A remark on an error analysis for classical and learned Tikhonov regularization schemes
This work addresses error analysis for regularization schemes in inverse problems, offering incremental insights for practitioners in computational mathematics and imaging.
The paper analyzes error in classical and learned Tikhonov regularization for inverse problems, showing that using a fixed regularization parameter across noise levels has only a mild impact on reconstruction error, and provides strategies for estimating unknown subspace dimensions and applying learned sparsity terms.
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.