Optimal Estimation of `1-regularization Prior from a Regularized Empirical Bayesian Risk Standpoint

We address the problem of prior matrix estimation for the solution of `1-regularized ill-posed inverse problems. From a Bayesian viewpoint, we show that such a matrix can be regarded as an influence matrix in a multivariate `1-Laplace density function. Assuming a training set is given, the prior matrix design problem is cast as a maximum likelihood term with an additional sparsity-inducing term. This formulation results in an unconstrained yet nonconvex optimization problem. Memory requirements as well as computation of the nonlinear, nonsmooth sub-gradient equations are prohibitive for large-scale problems. Thus, we introduce an iterative algorithm to design efficient priors for such large problems. We further demonstrate that the solutions of ill-posed inverse problems by incorporation of `1-regularization using the learned prior matrix perform generally better than commonly used regularization techniques where the prior matrix is chosen a-priori.