  This paper develops a convex approach for sparse one-dimensional
deconvolution that improves upon L1-norm regularization, the standard convex
approach. We propose a sparsity-inducing non-separable non-convex bivariate
penalty function for this purpose. It is designed to enable the convex
formulation of ill-conditioned linear inverse problems with quadratic data
fidelity terms. The new penalty overcomes limitations of separable
regularization. We show how the penalty parameters should be set to ensure that
the objective function is convex, and provide an explicit condition to verify
the optimality of a prospective solution. We present an algorithm (an instance
of forward-backward splitting) for sparse deconvolution using the new penalty.
