Topological Data Analysis with ϵ -net Induced Lazy Witness Complex
Paper i proceeding, 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.

Författare

Naheed Anjum Arafat

Universiti Kebangsaan Singapura (NUS)

Debabrota Basu

Chalmers, Data- och informationsteknik, Data Science

Stéphane Bressan

Universiti Kebangsaan Singapura (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

Ämneskategorier

Datavetenskap (datalogi)

DOI

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

Mer information

Senast uppdaterat

2020-01-10