Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions. (9th December 2016) - Chen, De-Han; Hofmann, Bernd; Zou, Jun
Abstract: We consider the ill-posed operator equation Ax   =   y with an injective and bounded linear operator A mapping between ℓ 2 and a Hilbert space Y, possessing the unique solution x † = { x † k } k = 1 ∞ . For the cases that sparsity x † ∈ ℓ 0 is expected but often slightly violated in practice, we investigate in comparison with the ℓ 1 -regularization the elastic-net regularization, where the penalty is a weighted superposition of the ℓ 1 -norm and the ℓ 2 -norm square, under the assumption that x † ∈ ℓ 1 . 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 ℓ 1 -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 x † k → 0 for k → ∞ and the classical smoothness properties of x † as an element in ℓ 2 .