An iterative algorithm for the square-root Lasso

In the framework of sparsity-enforcing regularisation for linear inverse problems, we consider the minimisation of a square-root Lasso cost function. To solve this problem we devise a simple modification (called SQRT-ISTA) of the Iterative Soft-Thresholding Algorithm (ISTA) for the Lasso problem and we prove convergence for this algorithm. Under some additional assumptions, we derive an upper bound on the convergence rate of the cost function. We also generalise these results to the case of the group square-root Lasso, where sparsity is enforced for groups of variables instead of individual ones.