A directed persistent homology theory for dissimilarity functions

Méndez, David and Sánchez-García, Rubén J. (2023) A directed persistent homology theory for dissimilarity functions. Journal of Applied and Computational Topology, 7 (4), 771-813. (doi:10.1007/s41468-023-00124-x).

Record type: Article

Abstract

We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of our approach. Our directed persistent homology theory is motivated by homology with semiring coefficients: by explicitly removing additive inverses, we are able to detect directed cycles algebraically.

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Accepted/In Press date: 4 May 2023

e-pub ahead of print date: 2 June 2023

Published date: 2 June 2023

Additional Information: Funding Information: This work was supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1 The first author was partially supported by Ministerio de Economía y Competitividad (Spain) Grants Nos. MTM2016-79661-P and MTM2016-78647-P and by Ministerio de Ciencia, Innovación y Universidades (Spain) Grant No. PGC2018-095448-B-I00.

Keywords: directed cycles, directed simplicial complexes, dissimilarity functions, persistent homology

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