# Persistent homology detects curvature

arXiv: https://arxiv.org/abs/1905.13196 · HTML: https://mlsanity.com/paper/1905.13196/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2019.

## Built on by (most cited first, 26 of 26)

- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [Metric Space Magnitude for Evaluating the Diversity of Latent Representations](https://mlsanity.com/paper/2311.16054/) (2023)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Topological Singularity Detection at Multiple Scales](https://mlsanity.com/paper/2210.00069/) (2022)
- [From Geometry to Topology: Inverse Theorems for Distributed Persistence](https://mlsanity.com/paper/2101.12288/) (2021)
- [Topologically penalized regression on manifolds](https://mlsanity.com/paper/2110.13749/) (2021)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds](https://mlsanity.com/paper/2308.02615/) (2023)
- [Statistical learning on measures: an application to persistence diagrams](https://mlsanity.com/paper/2303.08456/) (2023)
- [Local Topology Measures of Contextual Language Model Latent Spaces With Applications to Dialogue Term Extraction](https://mlsanity.com/paper/2408.03706/) (2024)
- [Reconstruction of univariate functions from directional persistence diagrams](https://mlsanity.com/paper/2203.01894/) (2022)
- [Visualizing Topological Importance: A Class-Driven Approach](https://mlsanity.com/paper/2309.13185/) (2023)
- [Topological data analysis of human vowels: Persistent homologies across representation spaces](https://mlsanity.com/paper/2310.06508/) (2023)
- [Discrete transforms of quantized persistence diagrams](https://mlsanity.com/paper/2312.17093/) (2023)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration](https://mlsanity.com/paper/2204.07821/) (2022)
- [Machine learning and Topological data analysis identify unique features of human papillae in 3D scans](https://mlsanity.com/paper/2307.06255/) (2023)
- [Topological Inductive Bias fosters Multiple Instance Learning in Data-Scarce Scenarios](https://mlsanity.com/paper/2307.14025/) (2023)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)
- [TUN: Detecting Significant Points in Persistence Diagrams with Deep Learning](https://mlsanity.com/paper/2512.14274/) (2025)
- [Measuring Dataset Diversity from a Geometric Perspective](https://mlsanity.com/paper/2602.09340/) (2026)
- [LandscapeSHAP: Which Persistent Homology Class Gets the Credit?](https://mlsanity.com/paper/2609.31469/) (2026)
- [PLDiv: Measuring Dataset Diversity and Semantic Collapse via Persistence Landscapes](https://mlsanity.com/paper/2610.14031/) (2026)

## Abstract (arXiv)

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.