  Inference by means of mathematical modeling from a collection of observations
remains a crucial tool for scientific discovery and is ubiquitous in
application areas such as signal compression, imaging restoration, and
supervised machine learning. The inference problems may be solved using
variational formulations that provide theoretically proven methods and
algorithms. With ever-increasing model complexities and growing data size, new
specially designed methods are urgently needed to recover meaningful quantifies
of interest. We consider the broad spectrum of linear inverse problems where
the aim is to reconstruct quantities with a sparse representation on some
vector space; often solved using the (generalized) least absolute shrinkage and
selection operator (lasso). The associated optimization problems have received
significant attention, in particular in the early 2000's, because of their
connection to compressed sensing and the reconstruction of solutions with
favorable sparsity properties using augmented Lagrangians, alternating
directions and splitting methods. We provide a new perspective on the
underlying l1 regularized inverse problem by exploring the generalized lasso
problem through variable projection methods. We arrive at our proposed variable
projected augmented Lagrangian (vpal) method. We analyze this method and
provide an approach for automatic regularization parameter selection based on a
degrees of freedom argument. Further, we provide numerical examples
demonstrating the computational efficiency for various imaging problems.
