Explainable topological data analysis using persistence heatmaps

Abstract

Topological data analysis (TDA) leverages tools from algebraic topology to aid in various machine learning tasks. Numerous TDA constructions are provably stable in the sense that changes in the input produce linearly bounded changes in the output, with a specified bound. We use representative cycles, which are unstable TDA constructions, to produce stable visualizations to aid in explaining TDA. For example, we produce stable heatmaps on images containing the data such that summing the values of the pixels gives the value of a learned regression function.

28 pages, 14 figures, completely rewritten, new title, new theorems, new examples