Joint Learning of Full-structure Noise in Hierarchical Bayesian Regression Models
Keywords: Full-structure Noise, Hierarchical Bayesian Regression Models, Sparse Bayesian Learning, Unsupervised Learning, Brain Source Imaging, Covariance Estimation.
Abstract: We consider hierarchical Bayesian (type-II maximum likelihood) models for observations with latent variables for source and noise, where both hyperparameters need to be estimated jointly from data. This problem has application in many domains in imaging including biomagnetic inverse problems. 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. Here, 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. As a departure from classical sparse Bayesan learning (SBL) models where across-sensor observations are assumed to be independent and identically distributed, we consider Gaussian noise with full covariance structure. Using Riemannian geometry, we derive an efficient algorithm for updating both source and noise covariance along the manifold of positive definite matrices. Using the majorization-maximization framework, we demonstrate that our algorithm has guaranteed and fast convergence. We validate the algorithm both in simulations and with real 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.
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