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Optimising the Topological Information of the \(A_\infty \)-Persistence Groups

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Abstract

Persistent homology typically studies the evolution of homology groups \(H_p(X)\) (with coefficients in a field) along a filtration of topological spaces. \(A_\infty \)-persistence extends this theory by analysing the evolution of subspaces such as \(V :=\text {Ker}\,{\Delta _n}_{| H_p(X)} \subseteq H_p(X)\), where \(\{\Delta _m\}_{m\ge 1}\) denotes a structure of \(A_\infty \)-coalgebra on \(H_*(X)\). In this paper we illustrate how \(A_\infty \)-persistence can be useful beyond persistent homology by discussing the topological meaning of V, which is the most basic form of \(A_\infty \)-persistence group. In addition, we explore how to choose \(A_\infty \)-coalgebras along a filtration to make the \(A_\infty \)-persistence groups carry more faithful information.

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Acknowledgements

I would like to thank Prof. Aniceto Murillo and Prof. Jim Stasheff for their valuable feedback on this work.

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  1. Institut de Robòtica i Informàtica Industrial, CSIC-UPC, Llorens i Artigas 4-6, 08028, Barcelona, Spain

    Francisco Belchí

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Correspondence to Francisco Belchí.

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This work has been supported by the Spanish MINECO Grants MTM2010-18089 and MTM2013-41762-P, by the Junta de Andalucía Grant FQM-213 and by the UK’s EPSRC Grant Joining the dots: from data to insight, EP/N014189/1.

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Belchí, F. Optimising the Topological Information of the \(A_\infty \)-Persistence Groups. Discrete Comput Geom 62, 29–54 (2019). https://doi.org/10.1007/s00454-019-00094-x

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