Overcomplete representation in a hierarchical Bayesian framework

<p style='text-indent:20px;'>A common task in inverse problems and imaging is finding a solution that sparse, the sense most of its components vanish. In framework compressed sensing, general results guaranteeing exact recovery have been proven. practice, sparse solutions are often computed combining <inline-formula><tex-math id="M1">\begin{document}$ \ell_1 $\end{document}</tex-math></inline-formula>-penalized least squares optimization with an appropriate numerical scheme to accomplish task. A computationally efficient alternative for linear provided by Bayesian hierarchical models, which sparsity encoded defining conditionally Gaussian prior model parameter obeying generalized gamma distribution. An iterative alternating sequential (IAS) algorithm has demonstrated lead scheme, combined Krylov subspace iterations early termination condition, approach particularly well suited large scale problems. Here extended whose allows coding overcomplete system such as composite frames. It shown among multiple possible representations unknown, IAS algorithm, particular, hybrid version it, effectively identifying solution. Computed examples show method not only traditional applications but also dictionary learning machine learning.</p>