Persistent homology detects curvature
This work addresses a fundamental assumption in topological data analysis, potentially improving geometric inference from data, though it appears incremental as it builds on existing persistent homology methods.
The paper challenges the conventional interpretation in topological data analysis that short intervals in persistent homology represent noise, providing evidence that they instead encode geometric curvature information. Specifically, it proves that persistent homology can detect the curvature of disks from sampled points and develops a computational framework using average persistence landscapes to solve such inverse problems.
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space. In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning.