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

TL;DR

This work reframes EEG source localization as a large-scale generalized elastic net problem regularized by graph total variation over the cortical mesh and time-dynamics, enabling sparse and smooth reconstructions. It introduces VPAL, a variable projection augmented Lagrangian method with nonlinear conjugate gradient updates, and a windowed variant VPAL_W for near real-time sequential reconstruction; the authors prove convergence for broad convex/non-smooth settings and demonstrate strong scalability against ADMM and FISTA, with real-time potential on large-scale meshes. Empirical results show competitive accuracy with substantially faster runtimes, while ADMM struggles at scale and sLORETA remains fastest for very small problems; VPAL_W stands out for large datasets and streaming data scenarios. The proposed approach broadens the practical use of sparsity-promoting regularization in EEG inverse problems and extends to other graph-based large-scale inverse problems, offering a viable path to real-time brain activity monitoring and broader applications in graph tomography.

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.

Figures (7)

• Figure 1: Illustration of the difference between the standard total variation operator TV for images in the left panel and our implementation of graphTV on the right. Note that in the image, the total variation is calculated as the difference between the middle pixel, $x_{ij}$ and adjacent pixels. This is defined analogously for graphs where adjacency is interpreted as having an edge instead.
• Figure 2: Visualization of sLORETA, ADMM, FISTA, VPAL and VPAL$_{\tt W}$ reconstructions, compared against ground truth comparison dataset, $\bfX_{\textnormal{true}}^{\textnormal{C}}$. An approximately optimal regularization parameter, $\lambda = 0.0059$, was used for the sLORETA reconstruction.
• Figure 3: Relative error and residual plot against time for VPAL, ADMM and FISTA (including the computations of residuals and errors in contrast to values reported in \ref{['table:cres']}).
• Figure 4: Visualization of ADMM, FISTA, VPAL and VPAL$_{\tt W}$ reconstructions, compared against ground truth scalability datasets, $\bfX_{\textnormal{true}}^{\textnormal{S}}$, $\bfX_{\textnormal{true}}^{\textnormal{M}}$, $\bfX_{\textnormal{true}}^{\textnormal{L}}$. Visualization is shown for the first of the three trial datasets. Since most interesting dynamics occur before $t = 40$, later time points are not shown.
• Figure 5: Average runtime comparison of ADMM, VPAL and VPAL$_{\tt W}$ as $n$ is increased. Error bars correspond to the standard deviation of runtime across three unique datasets.

Theorems & Definitions (10)

• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• theorem 1
• proof
• theorem 2
• proof