Fast $\ell_1$-Regularized EEG Source Localization Using Variable Projection

Electroencephalograms (EEG) are invaluable for treating neurological disorders, however, mapping EEG electrode readings to brain activity requires solving a challenging inverse problem. Due to the time series data, the use of $\ell_1$ regularization quickly becomes intractable for many solvers, and, despite the reconstruction advantages of $\ell_1$ regularization, $\ell_2$-based approaches such as sLORETA are used in practice. In this work, we formulate EEG source localization as a graphical generalized elastic net inverse problem and present a variable projected algorithm (VPAL) suitable for fast EEG source localization. We prove convergence of this solver for a broad class of separable convex, potentially non-smooth functions subject to linear constraints and include a modification of VPAL that reconstructs time points in sequence, suitable for real-time reconstruction. Our proposed methods are compared to state-of-the-art approaches including sLORETA and other methods for $\ell_1$-regularized inverse problems.