A novel monotonic fixed-point algorithm for l1-regularized least square vector and matrix problem

Least square problem with l1 regularization has been proposed as a promising method for sparse signal reconstruction (e.g., basis pursuit de-noising and compressed sensing) and feature selection (e.g., the Lasso algorithm) in signal processing, statistics, and related fields. These problems can be cast as l1-regularized least-square program (LSP). In this paper, we propose a novel monotonic fixed point method to solve large-scale l1-regularized LSP. And we also prove the stability and convergence of the proposed method. Furthermore we generalize this method to least square matrix problem and apply it in nonnegative matrix factorization (NMF). The method is illustrated on sparse signal reconstruction, partner recognition and blind source separation problems, and the method tends to convergent faster and sparser than other l1-regularized algorithms.