Kernel regression, minimax rates and effective dimensionality: beyond the regular case

TL;DR

This paper explores the capabilities of kernel regularization methods to attain optimal convergence rates under broad eigenvalue decay conditions, extending understanding beyond traditional polynomial decay assumptions.

Contribution

It introduces a framework for analyzing kernel regularization under weaker, more general eigenvalue decay assumptions, broadening the applicability of minimax rate results.

Findings

Achieves minimax convergence rates under generalized eigenvalue decay.

Extends theoretical understanding of kernel methods beyond polynomial spectrum decay.

Provides insights into data structure effects at multiple scales.

Abstract

We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.