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.
On the Computation and Applications of Large Dense Partial Correlation Networks