A Biconvex Analysis for Lasso $\ell _1$ Reweighting

Iterative l
₁
reweighting algorithms are very popular in sparse signal recovery and compressed sensing, since in the practice they have been observed to outperform classical 11 methods. Nevertheless, the theoretical analysis of their convergence is a critical point, and generally is limited to the convergence of the functional to a local minimum or to subsequence convergence. In this letter, we propose a new convergence analysis of a Lasso l
₁
reweighting method, based on the observation that the algorithm is an alternated convex search for a biconvex problem. Based on that, we are able to prove the numerical convergence of the sequence of the iterates generated by the algorithm. Furthermore, we propose an alternative iterative soft thresholding procedure, which is faster than the main algorithm.