Fast ℓ 1 -regularized EEG source localization using variable projection
Jack Michael Solomon, Rosemary Anne Renaut, Matthias Chung · Inverse Problems · 2025
Abstract Electroencephalograms (EEG) are invaluable for treating neurological disorders, however, mapping EEG electrode readings to brain activity requires solving a challenging inverse problem. For time series data, the use of ℓ 1 regularization quickly becomes intractable for many solvers, and, despite the reconstruction advantages of ℓ 1 regularization, ℓ 2 -based approaches such as standardized low-resolution brain electromagnetic tomography 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 augmented Lagrangian 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. Leveraging the efficiency of the proposed VPAL algorithm, we introduce a windowed variation, VPAL W , that computes time dynamics in sequence suitable for real-time reconstruction. Our proposed methods are compared to state-of-the-art approaches including sLORETA and other methods for ℓ 1 -regularized inverse problems.