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

Bekhti, Y; Lucka, F; Salmon, J; Gramfort, A; (2018) A hierarchical Bayesian perspective on majorization-minimization for non-convex sparse regression: application to M/EEG source imaging. Inverse Problems , 34 (8) , Article 085010. 10.1088/1361-6420/aac9b3. Green open access

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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 (see 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.

Type: Article
Title: A hierarchical Bayesian perspective on majorization-minimization for non-convex sparse regression: application to M/EEG source imaging
Open access status: An open access version is available from UCL Discovery
DOI: 10.1088/1361-6420/aac9b3
Publisher version: https://doi.org/10.1088/1361-6420/aac9b3
Language: English
Additional information: Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence(http://creativecommons.org/licenses/by/3.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Keywords: majorization-minimization, hierarchical Bayesian modeling, sparse models, M/EEG inverse problems, uncertainty quantification, MCMC sampling
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10052592
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