Fast Candidate Points Selection in the LASSO Path

The LASSO sparse regression method has recently received attention in a variety of applications from image compression techniques to parameter estimation problems. This paper addresses the problem of regularization parameter selection in this method in a general case of complex-valued regressors and bases. Generally, this parameter controls the degree of sparsity or equivalently, the estimated model order. However, with the same sparsity/model order, the smallest regularization parameter is desired. We relate such points to the nonsmooth points in the path of LASSO solutions and give an analytical expression for them. Then, we introduce a numerically fast method of approximating the desired points by a recursive algorithm. The procedure decreases the necessary number of solutions of the LASSO problem dramatically, which is an important issue due to the polynomial computational cost of the convex optimization techniques. We illustrate our method in the context of DOA estimation.