  While sparse inverse covariance matrices are very popular for modeling
network connectivity, the value of the dense solution is often overlooked. In
fact the L2-regularized solution has deep connections to a number of important
applications to spectral graph theory, dimensionality reduction, and
uncertainty quantification. We derive an approach to directly compute the
partial correlations based on concepts from inverse problem theory. This
approach also leads to new insights on open problems such as model selection
and data preprocessing, as well as new approaches which relate the above
application areas.
