Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions

Abstract

Recent advances in combinatorial dynamical systems that generalize the classic discrete Morse theory have prompted algorithmic studies of combinatorial vector fields. In this regard, authors in [7] recently proposed the concept of Conley-Morse persistence barcode that summarizes the continuation of invariant sets in an evolving vector field through homological persistence. They proposed an algorithm to compute this barcode using a filtration of the so called \emph{index pairs} on a poset called \emph{transition diagram}. The algorithm becomes costly due to multiple runs of zigzag persistence it executes on filtrations of `unwieldy' structures of index pairs. We overcome this difficulty by replacing the index pairs with \emph{blocks}, which are structurally much simpler. These replacements need reversal of certain relations in the transition diagram resulting in a much simpler algorithm. The algorithm works by updating matrix decompositions akin to computing `vineyard' in standard persistence.