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

TL;DR

This paper explores how hierarchical Bayesian models and MCMC sampling can enhance majorization-minimization techniques for non-convex sparse regression, particularly improving source imaging in M/EEG by better initialization and uncertainty quantification.

Contribution

It establishes a connection between hierarchical Bayesian modeling and MM, and introduces MCMC-based methods to improve optimization and uncertainty analysis in non-convex sparse regression.

Findings

MCMC samples improve MM initialization for better local minima

Posterior modes reveal neural source configurations in M/EEG

Sensor type and number influence uncertainty in source estimates

Abstract

Majorization-minimization (MM) is a standard iterative optimization technique which consists in minimizing a sequence of convex surrogate functionals. MM approaches have been particularly successful to tackle inverse problems and statistical machine learning problems where the regularization term is a sparsity-promoting concave function. However, due to non-convexity, the solution found by MM depends on its initialization. Uniform initialization is the most natural and often employed strategy as it boils down to penalizing all coefficients equally in the first MM iteration. Yet, this arbitrary choice can lead to unsatisfactory results in severely under-determined inverse problems such as source imaging with magneto- and electro-encephalography (M/EEG). The framework of hierarchical Bayesian modeling (HBM) is an alternative approach to encode sparsity. This work shows that for certain hierarchical models, a simple alternating 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 improve upon these MM techniques by probing the multimodal posterior density using Markov Chain Monte-Carlo (MCMC) techniques. Firstly, we show that these samples can provide well-informed initializations that help MM schemes to reach better local minima. Secondly, we demonstrate how it can reveal the different modes of the posterior distribution in order to explore and quantify the inherent uncertainty and ambiguity of such ill-posed inference procedure. In the context of 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 type of sensors affect the uncertainties on the estimates.