c-LASSO and its dual for sparse signal estimation from array data

We treat the estimation of a sparse set of sources emitting plane waves observed by a sensor array as a complex-valued LASSO (c-LASSO) problem where the usual ℓ1-norm constraint is replaced by the ℓ1-norm of a matrix D times the solution vector. When the sparsity order is given, algorithmically selecting a suitable value for the c-LASSO regularization parameter remains a challenging task. The corresponding dual problem is formulated and it is shown that the dual solution is useful for selecting the regularization parameter of the c-LASSO. The solution path of the c-LASSO is analyzed and this motivates an order-recursive algorithm for the selection of the regularization parameter and a faster iterative algorithm that is based on a further approximation. This greatly facilitates computation of the c-LASSO-path as we can predict the changes in the active indices as the regularization parameter is reduced. Using this regularization parameter, the directions of arrival for all sources are estimated.