  The introduction of graph theory in neuroimaging has pro- vided invaluable
tools for the study of brain connectivity. These methods require the definition
of a graph, which is typically derived by estimating the effective connectivity
between brain regions through the optimization of an ill-posed inverse problem.
Considerable efforts have been devoted to the development of methods extracting
sparse connectivity graphs. The present paper aims at highlighting the benefits
of an alternative ap- proach. We investigate low-rank L2 regularized matrices
recently intro- duced under the denomination of Riccati regularized precision
matrices. We demonstrate their benefits for the analysis of cortical thickness
map and for the extraction of functional biomarkers from resting state fMRI
scans. In addition, we explain how speed and result quality can be further
improved with random projections. The promising results obtained using the
Human Connectome Project dataset as well as the numerous possi- ble extensions
and applications suggest that Riccati precision matrices might usefully
complement current sparse approaches.
