Adaptive Nonparametric Estimation via Kernel Transport on Group Orbits: Oracle Inequalities and Minimax Rates
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917137916887040 |
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| author | Nembe, Jocelyn |
| author_facet | Nembe, Jocelyn |
| contents | We develop a unified framework for nonparametric functional estimation based on kernel transport along orbits of discrete group actions, which we term \emph{Twin Spaces}. Given a base kernel $K$ and a group $G = \langleφ\rangle$ acting isometrically on the input space $E$, we construct a hierarchy of transported kernels $\{K_j\}_{j\geq 0}$ and a penalized model selection scheme satisfying a Kraft inequality. Our main contributions are threefold: (i) we establish non-asymptotic oracle inequalities for the penalized twin-kernel estimator with explicit constants; (ii) we introduce novel twin-regularity classes that capture smoothness along group orbits and prove that our estimator adapts to these classes; (iii) we show that the framework recovers classical minimax-optimal rates in the Euclidean setting while enabling improved rates when the target function exhibits orbital structure. The effective dimension $d_{\mathrm{eff}}$ governing the rates is characterized in terms of the quotient $G/L$, where $L$ is the subgroup preserving the base operation. Connections to wavelet methods, geometric quantization, and adaptive computation are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Nonparametric Estimation via Kernel Transport on Group Orbits: Oracle Inequalities and Minimax Rates Nembe, Jocelyn Statistics Theory 2020: Primary 62G05, 62G20, Secondary 46E22, 22F05 We develop a unified framework for nonparametric functional estimation based on kernel transport along orbits of discrete group actions, which we term \emph{Twin Spaces}. Given a base kernel $K$ and a group $G = \langleφ\rangle$ acting isometrically on the input space $E$, we construct a hierarchy of transported kernels $\{K_j\}_{j\geq 0}$ and a penalized model selection scheme satisfying a Kraft inequality. Our main contributions are threefold: (i) we establish non-asymptotic oracle inequalities for the penalized twin-kernel estimator with explicit constants; (ii) we introduce novel twin-regularity classes that capture smoothness along group orbits and prove that our estimator adapts to these classes; (iii) we show that the framework recovers classical minimax-optimal rates in the Euclidean setting while enabling improved rates when the target function exhibits orbital structure. The effective dimension $d_{\mathrm{eff}}$ governing the rates is characterized in terms of the quotient $G/L$, where $L$ is the subgroup preserving the base operation. Connections to wavelet methods, geometric quantization, and adaptive computation are discussed. |
| title | Adaptive Nonparametric Estimation via Kernel Transport on Group Orbits: Oracle Inequalities and Minimax Rates |
| topic | Statistics Theory 2020: Primary 62G05, 62G20, Secondary 46E22, 22F05 |
| url | https://arxiv.org/abs/2512.10049 |