Going beyond linear dependencies to unveil connectivity of meshed grids

Partial correlations (PCs) are well suited for revealing linearly dependent (un)mediated connections in a graph when measurements (e.g., time courses) are available per node. Unfortunately, PC-based approaches to identifying the topology of a graph are less effective if nonlinear dependencies between given nodal measurements are present. To bypass this hurdle, nonlinear PCs relying on the ℓ2-norm regularized multi-kernel ridge regression (MKRR) have been recently proposed for brain network connectivity analysis. However, ℓ2-norm regularization limits the flexibility in combining kernels, which can compromise performance. For this reason, the present paper broadens the nonlinear PC approach to account for general ℓp-norm regularized MKRR, in which the user-selected parameter p ≥ 1 is attuned to the problem at hand. Aiming at a scalable algorithm, the Frank-Wolfe iterations are invoked to solve the ℓp-norm based MKRR, which not only features simple closed-form updates, but it is also fast convergent. The end result is a novel scheme that leverages nonlinear dependencies captured by the generalized PC model to identify the topology of not only radial but also meshed autonomous energy grids. Improved performance is achieved at affordable computational complexity relative to existing alternatives. Simulated tests showcase the merits of the proposed schemes.