Framework enables smooth differentiation of zigzag persistent homology
A differentiability framework for zigzag persistent homology via linear interpolation
Machine LearningComputational Geometry
Summary
Persistent homology is a tool used to study shapes and patterns that change over time, but existing methods only work when the changes are always in one direction. The authors developed a way to handle situations where these shapes and patterns can grow and shrink non-linearly, using something called zigzag persistence. They created a smooth mathematical method that links changes in data values to changes in the shape's features, allowing this information to be used in machine learning. Their framework is tested on sensor networks and dynamic graphs, showing it can help improve monitoring and classification tasks.
What this means in practice
- •For sensor network engineers: Optimize sensor coverage over time using differentiable zigzag persistence to track dynamic network changes effectively.
- •For data scientists analyzing dynamic graphs: Improve classification accuracy of time-evolving graph structures by integrating the differentiable zigzag persistence framework into learning models.
Authors
Enrico Maria Ferrari, Clemens Bannwart, Matteo Biagetti
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.