A regularized matrix factorization approach to induce structured sparse-low-rank solutions in the EEG inverse problem

Abstract

We consider the estimation of the Brain Electrical Sources (BES) matrix from noisy electroencephalographic (EEG) measurements, commonly named as the EEG inverse problem. We propose a new method to induce neurophysiological meaningful solutions, which takes into account the smoothness, structured sparsity, and low rank of the BES matrix. The method is based on the factorization of the BES matrix as a product of a sparse coding matrix and a dense latent source matrix. The structured sparse-low-rank structure is enforced by minimizing a regularized functional that includes the 21 -norm of the coding matrix and the squared Frobenius norm of the latent source matrix. We develop an alternating optimization algorithm to solve the resulting nonsmooth-nonconvex minimization problem. We analyze the convergence of the optimization procedure, and we compare, under different synthetic scenarios, the performance of our method with respect to the Group Lasso and Trace Norm regularizers when they are applied directly to the target matrix.

Granger Causality Inference in EEG Source Connectivity Analysis: A State-Space Approach

“…Our assumption on regarding inactive source from sparse rows in C is in agreement with the use of penalty (15) by [MMARPH14]. However, [MMARPH14] did not model a dynamic of z but rather estimated z(t) as a whole time series segment whereas our subspace approach estimates system parameters but not all related signals directly. The state-space model (4)-(5) by [YYR16] was close to our model (6a)-(6c) but the description of state variable (a time series in ROI level) and the system parameter P (a binary matrix) were different from ours.…”

“…Our assumption on inferring inactive sources from sparse rows in C is in agreement with the use of penalty (15) by [MMARPH14]. However, [MMARPH14] did not model source dynamics but rather estimated x ( t ) as a whole time series segment with a prior that some components of x i ( t )’s are entirely zero.…”

“…As a result, the source imaging problem becomes an underdetermined problem. [MMARPH14] proposed that the source time series matrix is factorized into coding matrix C and a latent source time series z ( t ), then x ( t ) = Cz ( t ) where C is assumed to be sparse. The relationship between sources and sensors is then explained by The problem of reconstructing x is now to estimate z and C instead.…”

“…The relationship between sources and sensors is then explained by The problem of reconstructing x is now to estimate z and C instead. [MMARPH14] applied an 𝓁 2,1 regularization method by penalizing the rows of the matrix with the 2-norm to induce a sparsity pattern in source time series. Then the regularized EEG inverse problem with 𝓁 2,1 -norm penalty term was proposed as The problem is non-convex in C and Z (the matrix of latent time series.)…”

Granger Causality Inference in EEG Source Connectivity Analysis: A State-Space Approach

“…As a result, the source imaging problem becomes an underdetermined problem. [MMARPH14] proposed that the source time series matrix is factorized into coding matrix C and a latent source time series z(t), then x(t) = Cz(t) where C is assumed to be sparse. The relationship between sources and sensors is then explained by y(t) = LCz(t) + v(t).…”