Affine scaling transformation based methods for computing low complexity sparse solutions

This paper presents affine scaling transformation based methods for finding low complexity sparse solutions to optimization problems. The methods achieve sparse solutions in a more general context, and generalize our earlier work on FOCUSS developed to deal with the underdetermined linear inverse problem. The key result is a theorem which shows a simple condition that a sequence has to satisfy for it to converge to a sparse limiting solution. Three approaches to incorporate this condition into optimization problems are presented. These consist of either imposing the condition as an additional optimization constraint, or suitably modifying the cost function, or using a combination of the two. The benefits of the methodology when applied to the linear inverse problem are twofold. Firstly, it allows for the treatment of the overdetermined problem in addition to the underdetermined problem, and secondly it enables establishing sufficient conditions under which regularized versions of FOCUSS are assured of convergence to sparse solutions