Model Consistency of the Iterative Regularization of Dual Ascent for Low-Complexity Regularization

Abstract

Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The former involves solving a variational optimization problem, which consists of a data-fidelity term and a regularization term, balanced by an appropriate weighting parameter. The latter mitigates overfitting to noise by selecting a suitable stopping time during the iterative process. A key topic in the study of regularization is the relationship between the regularized solution and the original ground truth. When the ground truth possesses a low-complexity structure referred to as the "model" it can be shown that under appropriate regularization promoting the same structure, the solution to the regularized problem is robust to small perturbations. This property is called "model consistency". For variational regularization, model consistency in linear inverse problems has been studied in [1]. However, for iterative regularization, model consistency remains an open question. In this paper, building on recent developments in partial smoothness [1], we show that when the noise level is sufficiently small and an appropriate stopping criterion is used, iterative regularization is model consistent as well. Moreover, we show that the considered algorithm exhibits local linear behavior of the regularization. We provide numerical simulations to support our theoretical findings.