On Tikhonov regularization with non-convex sparsity constraints

Abstract

This paper deals with a theoretical analysis of a novel regularization technique for (nonlinear) inverse problems, in the field of the so-called sparsity promoting regularizations. We investigate the well-posedness and the convergence rates of a particular Tikhonov-type regularization. The regularization term is chosen to be the canonical norm in the sequence spaces ℓp. In doing so we restrict ourselves to cases of 0 < p ⩽ 1, motivated by sparsity promoting regularization. For p < 1 the triangle inequality is not valid any more and we are facing a non-convex constraint in a quasi Banach space. We provide results on the existence of minimizers, stability and convergence in a classic general setting. In addition we give convergence rates results in the respective Hilbert space topology under classic assumptions.