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

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.

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BibTeX RIS

BibTeX

@misc{woytowich2026updating,
  title = {{Updating a Discrete Morse Vector Field for a Lower Star Filtration Vineyard}},
  author = {Kevin Woytowich and Nkechi Nnadi and Elizabeth Munch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31954},
  eprint = {2609.31954},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31954},
}

RIS

TY  - UNPB
TI  - Updating a Discrete Morse Vector Field for a Lower Star Filtration Vineyard
AU  - Kevin Woytowich
AU  - Nkechi Nnadi
AU  - Elizabeth Munch
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31954
UR  - https://arxiv.org/abs/2609.31954
AB  - 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.
ER  -

Abstract

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