  The Lasso (Least Absolute Shrinkage and Selection Operator) has been a
popular technique for simultaneous linear regression estimation and variable
selection. In this paper, we propose a new novel approach for robust Lasso that
follows the spirit of M-estimation. We define $M$-Lasso estimates of regression
and scale as solutions to generalized zero subgradient equations. Another
unique feature of this paper is that we consider complex-valued measurements
and regression parameters, which requires careful mathematical characterization
of the problem. An explicit and efficient algorithm for computing the $M$-Lasso
solution is proposed that has comparable computational complexity as
state-of-the-art algorithm for computing the Lasso solution. Usefulness of the
$M$-Lasso method is illustrated for direction-of-arrival (DoA) estimation with
sensor arrays in a single snapshot case.
