Adaptive estimation of Sobolev-type energy functionals on the sphere
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912877729808384 |
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| author | Durastanti, Claudio |
| author_facet | Durastanti, Claudio |
| contents | We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere.
We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias--variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04823 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adaptive estimation of Sobolev-type energy functionals on the sphere Durastanti, Claudio Statistics Theory 62G05, 62G20, 42C40 We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere. We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias--variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods. |
| title | Adaptive estimation of Sobolev-type energy functionals on the sphere |
| topic | Statistics Theory 62G05, 62G20, 42C40 |
| url | https://arxiv.org/abs/2602.04823 |