Adaptive Nonparametric Estimation via Kernel Transport on Group Orbits: Oracle Inequalities and Minimax Rates

TL;DR

This paper introduces a unified kernel transport framework for nonparametric estimation on group orbits, providing oracle inequalities, adaptive regularity classes, and minimax rates that improve with orbital structure.

Contribution

It develops a novel twin-kernel estimation method with theoretical guarantees, capturing smoothness along group orbits and achieving optimal rates in Euclidean and structured settings.

Findings

Establishes non-asymptotic oracle inequalities with explicit constants.

Defines twin-regularity classes for orbital smoothness and proves adaptation.

Recovers classical minimax rates and improves them using orbital structure.

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.