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Persistent homology detects curvature

Peter Bubenik, Michael Hull, Dhruv Patel, Benjamin Whittle

TL;DR

This work investigates how persistent homology, traditionally viewed as a topological proxy, encodes geometry such as curvature. It develops a framework that uses short persistence intervals and the average persistence landscape to solve inverse problems, notably estimating curvature from samples on disks of constant curvature $K\in[-2,2]$. Theoretical results show that equilateral triangles maximize persistence and that the persistence for equilateral triangles yields invertible dependence on $K$, while a continuous map from metric-measure spaces to $L^2(\mathbb{N}\times\mathbb{R})$ enables learning curvature from data. Computational demonstrations with Euclidean, spherical, and hyperbolic disks show accurate curvature estimation using supervised and unsupervised methods, and robustness to monotone-distance transforms via ordinal data. Overall, the paper provides a principled pipeline marrying persistent homology with statistical learning to extract geometric information from topological summaries and to address inverse problems in geometry.

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.

Persistent homology detects curvature

TL;DR

This work investigates how persistent homology, traditionally viewed as a topological proxy, encodes geometry such as curvature. It develops a framework that uses short persistence intervals and the average persistence landscape to solve inverse problems, notably estimating curvature from samples on disks of constant curvature . Theoretical results show that equilateral triangles maximize persistence and that the persistence for equilateral triangles yields invertible dependence on , while a continuous map from metric-measure spaces to enables learning curvature from data. Computational demonstrations with Euclidean, spherical, and hyperbolic disks show accurate curvature estimation using supervised and unsupervised methods, and robustness to monotone-distance transforms via ordinal data. Overall, the paper provides a principled pipeline marrying persistent homology with statistical learning to extract geometric information from topological summaries and to address inverse problems in geometry.

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.

Paper Structure

This paper contains 33 sections, 14 theorems, 27 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical results: short bars detect geometry
  3. A framework for solving inverse problems: inference using average persistence landscapes
  4. Computational results
  5. Expected impact
  6. Related work
  7. Background
  8. Filtered simplicial complexes from points
  9. Persistent homology
  10. Geometries of constant curvature
  11. Triangles
  12. Circumcircles
  13. Areas of disks
  14. Distances between points on a unit disk
  15. Laws of sines and cosines
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $p(K)$ denote the maximum (Čech) persistence for three points on a surface of constant curvature $K$ with pairwise distances at most some fixed constant. Then $p(K)$ is an invertible function.

Figures (11)

  • Figure 1: Plots showing actual curvature and estimated curvature using $H_0$ and $H_1$ from distance data, for nearest neighbors (left), support vector regression (center), and the first principal component (right).
  • Figure 2: $\triangle ABC$.
  • Figure 3: Replacing $\triangle ABC$ with an isosceles triangle $\triangle A'BC$.
  • Figure 4: Replacing an isosceles triangle $\triangle ABC$ with an equilateral triangle $\triangle A'BC$.
  • Figure 5: $b(T)=d(A, M)$ and $d(T)=d(A, P)$.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 3.1
  • ...and 14 more