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

Abstract

Majorization-minimization (MM) is a standard iterative optimization technique\nwhich consists in minimizing a sequence of convex surrogate functionals. MM\napproaches have been particularly successful to tackle inverse problems and\nstatistical machine learning problems where the regularization term is a\nsparsity-promoting concave function. However, due to non-convexity, the\nsolution found by MM depends on its initialization. Uniform initialization is\nthe most natural and often employed strategy as it boils down to penalizing all\ncoefficients equally in the first MM iteration. Yet, this arbitrary choice can\nlead to unsatisfactory results in severely under-determined inverse problems\nsuch as source imaging with magneto- and electro-encephalography (M/EEG). The\nframework of hierarchical Bayesian modeling (HBM) is an alternative approach to\nencode sparsity. This work shows that for certain hierarchical models, a simple\nalternating scheme to compute fully Bayesian maximum a posteriori (MAP)\nestimates leads to the exact same sequence of updates as a standard MM strategy\n(cf. the Adaptive Lasso). With this parallel outlined, we show how to improve\nupon these MM techniques by probing the multimodal posterior density using\nMarkov Chain Monte-Carlo (MCMC) techniques. Firstly, we show that these samples\ncan provide well-informed initializations that help MM schemes to reach better\nlocal minima. Secondly, we demonstrate how it can reveal the different modes of\nthe posterior distribution in order to explore and quantify the inherent\nuncertainty and ambiguity of such ill-posed inference procedure. In the context\nof M/EEG, each mode corresponds to a plausible configuration of neural sources,\nwhich is crucial for data interpretation, especially in clinical contexts.\nResults on both simulations and real datasets show how the number or the type\nof sensors affect the uncertainties on the estimates.\n