Statistical topology via Morse theory, persistence and nonparametric estimation

By Peter Bubenik, Gunnar E. Carlsson, Peter T. Kim, Zhi-Ming Luo. 2009

56 citations recorded by the source corpus.

In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance between the estimated persistent homology and the true persistent homology. The connection to statistics comes from the fact that when viewed as a nonparametric regression problem, the bottleneck distance is bounded by the sup-norm loss. Consequently, a sharp asymptotic minimax bound is determined under the sup-norm risk over Holder classes of functions for the nonparametric regression problem on manifolds. This provides good convergence properties for the persistent homology estimator in terms of the expected bottleneck distance.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46a9457fac0f05766db736789ec8e1afb75731f7

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Probability measures on the space of persistence diagrams (2011, background): “Most of the related work on stochastic aspects of topological summaries can be subdivided into two categories: the study of random abstract simplicial complexes generated from stochastic processes [5, 6, 7, 8, 9, 10] and non-asymptotic bounds on the convergence or consistency of topological summarie”
On the topology of random complexes built over stationary point processes. (2012, background): “beginning to appreciate the fact that stochasticity underlies their data as a consequence of sampling, and are beginning to build statistical models to allow parameter estimation and inference (e.g., [8, 9, 13, 30, 40]). The results of this paper show that small changes in model structure (such as t”
Topological Consistency via Kernel Estimation (2014, methodology): “Asymptotic and non-asymptotic analysis of consistency and convergence of topological summaries as the number of observations increase has been examined for a variety of geometric objects using a variety of statistical and probabilistic tools [1, 2, 3, 4, 5, 8, 9, 12, 16, 30, 31, 36, 37].”
Statistical topological data analysis using persistence landscapes (2012, background): “We consider our data X as a random variable on some underlying probability space (Ω,F , P ), and so the corresponding persistence landscape λ(X) is a Borel random variable with values in the separable Banach space Lp(S).”
Persistent homology for random fields and complexes (2010, background): “In a complementary but related direction [8, 9] apply persistence techniques to the nonparametric study of functions on a given manifold.”