Topological Data Analysis with ϵ -net Induced Lazy Witness Complex
Paper in proceedings, 2019

Topological data analysis computes and analyses topological features of the point clouds by constructing and studying a simplicial representation of the underlying topological structure. The enthusiasm that followed the initial successes of topological data analysis was curbed by the computational cost of constructing such simplicial representations. The lazy witness complex is a computationally feasible approximation of the underlying topological structure of a point cloud. It is built in reference to a subset of points, called landmarks, rather than considering all the points as in the Čech and Vietoris-Rips complexes. The choice and the number of landmarks dictate the effectiveness and efficiency of the approximation. We adopt the notion of ϵ -cover to define ϵ -net. We prove that ϵ -net, as a choice of landmarks, is an ϵ -approximate representation of the point cloud and the induced lazy witness complex is a 3-approximation of the induced Vietoris-Rips complex. Furthermore, we propose three algorithms to construct ϵ -net landmarks. We establish the relationship of these algorithms with the existing landmark selection algorithms. We empirically validate our theoretical claims. We empirically and comparatively evaluate the effectiveness, efficiency, and stability of the proposed algorithms on synthetic and real datasets.

Author

Naheed Anjum Arafat

National University of Singapore (NUS)

Debabrota Basu

Chalmers, Computer Science and Engineering (Chalmers), Data Science

Stéphane Bressan

National University of Singapore (NUS)

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

03029743 (ISSN) 16113349 (eISSN)

Vol. 11707 LNCS 376-392

Subject Categories

Computer Science

DOI

10.1007/978-3-030-27618-8_28

More information

Latest update

1/10/2020