LEMUR: Latent EM Unsupervised Regression for Sparse Inverse Problems

Most methods for sparse signal recovery require setting one or several hyperparameters. We propose an unsupervised method to estimate the parameters of a Bernoulli-Gaussian (BG) model describing sparse signals. The proposed method is first derived for denoising problems using a maximum likelihood (ML) approach. Then, an extension to general inverse problems is achieved through a latent variable formulation. Two expectation-maximization (EM) algorithms are then proposed to estimate the signal together with the BG model parameters. Combining these two approaches leads to the proposed LEMUR algorithm. LEMUR is then evaluated on extensive simulations regarding the ability to recover the parameters and provide accurate sparse signal estimates.