Generalized Fixed-Point Continuation Method: Convergence and Application

Abstract

In this paper, we consider a class of minimization problems with the objective functions having a form of summation of a penalized differentiable convex function, and a weighted $\ell _1$ -norm. However, different from the common assumption of positive weights in existing studies, we shall address a general case where the weights can be either positive or negative, motivated by the fact that negative weights are also capable of inducing sparsity, and even achieving outstanding performance. To deal with the resulting problem, a generalized fixed-point continuation (GFPC) method is introduced, and an accelerated variant is developed. More importantly, the convergence of this algorithm is analyzed in detail, and its application to compressing sensing problems that employ the Shannon entropy function (SEF) for sparsity promotion is also studied. Numerical examples are carried out to demonstrate the effectiveness of the GFPC algorithm.