  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.
