Kalisnik, Sara, Kurlin, Vitaliy ORCID: 0000-0001-5328-5351 and Lesnik, Davorin
(2019)
A Higher-Dimensional Homologically Persistent Skeleton.
Advances in Applied Mathematics, 102.
113 - 142.
Text
Skeleton-final-AAM.pdf - Author Accepted Manuscript Download (584kB) |
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 |
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Depositing User: | Symplectic Admin |
Date Deposited: | 10 Sep 2018 07:09 |
Last Modified: | 19 Jan 2023 01:25 |
DOI: | 10.1016/j.aam.2018.07.004 |
Related URLs: | |
URI: | https://livrepository.liverpool.ac.uk/id/eprint/3025921 |