Parameter Estimation in Sparse Inverse Problems using Bernoulli-Gaussian Prior

Sparse coding is now one of the state-of-art approaches for solving inverse problems. In combination with (Fast) Iterative Shrinkage Thresholding Algorithm (ISTA), among other algorithms, one can efficiently get a nice estimator of the sought sparse signal. However, the major drawback of these methods is the tuning of the so-called hyperparameter. In this paper, we first provide an Expectation-Maximization (EM) algorithm to estimate the parameters of a Bernoulli-Gaussian model for denoising a signal corrupted by a white Gaussian noise. Then, building on the Expectation-Maximization interpretation of ISTA, we provide a simple iterative algorithm to blindly estimate all the model parameters in the linear inverse problem context, including the hyperparameter involved in the popular L0 regularized minimization. Moreover, the algorithm directly yields an estimator of the sparse signal.