes (2015, methodology): “stence homology of either the Cech or the Rips complex. The rst study of persistent homology in degree k1 for a random setting was for npoints chosen uniformly i.i.d. on a circle by Bubenik and Kim [15]. In this setting, they used the theory of order statistics to describe the limiting distribution ”
Statistical topology via Morse theory, persistence and nonparametric estimation (2009, methodology): “To date the most comprehensive parametric statistical approach is contained in [4].”
Barcodes: The persistent topology of data (2007, background): “This (or a discretized version thereof) is one commonly investigated setting [1, 7], as is filtration by means of curvature data [5].”
Persistent homology for random fields and complexes (2010, background): “A concise, yet very clear introduction to the topics that concern us can be found in [8, 23], while [25, 36] are good examples of a thorough coverage of homology theory.”