Stabilizing Localization of Representative cycles
Himanshu Yadav, Alexander Wagner, Peter Bubenik
TL;DR
This work addresses the instability of visualizing representative cycles from persistence diagrams by introducing the persistence heatmap, a parametrized, interpretable heat distribution over a simplicial complex. The core idea is to compute an annotated persistence diagram and then average across filtrations to form the expected persistence heatmap, ensuring stability through kernel smoothing and Lipschitz bounds. The authors establish uniform continuity and Lipschitz stability for several kernels (triangular, Epanechnikov, Gaussian) and integrate machine learning (SVM) to learn task-specific parameterizations that emphasize discriminative topological features. Computational experiments on synthetic examples and dynamical systems demonstrate that the resulting stable heatmaps improve explainability and localization of topological structure for classification and regression tasks in TDA.
Abstract
We introduce the persistence heatmap, a parametrized summary based on representative cycles in persistence diagrams, designed to enhance stability and explainability in topological data analysis. Algorithms to compute persistence diagrams produce representative cycles and boundaries. These chains are difficult to use because they are unstable to perturbations of the input. Instead, we average to produce chains with real-valued coefficients. We prove Lipschitz stability and uniform continuity of our heatmap. Moreover, we use machine learning to learn a task-specific parametrization of the heatmap.