Inducing sparsity via the horseshoe prior in imaging problems

A problem typically arising in imaging applications is the reconstruction task under sparsity constraints. A computationally efficient strategy to address this problem is to recast it in a hierarchical Bayesian framework coupled with a Maximum A Posteriori (MAP) estimation approach. More specifically, the original unknown is modeled as a conditionally Gaussian random variable with an unknown variance. Here, the expected behavior on the variance is encoded in a half-Cauchy hyperprior. The latter, coupled to the conditioned Gaussian prior, yields the horseshoe shrinkage prior, particularly popular within the statistics community and here introduced into the context of imaging problems. The arising non-convex MAP estimation problem is tackled via an alternating minimization scheme for which the global convergence to a stationary point is guaranteed. Experimental results prove that the derived hypermodel is competitive with classical variational methods as well as with other hierarchical Bayesian models typically employed for sparse recovery problems.