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Register a new userComputing Topological Persistence for Simplicial Maps
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Added by
Tamal Dey
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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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 $mathbb{Z}_2$ 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.
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Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general $O(m^omega)$ time complexity are not known, where $omega< 2.37286$ is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with $m$ additions and deletions on a graph with $n$ vertices and edges, the algorithm for $0$-dimension runs in $O(mlog^2 n+mlog m)$ time and the algorithm for 1-dimension runs in $O(mlog^4 n)$ time. The algorithm for $0$-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on $2$-manifolds. The algorithm for $1$-dimension pairs a negative edge with the earliest positive edge so that a $1$-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for $0$-dimension to compute the $(p-1)$-dimensional zigzag persistence for $mathbb{R}^p$-embedded complexes in $O(mlog^2 n+mlog m+nlog n)$ time.
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We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link condition preserves homeomorphism in low dimensional complexes, and homotopy in general. But, checking the link condition involves computation in all dimensions, and hence can be costly, especially in high dimensional complexes. We define a weaker and more local condition called the p-link condition for each dimension p, and study its effect on edge contractions. We prove the following: (i) For homology groups, edges satisfying the p- and (p-1)-link conditions can be contracted without disturbing the p-dimensional homology group. (ii) For relative homology groups, the (p-1)-, and the (p-2)-link conditions suffice to guarantee that the contraction does not introduce any new class in any of the resulting relative homology groups, though some of the existing classes can be destroyed. Unfortunately, the surjection in relative homolgy groups does not guarantee that no new relative torsion is created. (iii) For torsions, edges satisfying the p-link condition alone can be contracted without creating any new relative torsion and the p-link condition cannot be avoided. The results on relative homology and relative torsion are motivated by recent results on computing optimal homologous chains, which state that such problems can be solved by linear programming if the complex has no relative torsion. Edge contractions that do not introduce new relative torsions, can safely be availed in these contexts.
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