Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions

Abstract

We consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between and a Hilbert space Y, possessing the unique solution . For the cases that sparsity is expected but often slightly violated in practice, we investigate in comparison with the -regularization the elastic-net regularization, where the penalty is a weighted superposition of the -norm and the -norm square, under the assumption that . There occur two positive parameters in this approach, the weight parameter η and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in -regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay for and the classical smoothness properties of as an element in .