Persistence Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909630643306496 |
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| author | Henneuse, Hugo |
| author_facet | Henneuse, Hugo |
| contents | To our knowledge, the analysis of convergence rates for persistence diagrams estimation from noisy signals has predominantly relied on lifting signal estimation results through sup-norm (or other functional norm) stability theorems. We believe that moving forward from this approach can lead to considerable gains. We illustrate it in the setting of nonparametric regression. From a minimax perspective, we examine the inference of persistence diagrams (for the sublevel sets filtration). We show that for piecewise Hölder-continuous functions, with control over the reach of the set of discontinuities, taking the persistence diagram coming from a simple histogram estimator of the signal permits achieving the minimax rates known for Hölder-continuous functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_19396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Persistence Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals Henneuse, Hugo Statistics Theory Algebraic Topology To our knowledge, the analysis of convergence rates for persistence diagrams estimation from noisy signals has predominantly relied on lifting signal estimation results through sup-norm (or other functional norm) stability theorems. We believe that moving forward from this approach can lead to considerable gains. We illustrate it in the setting of nonparametric regression. From a minimax perspective, we examine the inference of persistence diagrams (for the sublevel sets filtration). We show that for piecewise Hölder-continuous functions, with control over the reach of the set of discontinuities, taking the persistence diagram coming from a simple histogram estimator of the signal permits achieving the minimax rates known for Hölder-continuous functions. |
| title | Persistence Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals |
| topic | Statistics Theory Algebraic Topology |
| url | https://arxiv.org/abs/2403.19396 |