Spectral Equivariance and Geometric Transport in Reproducing Kernel Hilbert Spaces: A Unified Framework for Orthogonal Polynomial and Kernel Estimation

TL;DR

This paper introduces a unified geometric framework for nonparametric estimation using Twin Kernel Spaces, revealing spectral equivariance and transport properties that unify various spectral and kernel methods.

Contribution

It develops a Spectral Equivariance Theorem and a geometric framework that unifies orthogonal polynomial estimators, kernel estimators, and spectral smoothing through group actions and transport.

Findings

Eigenfunctions of transported kernels are unitarily transported from base eigenfunctions.

Orthogonal polynomial estimators are equivariant under geometric deformation.

Minimax rates and bias-variance tradeoffs are invariant under transport.

Abstract

We develop a unified geometric framework for nonparametric estimation based on the notion of Twin Kernel Spaces, defined as orbits of a reproducing kernel under a group action. This structure induces a family of transported RKHS geometries in which classical orthogonal polynomial estimators, kernel estimators, and spectral smoothing methods arise as projections onto transported eigenfunction systems. Our main contribution is a Spectral Equivariance Theorem showing that the eigenfunctions of any transported kernel are obtained by unitary transport of the base eigenfunctions. As a consequence, orthogonal polynomial estimators are equivariant under geometric deformation, kernel estimators correspond to soft spectral filtering in a twin space, and minimax rates and bias--variance tradeoffs are invariant under transport. We provide examples based on Hermite and Legendre polynomials, affine and Gaussian groups, and illustrate the effectiveness of twin transport for adaptive and multimodal estimation. The framework reveals a deep connection between group actions, RKHS geometry, and spectral nonparametrics, offering a unified perspective that encompasses kernel smoothing, orthogonal series, splines, and multiscale methods.