A Higher-Dimensional Homologically Persistent Skeleton

Kalisnik, Sara, Kurlin, Vitaliy ORCID: 0000-0001-5328-5351 and Lesnik, Davorin (2019) A Higher-Dimensional Homologically Persistent Skeleton. Advances in Applied Mathematics, 102 (Januar). pp. 113-142.

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Abstract

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data — for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when data points split into good clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a large interval in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimum spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

Item Type:	Article
Uncontrolled Keywords:	math.AT, math.AT
Depositing User:	Symplectic Admin
Date Deposited:	17 Dec 2020 10:11
Last Modified:	18 Jan 2023 23:06
DOI:	10.1016/j.aam.2018.07.004
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3110570