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

TL;DR

This paper introduces a fast, convergent algorithm for EEG source localization that efficiently handles $ ext{l}_1$ regularization, enabling real-time brain activity reconstruction from EEG data.

Contribution

The paper proposes a novel variable projected algorithm (VPAL) for $ ext{l}_1$-regularized EEG source localization, with proven convergence and real-time reconstruction capabilities.

Findings

VPAL outperforms existing methods in speed and accuracy.

The algorithm is suitable for real-time EEG source reconstruction.

Compared to sLORETA, VPAL provides more precise localization.

Abstract

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.