# A differentiability framework for zigzag persistent homology via linear interpolation

arXiv: https://arxiv.org/abs/2609.39242 · HTML: https://mlsanity.com/paper/2609.39242/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 54 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 28 of 28)

- [Temporal Graph Networks for Deep Learning on Dynamic Graphs](https://mlsanity.com/paper/2006.10637/) (2020)
- [Inductive Representation Learning on Temporal Graphs](https://mlsanity.com/paper/2002.07962/) (2020)
- [Predicting Dynamic Embedding Trajectory in Temporal Interaction Networks](https://mlsanity.com/paper/1908.01207/) (2019)
- [Towards Better Dynamic Graph Learning: New Architecture and Unified Library](https://mlsanity.com/paper/2303.13047/) (2023)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting](https://mlsanity.com/paper/2105.04100/) (2021)
- [Event-Based Backpropagation can compute Exact Gradients for Spiking Neural Networks](https://mlsanity.com/paper/2009.08378/) (2020)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Uncovering the Topology of Time-Varying fMRI Data using Cubical Persistence](https://mlsanity.com/paper/2006.07882/) (2020)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [A Fast and Robust Method for Global Topological Functional Optimization](https://mlsanity.com/paper/2009.08496/) (2020)
- [Continuation of Point Clouds via Persistence Diagrams](https://mlsanity.com/paper/1506.03147/) (2015)
- [Temporal Network Analysis Using Zigzag Persistence](https://mlsanity.com/paper/2205.11338/) (2022)
- [Algebraic Stability of Zigzag Persistence Modules](https://mlsanity.com/paper/1604.00655/) (2016)
- [Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems](https://mlsanity.com/paper/2009.08972/) (2020)
- [Capturing Dynamics of Time-Varying Data via Topology](https://mlsanity.com/paper/2010.05780/) (2020)
- [Fast Computation of Zigzag Persistence](https://mlsanity.com/paper/2204.11080/) (2022)
- [Differentiability and Optimization of Multiparameter Persistent Homology](https://mlsanity.com/paper/2406.07224/) (2024)
- [Spatio-temporal Persistent Homology for Dynamic Metric Spaces](https://mlsanity.com/paper/1812.00949/) (2018)
- [Computing Zigzag Vineyard Efficiently Including Expansions and Contractions](https://mlsanity.com/paper/2307.07462/) (2023)
- [Coordinate-Free Quantification of Coverage in Dynamic Sensor Networks](https://mlsanity.com/paper/1411.7337/) (2014)
- [Weighted Combinatorial Laplacian and its Application to Coverage Repair in Sensor Networks](https://mlsanity.com/paper/2312.04825/) (2023)
- [Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data](https://mlsanity.com/paper/2502.16049/) (2025)
- [T3former: Temporal Graph Classification with Topological Machine Learning](https://mlsanity.com/paper/2510.13789/) (2025)
- [Zigzag Persistence of Neural Responses to Time-Varying Stimuli](https://mlsanity.com/paper/2603.03037/) (2026)

## Abstract (arXiv)

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.