Shape Parameter and Sparse Representation Recovery under Generalized Gaussian Noise

We tackle the sparse representation problem with Generalized Gaussian noise from a new angle: we estimate the unknown shape parameter $p$ of the noise distribution, not only compute the sparse representation. The procedure alternates between computing sparse representations for the current $p$ and re-estimating $p$ based on the empirical representation residual, until convergence. As basic sparse representation algorithm we propose a version of Feasibility Pump and show that it gives better results than $\ell_{p}$ versions of Orthogonal Matching Pursuit and $\ell_{1}$ regularization. The results are comparable to those of the algorithms that know the true shape parameter value.