Topological Data Analysis with $$\epsilon $$-net Induced Lazy Witness Complex

Abstract

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 theČech and Vietoris-Rips complexes. The choice and the number of landmarks dictate the effectiveness and efficiency of the approximation. We adopt the notion of -cover to define -net. We prove that -net, as a choice of landmarks, is an -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 -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.

Adaptive Approximation of Persistent Homology

“…Approaches used in these techniques include repeatedly calculating the persistent homology on subsamples (Solomon et al, 2022;Chazal et al, 2015a,b) as well as using witness complexes (de Silva and Carlsson, 2004;Guibas and Oudot, 2008;Arafat et al, 2019), scaled grids (Choudhary et al, 2021;Kerber and Sharathkumar, 2013), or net-trees (Sheehy, 2013). Our approach falls into this category.…”

Topological Distances Between Brain Networks

“…The upper bound |Z|/|L| ≤ 20 was suggested for 2D surface data (de Silva and Carlsson, 2004). Taking |Z|/|L| = 20 and denoting by N W , N R the sizes of the corresponding witness complex and Rips complex respectively, the worst-case scaling of the number of simplices in the witness complex is (Otter et al, 2017;Arafat et al, 2019)…”