Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered diagonal frame decompositions with general non-linear filters. In particular, our results generalize SVD-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.

翻译：反问题是信号处理、医学成像等多个科学领域的关键问题。由于反问题通常存在对数据扰动的不稳定性，研究者提出了多种正则化技术。特别地，滤波对角框架分解已被证明有效且计算高效。然而，现有收敛性分析仅适用于线性滤波器和软阈值等少数非线性滤波器。本文研究了具有一般非线性滤波器的滤波对角框架分解。作为特例，我们的成果将基于SVD的谱滤波从线性推广至非线性。第一种方法建立了非线性对角框架滤波与变分正则化之间的联系，从而可借助变分正则化结果推导非线性谱滤波的收敛性。第二种方法作为主要理论结果，在保持收敛性的同时放松了变分情形中的假设条件。此外，我们讨论了非线性滤波与即插即用正则化之间的关联，并探索了这种关系可能带来的益处。