Adaptation in a class of linear inverse problems

TL;DR

This paper introduces a wavelet-based adaptive estimation method for linear inverse problems, achieving optimal rates over various Besov classes by penalized regression in a Gaussian sequence framework.

Contribution

It develops a level-by-level complexity penalized regression approach that is exactly rate-adaptive and optimal for a broad class of functions in linear inverse problems with wavelet-vaguelette decomposition.

Findings

Estimator achieves exact rate-adaptive optimality.

Method performs well across a wide range of Besov function classes.

The approach is based on penalized regression in the Gaussian sequence model.

Abstract

We consider the linear inverse problem of estimating an unknown signal $f$ from noisy measurements on $Kf$ where the linear operator $K$ admits a wavelet-vaguelette decomposition (WVD). We formulate the problem in the Gaussian sequence model and propose estimation based on complexity penalized regression on a level-by-level basis. We adopt squared error loss and show that the estimator achieves exact rate-adaptive optimality as $f$ varies over a wide range of Besov function classes.