Computing Topological Persistence for Simplicial Maps

Abstract

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under Z 2 coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis.First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps.Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses-two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally.Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.

Polynomial-Sized Topological Approximations Using the Permutahedron

“…The approximation tower of [28] consists of a filtration of simplicial complexes, whose k-skeleton has size only n(1/ε) O(λk) , where λ is the doubling dimension of the metric. Since then, several alternative techniques have been explored for Rips [12] andČech complexes [5,22], all arriving at the same complexity bound.…”

“…Any simplicial tower can be written in the form K 1 → · · · → K M where each K i differs from K i−1 in an elementary fashion [12,21]. Either each K i contains one more simplex than K i−1 (which is called an elementary inclusion), or a pair of vertices of K i−1 collapse into one in K i (elementary contraction).…”

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

“…The cohomology algorithm has been reported to work better in practice than the standard homology algorithm [9] but this advantage seems to fade away when optimizations are employed to the homology algorithms [2]. An elegant description of the cohomology algorithm, using the notion of annotations [5], has been introduced in [11] and used to design more general algorithms for maintaining cohomology groups under simplicial maps.…”

Big Data Reduction Methods: A Survey

“…The concept of selective collapse algorithm is introduced to reduce the computational complexity of processing persistent homology. The proposed framework keeps computation traces of collapsing the complex using persistence algorithm proposed in [42]. In addition, the strong collapses are represented by forest that facilitates in easy processing of persistence across all nodes of the complex.…”

“…The solutions like selective collapse algorithms represent datasets in the form of forests, and the nodes are collapsed in a way to improve the speed of persistent homology and maintain strong collapse. The persistent homology tools for reducing and analyzing big data still need to be further explored in the future research [18,42]. Similarly, the automated extraction of events and their representation in network structures is an emerging research area.…”