Unsupervised Expectation Propagation Method for Large-Scale Sparse Linear Inverse Problems

This paper addresses the estimation of large-scale sparse coefficients from noisy linear measurements using Expectation Propagation (EP) method for unsupervised approximate Bayesian inference. In the Bayesian model, the Laplace prior, Mixture of two Gaussians (MoG2) prior, and Spike-and-Slab (SaS) prior are adopted respectively as the sparsity-promoting priors of the unknown sparse parameter. In solving high-dimensional linear inverse problems, the proposed EP method directly provides the approximate minimum mean squared error (MMSE) estimate and the approximate posterior uncertainty by an approximating posterior distribution. Furthermore, to tackle the challenging problem of hyperparameter tuning, the EP posterior approximation is embedded in a variational Expectation Maximization (EM) approach to allow for unsupervised hyperparameter estimation. Experiments are conducted on synthetic datasets, including an imaging deconvolution problem, to illustrate the efficiency of the proposed unsupervised EP method and the advantage of using MoG2 and SaS priors in solving sparse linear inverse problems.