Wavelet-Based Density Estimation for Persistent Homology
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911849312681984 |
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| author | Häberle, Konstantin Bravi, Barbara Monod, Anthea |
| author_facet | Häberle, Konstantin Bravi, Barbara Monod, Anthea |
| contents | Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram -- a multiset of points supported on the upper half plane -- that is often used as a statistical summary of the topological features of data. In this paper, we study the random nature of persistent homology and estimate the density of expected persistence diagrams from observations using wavelets; we show that our wavelet-based estimator is optimal. Furthermore, we propose an estimator that offers a sparse representation of the expected persistence diagram that achieves near-optimality. We demonstrate the utility of our contributions in a machine learning task in the context of dynamical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_08999 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wavelet-Based Density Estimation for Persistent Homology Häberle, Konstantin Bravi, Barbara Monod, Anthea Statistics Theory 62G07, 62R40, 55N31 Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram -- a multiset of points supported on the upper half plane -- that is often used as a statistical summary of the topological features of data. In this paper, we study the random nature of persistent homology and estimate the density of expected persistence diagrams from observations using wavelets; we show that our wavelet-based estimator is optimal. Furthermore, we propose an estimator that offers a sparse representation of the expected persistence diagram that achieves near-optimality. We demonstrate the utility of our contributions in a machine learning task in the context of dynamical systems. |
| title | Wavelet-Based Density Estimation for Persistent Homology |
| topic | Statistics Theory 62G07, 62R40, 55N31 |
| url | https://arxiv.org/abs/2305.08999 |