# Persistent homology detects curvature

## TL;DR

This paper demonstrates that persistent homology, traditionally viewed as capturing topological features, also encodes geometric information such as curvature, by analyzing sampled points from curved disks.

## Contribution

It provides theoretical evidence that short persistent homology intervals contain geometric information and introduces a computational framework for inverse problems using average persistence landscapes.

## Key findings

- Persistent homology detects curvature of sampled disks.
- Short intervals encode geometric, not just noise.
- A framework for learning curvature from persistence landscapes.

## Abstract

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.