Persistent homology detects curvature
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866909497875759104 |
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| author | Bubenik, Peter Hull, Michael Patel, Dhruv Whittle, Benjamin |
| author_facet | Bubenik, Peter Hull, Michael Patel, Dhruv Whittle, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1905_13196 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Persistent homology detects curvature Bubenik, Peter Hull, Michael Patel, Dhruv Whittle, Benjamin Computational Geometry Machine Learning Algebraic Topology 55N99 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. |
| title | Persistent homology detects curvature |
| topic | Computational Geometry Machine Learning Algebraic Topology 55N99 |
| url | https://arxiv.org/abs/1905.13196 |