Adaptive complexity regularization for linear inverse problems

TL;DR

This paper introduces an adaptive regularization method for ill-posed inverse problems that automatically selects optimal smoothing parameters, achieving near-optimal convergence without prior knowledge of the function's regularity.

Contribution

It develops a penalized selection procedure for regularization parameters applicable to Tikhonov and projection methods, with proven oracle inequalities and optimal convergence rates.

Findings

The method adapts to unknown regularity of the target function.

Achieves oracle inequalities ensuring near-optimal performance.

Applicable to Tikhonov and projection regularization techniques.

Abstract

We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence without prior knowledge of the regularity of the function to be estimated. We provide for such estimators oracle inequalities and optimal rates of convergence. This penalized approach is applied to Tikhonov regularization and to regularization by projection.