Optimising the topological information of the $A_\infty$-persistence groups

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 2017Embargo end date: 01 Jun 2017 Spain Publisher:Springer NatureJournal:Discret. Comput. Geom., volume 62, pages 29-54

Authors: Francisco Belchí Guillamón;

doi: 10.48550/arxiv.1706.06019

arXiv: 1706.06019

handle: 10261/202286

Optimising the topological information of the $A_\infty$-persistence groups

- Summary
- Subjects
- Metrics

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 $V := \text{Ker}\, {Δ_n}_{| H_p(X)} \subseteq H_p(X)$, where $\{Δ_m\}_{m\geq1}$ 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.

26 pages, 3 figures

Country

Spain

Related Organizations

Spanish National Research Council
Spain

Keywords

Computational Geometry (cs.CG), FOS: Computer and information sciences, Applied homological algebra and category theory in algebraic topology, Computer Vision and Pattern Recognition (cs.CV), Topological data analysis, Computer Science - Computer Vision and Pattern Recognition, topological data analysis, Spectral sequences, 16E45, 18G55, 55S30, 57M25, 57Q45, 18G40, 55P62, 55U99, FOS: Mathematics, Algebraic Topology (math.AT), zigzag persistence, Persistent homology, Mathematics - Algebraic Topology, Massey products, A∞-(co)algebras, A∞-persistence, Rational homotopy theory, knot theory, Differential graded algebras and applications (associative algebraic aspects), Knot theory, Persistent homology and applications, topological data analysis, persistent homology, $A_\infty $-persistence, Spectral sequences, hypercohomology, rational homotopy theory, spectral sequences, $A_\infty $-(co)algebras, Computer Science - Computational Geometry, Zigzag persistence

Impact byBIP!

	selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).	0
	popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.	Average
	influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).	Average
	impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.	Average