An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography

TL;DR

This paper introduces an explicit iterative algorithm for sparse, constrained least squares problems, demonstrating its application to divergence-free current reconstruction in magneto-encephalography with promising results.

Contribution

It presents a fully explicit iterative method for $\, ext{l}_1$-penalized constrained minimization, applicable even when involved matrices are non-invertible.

Findings

Effective reconstruction of divergence-free currents in synthetic magneto-encephalography data.

Imposing zero divergence and joint sparsity influences reconstruction quality.

Algorithm converges in finite-dimensional settings.

Abstract

We propose an iterative algorithm for the minimization of a $\ell_1$-norm penalized least squares functional, under additional linear constraints. The algorithm is fully explicit: it uses only matrix multiplications with the three matrices present in the problem (in the linear constraint, in the data misfit part and in penalty term of the functional). None of the three matrices must be invertible. Convergence is proven in a finite-dimensional setting. We apply the algorithm to a synthetic problem in magneto-encephalography where it is used for the reconstruction of divergence-free current densities subject to a sparsity promoting penalty on the wavelet coefficients of the current densities. We discuss the effects of imposing zero divergence and of imposing joint sparsity (of the vector components of the current density) on the current density reconstruction.