Sparse optimization for estimating the cross-power spectrum in linear inverse models : from theory to the application in brain connectivity

In this work we present a computationally efficient linear optimization approach for estimating the cross--power spectrum of an hidden multivariate stochastic process from that of another observed process. Sparsity in the resulting estimator of the cross--power is induced through $\ell_1$ regularization and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is used for computing such an estimator. With respect to a standard implementation, we prove that a proper initialization step is sufficient to guarantee the required symmetric and antisymmetric properties of the involved quantities. Further, we show how structural properties of the forward operator can be exploited within the FISTA update in order to make our approach adequate also for large--scale problems such as those arising in context of brain functional connectivity. The effectiveness of the proposed approach is shown in a practical scenario where we aim at quantifying the statistical relationships between brain regions in the context of non-invasive electromagnetic field recordings. Our results show that our method provide results with an higher specificity that classical approaches based on a two--step procedure where first the hidden process describing the brain activity is estimated through a linear optimization step and then the cortical cross--power spectrum is computed from the estimated time--series.