Beamforming of the Residuals is the LASSO's Dual

Waves hidden in additive noise are observed by a sensor array. The waves are assumed to originate from a sparse set of sources. We treat the estimation of the sparse set of sources as a generalized complex-valued LASSO problem. The generalized complex-valued LASSO problem is strictly convex and strong duality holds. The corresponding dual problem is interpretable as a weighted conventional beamformer acting on the residuals of the LASSO. Moreover, this establishes a simple linear-affine relation between the dual and primal vectors. The solution path of the complex-valued LASSO is analyzed and three procedures for signal processing are proposed and evaluated which are based on the generalized LASSO and its dual: An order-recursive procedure and two iterative procedures which are based on a further approximation.