Maxisets for Model Selection

TL;DR

This paper characterizes the maximal function spaces (maxisets) where model selection procedures achieve specific convergence rates, using approximation spaces and wavelet collections, with attention to practical computability.

Contribution

It introduces a framework for identifying maxisets in model selection, linking them to approximation spaces and analyzing their computational aspects.

Findings

Maxisets are characterized by approximation spaces.

Classical wavelet collections' maxisets are described by functional spaces.

Calculability issues and performance loss are analyzed.

Abstract

We address the statistical issue of determining the maximal spaces (maxisets) where model selection procedures attain a given rate of convergence. By considering first general dictionaries, then orthonormal bases, we characterize these maxisets in terms of approximation spaces. These results are illustrated by classical choices of wavelet model collections. For each of them, the maxisets are described in terms of functional spaces. We take a special care of the issue of calculability and measure the induced loss of performance in terms of maxisets.