Joint Learning of Full-structure Noise in Hierarchical Bayesian Regression Models

Abstract

We consider the reconstruction of brain activity from electroencephalography (EEG). This inverse problem can be formulated as a linear regression with independent Gaussian scale mixture priors for both the source and noise components. Crucial factors influencing accuracy of source estimation are not only the noise level but also its correlation structure, but existing approaches have not addressed estimation of noise covariance matrices with full structure. To address this shortcoming, we develop hierarchical Bayesian (type-II maximum likelihood) models for observations with latent variables for source and noise, which are estimated jointly from data. As an extension to classical sparse Bayesian learning (SBL), where across-sensor observations are assumed to be independent and identically distributed, we consider Gaussian noise with full covariance structure. Using the majorization-maximization framework and Riemannian geometry, we derive an efficient algorithm for updating the noise covariance along the manifold of positive definite matrices. We demonstrate that our algorithm has guaranteed and fast convergence and validate it in simulations and with real MEG data. Our results demonstrate that the novel framework significantly improves upon state-of-the-art techniques in the real-world scenario where the noise is indeed non-diagonal and fully-structured. Our method has applications in many domains beyond biomagnetic inverse problems.

Robust Evaluation and Comparison of EEG Source Localization Algorithms for Accurate Reconstruction of Deep Cortical Activity

“…On the other hand, eLORETA (exact low-resolution brain electromagnetic tomography) further applies weighting for different sources and adopts regularization in its search for the most appropriate weighted minimum norm solution [20]. For eLORETA, we employ a weighted L2-norm minimization with 5% regularization [11].…”

“…EEG source localization is the process of estimating the electrical activity based on EEG signals recorded from the scalp. Typically, this is accomplished by using a linear regression model, which is commonly represented as [11,12]:…”

“…Recently, there have been advancements in learning algorithms for EEG source localization, including full Dugh, thin Dugh, and Mackay. These state-of-the-art algorithms utilize empirical Bayesian frameworks with fully updatable parameters, which give them an advantage over the theoretical designs of classical algorithms [10,11].…”

“…Both the full Dugh and thin Dugh algorithms integrate spatiotemporal dynamics by incorporating temporal covariance in their Type-II Maximum Likelihood (ML) estimation Bayesian learning processes [11]. These algorithms separate the spatial and temporal covariance of the EEG source and noise by assuming that neuronal sources from different brain regions exhibit similar autocorrelation spectra.…”

“…Although full Dugh, thin Dugh, and Mackay assert their superiority over classical source localization algorithms [11], their capability to accurately estimate source activity has not been thoroughly validated. This study addresses this gap by conducting a comprehensive mathematical analysis of the methodologies employed by these algorithms.…”