Rigorous cubical approximation and persistent homology of continuous functions

Pawel Dlotko 1 Thomas Wanner 2
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.
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Article dans une revue
Computers and Mathematics with Applications, Elsevier, 2018
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Soumis le : mardi 13 février 2018 - 10:26:26
Dernière modification le : mercredi 2 mai 2018 - 01:10:30

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Pawel Dlotko, Thomas Wanner. Rigorous cubical approximation and persistent homology of continuous functions. Computers and Mathematics with Applications, Elsevier, 2018. 〈hal-01706695〉

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