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MiS Preprint

A Higher-Dimensional Homologically Persistent Skeleton

Sara Kališnik Verovšek, Vitaliy Kurlin and Davorin Lešnik


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.

Oct 6, 2017
Oct 6, 2017

Related publications

2019 Repository Open Access
Sara Kališnik Verovšek, Vitaliy Kurlin and Davorin Lešnik

A higher-dimensional homologically persistent skeleton

In: Advances in applied mathematics, 102 (2019), pp. 113-142