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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On $\varepsilon$ approximations of persistence diagrams
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by Jonathan Jaquette and Miroslav Kramár PDF
Math. Comp. 86 (2017), 1887-1912 Request permission

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
  • Email: jaquette@math.rutgers.edu
  • 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
  • MR Author ID: 747856
  • Email: kramar.miroslav.e1@tohoku.ac.jp
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1887-1912
  • MSC (2010): Primary 55-04, 55N99
  • DOI: https://doi.org/10.1090/mcom/3137
  • MathSciNet review: 3626542