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 sparse 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 ℓ
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regularized minimization. Moreover, the algorithm directly yields an estimator of the sparse signal.