<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:mi>α</mml:mi> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mi>β</mml:mi> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> regularization for sparse recovery

Abstract

Abstract This paper presents a novel regularization with a non-convex, non-smooth term of the form with parameters to solve ill-posed linear problems with sparse solutions. We investigate the existence, stability and convergence of the regularized solution. It is shown that this type of regularization is well-posed and yields sparse solutions. Under an appropriate source condition, we get the convergence rate in the -norm for a priori and a posteriori parameter choice rules, respectively. A numerical algorithm is proposed and analyzed based on an iterative threshold strategy with the generalized conditional gradient method. We prove the convergence even though the regularization term is non-smooth and non-convex. The algorithm can easily be implemented because of its simple structure. Some numerical experiments are performed to test the efficiency of the proposed approach. The experiments show that regularization with performs better in comparison with the classical sparsity regularization and can be used as an alternative to the regularizer.