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On $ \varepsilon$ approximations of persistence diagrams

Authors: Jonathan Jaquette and Miroslav Kramár
Journal: Math. Comp. 86 (2017), 1887-1912
MSC (2010): Primary 55-04, 55N99
Published electronically: October 26, 2016
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Abstract: Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function $ f \colon X \to \mathbb{R}$, where $ X$ is a CW-complex. In the special case $ X = [0,1]^N$, $ N \in \mathbb{N}$, we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.

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Additional Information

Jonathan Jaquette
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghusen Road, Piscataway, New Jersey 08854-8019

Miroslav Kramár
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghusen Road, Piscataway, New Jersey 08854-8019
Address at time of publication: Advanced Institute for Material Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577 Japan

Received by editor(s): December 4, 2014
Received by editor(s) in revised form: September 15, 2015, and December 27, 2015
Published electronically: October 26, 2016
Additional Notes: The first author’s research was funded in part by AFOSR Grant FA9550-09-1-0148 and NSF Grant DMS-0915019.
The second author’s research was funded in part by NSF Grants DMS-1125174 and DMS-0835621.
Article copyright: © Copyright 2016 American Mathematical Society

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