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 more 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. Due to its similarity to the least angle regression (LARS) algorithm, synthetic data were used for a preliminary comparison between solutions obtained using both AMNR and LARS for the same models, according to well-known quality measures. A visual event-related EEG from healthy young subjects and a resting-state EEG study associated to walking speed decline in active elders were used to illustrate its usefulness in the analysis of real experimental data.