Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions

2026-08-06

AI Summary

Existing methods for computing Conley–Morse persistent barcodes suffer from inefficiency due to their reliance on intricate index pair structures and repeated zigzag persistent homology computations. This work proposes an efficient algorithm that replaces index pairs with simpler, more structured “blocks” to reconstruct relationships within the transition graph. By adapting the vineyard paradigm from standard persistent homology, the algorithm enables dynamic updates of matrix factorizations, thereby circumventing redundant zigzag persistent homology calculations. This approach substantially reduces time complexity and significantly accelerates the computation of Conley–Morse barcodes, making it well-suited for tracking homological features of invariant sets in evolving vector fields.

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.

Problem

• Conley-Morse persistence barcode
• combinatorial vector fields
• zigzag persistence
• index pairs
• computational efficiency

Innovation

• Conley-Morse persistence barcode
• matrix decomposition update
• combinatorial vector fields
• index pairs replacement
• zigzag persistence