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 l(1)-norm constraint is replaced by the l(1)-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. (C) 2016 Elsevier B.V. All rights reserved.