Nonparametric regression on random geometric graphs sampled from submanifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916465440980992 |
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| author | Rosa, Paul Rousseau, Judith |
| author_facet | Rosa, Paul Rousseau, Judith |
| contents | We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_20909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonparametric regression on random geometric graphs sampled from submanifolds Rosa, Paul Rousseau, Judith Statistics Theory Machine Learning 62E20, 62R30 G.3 We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index. |
| title | Nonparametric regression on random geometric graphs sampled from submanifolds |
| topic | Statistics Theory Machine Learning 62E20, 62R30 G.3 |
| url | https://arxiv.org/abs/2405.20909 |