{"ID":2830879,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.10049","arxiv_id":"2512.10049","title":"Adaptive Nonparametric Estimation via Kernel Transport on Group Orbits: Oracle Inequalities and Minimax Rates","abstract":"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 $G = \\langle\\varphi\\rangle$ 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.","short_abstract":"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 $G = \\langle\\varphi\\rangle$ acting isometrically on the input space $E$, we construct a hierarchy of transported...","url_abs":"https://arxiv.org/abs/2512.10049","url_pdf":"https://arxiv.org/pdf/2512.10049v1","authors":"[\"Jocelyn Nembe\"]","published":"2025-12-10T20:07:16Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}