Hyperparameter estimation in maximum a posteriori regression using group sparsity with an application to brain imaging

Hyperparameter estimation is a recurrent problem in the signal and statistics literature. Popular strategies are crossvalidation or Bayesian inference, yet it remains an active topic of research in order to offer better or faster algorithms. The models considered here are sparse regression models with convex or non-convex group-Lasso-like penalties. Following the recent work of Pereyra et al. [1] we study the fixed point iteration algorithm they propose and show that, while it may be suitable for an analysis prior, it suffers from limitations when using high-dimensional sparse synthesis models. The first contribution of this paper is to show how to overcome this issue. Secondly, we demonstrate how one can extend the model to estimate a vector of regularization parameters. We illustrate this on models with group sparsity reporting improved support recovery and reduced amplitude bias on the estimated coefficients. This approach is compared with an alternative method that uses a single parameter but a non-convex penalty. Results are presented on simulations and an inverse problem relevant for neuroscience which is the localization of brain activations using magneto/electroencephalography.