Multiple penalized least squares and sign constraints with modified Newton-Raphson algorithms: application to EEG source imaging

Multiple penalized least squares (MPLS) models are a flexible approach to find adaptive least squares solutions required to be simultaneously sparse and smooth. This is particularly important when addressing real-life inverse problems where there is no ground truth available, such as electrophysiological source imaging. In this work we formalize a modified Newton-Raphson (MNR) algorithm to estimate general MPLS models and propose its extension to perform efficient optimization over the active set of selected features (AMNR). This algorithm can be used to minimize continuously differentiable objective functions with multiple restrictions, including sign constraints. We show that these algorithms provide solutions with acceptable reconstruction in simulated scenarios that do not cope with model assumptions, and for low n/p ratios. We then use both algorithms for estimating different electroencephalography (EEG) inverse models with multiple penalties. We also show how the AMNR allows us to estimate new models in the EEG inverse problem context, such as nonnegative versions of Smooth Garrote and Smooth LASSO. Synthetic data were used for a comparison between solutions obtained using the two algorithms proposed here and the least angle regression (LARS) algorithm, according to well-known quality measures. A visual event-related EEG from healthy young subjects and a resting-state EEG study on the relationship between cognitive ageing and walking speed decline in active elders, were used to illustrate the usefulness of the proposed methods in the analysis of real experimental data.