Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting

TL;DR

This paper introduces dilinear mappings and tensorial lifting to extend regularization theory to non-linear inverse problems like blind deconvolution and phase retrieval, providing new analysis tools and convergence guarantees.

Contribution

It develops a framework for non-linear inverse problems using tensorial lifting and diconvex regularization, extending linear theory to bilinear and quadratic cases.

Findings

Established convergence rates for dilinear inverse problems.

Validated theoretical results with numerical experiments on deautoconvolution.

Extended convex analysis concepts to the dilinear framework.

Abstract

Considering the question: how non-linear may a non-linear operator be in order to extend the linear regularization theory, we introduce the class of dilinear mappings, which covers linear, bilinear, and quadratic operators between Banach spaces. The corresponding dilinear inverse problems cover blind deconvolution, deautoconvolution, parallel imaging in MRI, and the phase retrieval problem. Based on the universal property of the tensor product, the central idea is here to lift the non-linear mappings to linear representatives on a suitable topological tensor space. At the same time, we extend the class of usually convex regularization functionals to the class of diconvex functionals, which are likewise defined by a tensorial lifting. Generalizing the concepts of subgradients and Bregman distances from convex analysis to the new framework, we analyse the novel class of dilinear inverse problems and establish convergence rates under similar conditions than in the linear setting. Considering the deautoconvolution problem as specific application, we derive satisfiable source conditions and validate the theoretical convergence rates numerically.