A Weighted Approach for Sparse Signal Support Estimation with Application to EEG Source Localization

In sparse signal recovery problems, <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula> -norm minimization is typically used as an alternative to more complex <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula>-norm minimization. The range space property (RSP) provides the conditions under which the least <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula>-norm solution is equal to at most one of the least <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula>-norm solutions. These conditions depend on the sensing matrix and the support of the underlying sparse solution. In this paper, we address the problem of recovering sparse signals by weighting the corresponding sensing matrix with a diagonal matrix. We show that by appropriately choosing the weights, we can formulate an <inline-formula> <tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula>-norm minimization problem that satisfies the RSP, even if the original problem does not. By solving the weighted problem we can obtain the support of the original problem. We provide the conditions which the weights must satisfy, for both noise free and noisy cases. Although the precise conditions involve information about the support of the sparse vector, the class of good weights is very wide, and in most cases encompasses an estimate of the underlying vector obtained via a conventional method, i.e., a method that does not encourage sparsity. The proposed approach is a good candidate for Electroencephalography (EEG) sparse source localization, where the corresponding sensing matrix has high coherence. The performance of the proposed approach is evaluated via simulations and also via experiments on localizing active sources in the brain corresponding to an auditory task from EEG recordings of a human subject.