A remark on an error analysis for classical and learned Tikhonov regularization schemes

TL;DR

This paper analyzes classical and learned Tikhonov regularization for inverse problems, showing that fixed regularization parameters have mild effects and proposing a data-driven dimension estimation strategy.

Contribution

It provides a theoretical and numerical error analysis, including a new strategy for estimating unknown data subspace dimensions and insights into learned sparsity-promoting methods.

Findings

Fixed regularization parameters have a mild impact on error across noise levels.

A data-driven strategy for estimating the unknown data subspace dimension is supported by numerical experiments.

Error analysis of learned sparsity-promoting methods is refined through discretized setting investigation.

Abstract

This paper presents an error analysis of classical and learned Tikhonov regularization schemes for inverse problems. We first demonstrate, both theoretically and numerically, that using a fixed regularization parameter across varying noise levels-which is a common miss-specification in practice-has only a mild impact on the reconstruction error. As a special case, we then investigate scenarios where the true data resides in an unknown finite-dimensional subspace. Here, our results lead to an empirically supported strategy for estimating the unknown dimension based on numerical experiments. Finally, we examine the approach that motivated this study: a method where a sparsity-promoting term is learned from denoising tasks and subsequently applied to general inverse problems via a simple heuristic parameter selection. The corresponding error analysis is initially developed using classical concepts and subsequently refined through a more detailed investigation of the discretized setting.