A differentiability framework for zigzag persistent homology via linear interpolation
Abstract
Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.
Related
- Geometric Realizations of Capability Spaces of Large Models and Zigzag Persistent Homology
- RedZeD: Computing persistent homology by Reduction to Zero Differentials
- Persistent Homology of Biquandle Coloring Quivers
- The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology
- Limit Theorems for Persistent Homology of Multiparameter Complex Processes
BibTeX
@misc{ferrari2026differentiability,
title = {{A differentiability framework for zigzag persistent homology via linear interpolation}},
author = {Enrico Maria Ferrari and Clemens Bannwart and Matteo Biagetti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39242},
eprint = {2609.39242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39242},
}
RIS
TY - UNPB
TI - A differentiability framework for zigzag persistent homology via linear interpolation
AU - Enrico Maria Ferrari
AU - Clemens Bannwart
AU - Matteo Biagetti
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39242
UR - https://arxiv.org/abs/2609.39242
AB - Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.
ER -