Stabilizing Localization of Representative cycles

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.