A Higher-Dimensional Homologically Persistent Skeleton

Sara Kališnik Verovšek, Vitaliy Kurlin, and Davorin Lešnik

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Submission date: 06. Oct. 2017
Pages: 41
published in: Advances in applied mathematics, 102 (2019), p. 113-142 
DOI number (of the published article): 10.1016/j.aam.2018.07.004
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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 given data points split into well-separated 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 long time 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 minimal 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.