Computing Topological Persistence for Simplicial Maps

Algorithms for persistent homology are well-studied where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under Z2 coefficients for a (monotone) sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. 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.