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Majorization-minimization (MM) is a standard iterative optimization techniquewhich consists in minimizing a sequence of convex surrogate functionals. MMapproaches have been particularly successful to tackle inverse problems andstatistical machine learning problems where the regularization term is asparsity-promoting concave function. However, due to non-convexity, thesolution found by MM depends on its initialization. Uniform initialization isthe most natural and often employed strategy as it boils down to penalizing allcoefficients equally in the first MM iteration. Yet, this arbitrary choice canlead to unsatisfactory results in severely under-determined inverse problemssuch as source imaging with magneto- and electro-encephalography (M/EEG). Theframework of hierarchical Bayesian modeling (HBM) is an alternative approach toencode sparsity. This work shows that for certain hierarchical models, a simplealternating scheme to compute fully Bayesian maximum a posteriori (MAP)estimates leads to the exact same sequence of updates as a standard MM strategy(cf. the Adaptive Lasso). With this parallel outlined, we show how to improveupon these MM techniques by probing the multimodal posterior density usingMarkov Chain Monte-Carlo (MCMC) techniques. Firstly, we show that these samplescan provide well-informed initializations that help MM schemes to reach betterlocal minima. Secondly, we demonstrate how it can reveal the different modes ofthe posterior distribution in order to explore and quantify the inherentuncertainty and ambiguity of such ill-posed inference procedure. In the contextof M/EEG, each mode corresponds to a plausible configuration of neural sources,which is crucial for data interpretation, especially in clinical contexts.Results on both simulations and real datasets show how the number or the typeof sensors affect the uncertainties on the estimates.

A hierarchical Bayesian perspective on majorization-minimization for non-convex sparse regression: application to M/EEG source imaging