Open Access
February 2017 Topological consistency via kernel estimation
Omer Bobrowski, Sayan Mukherjee, Jonathan E. Taylor
Bernoulli 23(1): 288-328 (February 2017). DOI: 10.3150/15-BEJ744


We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: (1) inferring the homology structure of manifolds from noisy observations, (2) inferring the persistent homology (a multi-scale extension of homology) of either density or regression functions. We prove consistency for both of these problems. In addition to the theoretical results, we demonstrate these methods on simulated data for binary regression and clustering applications.


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Omer Bobrowski. Sayan Mukherjee. Jonathan E. Taylor. "Topological consistency via kernel estimation." Bernoulli 23 (1) 288 - 328, February 2017.


Received: 1 December 2014; Revised: 1 April 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 06673479
MathSciNet: MR3556774
Digital Object Identifier: 10.3150/15-BEJ744

Keywords: clustering , homology , kernel density estimation , topological data analysis

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
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