Updating a Discrete Morse Vector Field for a Lower Star Filtration Vineyard

2026-09-25

AI Summary

This study addresses the inefficiency of recomputing persistent homology from scratch when lower-star filtrations change, proposing an incremental update framework grounded in discrete Morse theory. By constructing a colex vector field compatible with a total order, the method leverages vertex transpositions to rapidly reconstruct the vector field and introduces an optimized path-counting storage and update mechanism that avoids global recomputation. This approach enables the efficient maintenance of persistent homology under dynamically changing filtrations. Open-source proof-of-concept implementations demonstrate that the proposed algorithm achieves runtime performance comparable to standard matrix reduction methods on persistent homology transformation tasks.

Abstract

In this paper, we provide a construction of an acyclic discrete vector field that is compatible with a total order associated with a lower star filtration on a simplicial complex, called the colex vector field. We show that the colex vector field induces a filtered acyclic vector field, whose resulting Morse complex computes the persistent homology of the lower star filtration on the underlying simplicial complex. We show that the colex vector field can be recomputed quickly when the vertex function that induces the lower star filtration is modified via order-adjacent vertex swaps. We provide a framework for storing and computing the number of paths between cells in the simplicial complex, as well as a method to update these values quickly when the colex vector field changes. Finally, we provide publicly available proof-of-concept code for the ideas shown. When applying it to the Persistent Homology Transform, we show that its runtime is comparable to a more standard matrix reduction approach.

Problem

• Discrete Morse Vector Field
• Lower Star Filtration
• Persistent Homology
• Filtration Vineyard
• Simplicial Complex

Innovation

• Discrete Morse Vector Field
• Lower Star Filtration
• Persistent Homology
• Colex Vector Field
• Persistent Homology Transform