Zigzag Persistence via Reflections and Transpositions

Abstract

We introduce a new algorithm for computing zigzag persistence, designed in the same spirit as the standard persistence algorithm. Our algorithm reduces a single matrix, maintains an explicit set of chains encoding the persistent homology of the current zigzag, and updates it under simplex insertions and removals. The total worst-case running time matches the usual cubic bound.A noticeable difference with the standard persistence algorithm is that we do not insert or remove new simplices "at the end" of the zigzag, but rather "in the middle". To do so, we use arrow reflections and transpositions, in the same spirit as reflection functors in quiver theory. Our analysis introduces new kinds of reflections in quiver representation theory: the "injective and surjective diamonds". It also introduces the "transposition diamond" which models arrow transpositions. For each type of diamond we are able to predict the changes in the interval decomposition and associated compatible bases. Arrow transpositions have been studied previously in the context of standard persistent homology, and we extend the study to the context of zigzag persistence. For both types of transformations, we provide simple procedures to update the interval decomposition and associated compatible homology basis.

Discrete Morse Theory for Computing Zigzag Persistence

“…In this section, we describe an algorithm to compute the persistent homology of a zigzag Morse filtration as defined in Section 3. For the sake of concision, and for its favorable practical performance (see Section 6), we choose to work with the algorithm for zigzag persistence described in [32] ; our approach could be adapted for implementing algorithm [9,10]. Using notations from Section 3, let F be a general zigzag filtration…”

“…Let F be the associated atomic zigzag filtration of complexes where all maps are forward or backward inclusions of a single cell: F : [32] maintains, at step j of the computation, a homology matrix H(X j ) at the complex X j , that is compatible (defined later) with the following filtration F j ,…”

“…The theory of zigzag persistence was introduced in [9], and theoretical [33] and practical [10,32] algorithms have been introduced to compute it. Zigzag persistence has great applicative potential, considering it provably produces better topological information in topology inference [37], while maintaining the homology of smaller spaces X i thanks to deletions of faces, and more generally allows a finer approach to data analysis, such as density estimation and topological bootstrapping [9].…”

“…Zigzag persistence algorithms. There are currently two practical 2 approaches to compute zigzag persistent homology [9,10,32]. They both can be formulated in a unified framework [31].…”

“…where each arrow represents the insertion of a cell. These transformations are called reflection diamonds for (9) and (10), and transposition diamond for (11), and their effect on the interval decomposition of the zigzag module have been characterized for general zigzag filtrations of complexes in [31,32]. We give details on algorithm [32], that we use later in Section 5, in Appendix A.…”