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Weighted Group Lasso for a static EEG problem

Ole Løseth Elvetun, Bjørn Fredrik Nielsen, Niranjana Sudheer

TL;DR

This work develops a weighted Group Lasso approach for the static EEG inverse problem by grouping the three orthogonal dipole components at each spatial location into a single unit, mitigating depth and angle biases without sacrificing sparsity. It provides concise recovery guarantees for both single and multiple group sources and clarifies the role of the weighting operator in theory versus practice. Through simulations using NY Head lead fields, the authors show that a truncated Moore-Penrose pseudoinverse weighting markedly improves localization and orientation accuracy compared to standard weighting. The results indicate that weighted group sparsity is a robust and physiologically plausible method for EEG source imaging, with potential for extension to dynamic or multi-modal settings.

Abstract

We investigate the weighted Group Lasso formulation for the static inverse electroencephalography (EEG) problem, aiming at reconstructing the unknown underlying neuronal sources from voltage measurements on the scalp. By modelling the three orthogonal dipole components at each location as a single coherent group, we demonstrate that depth bias and orientation bias can be effectively mitigated through the proposed regularization framework. On the theoretical front, we provide concise recovery guarantees for both single and multiple group sources. Our numerical experiments highlight that while theoretical bounds hold for a broad range of weight definitions, the practical reconstruction quality, for cases not covered by the theory, depends significantly on the specific weighting strategy employed. Specifically, employing a truncated Moore-Penrose pseudoinverse for the involved weighting matrix gives a small Dipole Localization Error (DLE). The proposed method offers a robust approach for inverse EEG problems, enabling improved spatial accuracy and a more physiologically realistic reconstruction of neural activity.

Weighted Group Lasso for a static EEG problem

TL;DR

This work develops a weighted Group Lasso approach for the static EEG inverse problem by grouping the three orthogonal dipole components at each spatial location into a single unit, mitigating depth and angle biases without sacrificing sparsity. It provides concise recovery guarantees for both single and multiple group sources and clarifies the role of the weighting operator in theory versus practice. Through simulations using NY Head lead fields, the authors show that a truncated Moore-Penrose pseudoinverse weighting markedly improves localization and orientation accuracy compared to standard weighting. The results indicate that weighted group sparsity is a robust and physiologically plausible method for EEG source imaging, with potential for extension to dynamic or multi-modal settings.

Abstract

We investigate the weighted Group Lasso formulation for the static inverse electroencephalography (EEG) problem, aiming at reconstructing the unknown underlying neuronal sources from voltage measurements on the scalp. By modelling the three orthogonal dipole components at each location as a single coherent group, we demonstrate that depth bias and orientation bias can be effectively mitigated through the proposed regularization framework. On the theoretical front, we provide concise recovery guarantees for both single and multiple group sources. Our numerical experiments highlight that while theoretical bounds hold for a broad range of weight definitions, the practical reconstruction quality, for cases not covered by the theory, depends significantly on the specific weighting strategy employed. Specifically, employing a truncated Moore-Penrose pseudoinverse for the involved weighting matrix gives a small Dipole Localization Error (DLE). The proposed method offers a robust approach for inverse EEG problems, enabling improved spatial accuracy and a more physiologically realistic reconstruction of neural activity.

Paper Structure

This paper contains 9 sections, 3 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. The Group Lasso formulation
  4. Application to EEG Source Localization
  5. Analysis
  6. Group pursuit
  7. Group Lasso
  8. Numerical experiments
  9. Conclusion

Key Result

Theorem 3.2

Assume that $\mathbf{x}^*$ is a single group source, i.e., $\textnormal{supp}(\mathbf{x}^*) = g^*$ for some $g^* \in G$. Then If, in addition, Assumption assumption_uniqueness is satisfied, then $\mathbf{x}^*$ is the unique solution of this single group pursuit problem.

Figures (3)

  • Figure 1: Scatter plots for the true source depth versus the estimated source depth. If all points are located on the dashed red line $x = y$, there is no depth bias. Simulations for two different choices of ${\mathsf{B}}$.
  • Figure 2: Two-source reconstruction with average DOE values of $0.0455$ rad $(2.60^\circ)$ and $0.1378$ rad $(7.89^\circ)$ when ${\mathsf{B}} = \mathsf{A}_k^\dagger$ and ${\mathsf{B}} = {\mathsf{I}}$, respectively.
  • Figure 3: Three-source reconstruction with average DOE values of $0.2985$ rad $(17.1^\circ)$ and $0.5931$ rad $(33.9^\circ)$ when ${\mathsf{B}} = \mathsf{A}_k^\dagger$ and ${\mathsf{B}} = {\mathsf{I}}$, respectively.

Theorems & Definitions (6)

  • Theorem 3.2: Single group source
  • proof
  • Theorem 3.4: Recovery of disjoint groups
  • proof
  • Theorem 3.5
  • proof