Persistent homology detects curvature

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.

arXiv comment22 pages, corrections thanks to anonymous referees

References in corpus (14)

• Hierarchical structures of amorphous solids characterized by persistent homology
• Clique topology reveals intrinsic geometric structure in neural correlations
• Topological Data Analysis of Financial Time Series: Landscapes of Crashes
• A persistence landscapes toolbox for topological statistics
• A statistical approach to persistent homology
• Using persistent homology and dynamical distances to analyze protein binding
• The Vietoris-Rips complexes of a circle
• Persistent Homology analysis of Phase Transitions
• Analysis of Kolmogorov Flow and Rayleigh-Bénard Convection using Persistent Homology
• Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and Colloids
• Quantifying force networks in particulate systems
• Topological Microstructure Analysis Using Persistence Landscapes
• Structure of force networks in tapped particulate systems of disks and pentagons (Part 2): Persistence analysis
• Topological Data Analysis for Object Data

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• Rigidity of terminal simplices in persistent homology
• Geometric Bounds for Persistence