  The ability of many powerful machine learning algorithms to deal with large
data sets without compromise is often hampered by computationally expensive
linear algebra tasks, of which calculating the log determinant is a canonical
example. In this paper we demonstrate the optimality of Maximum Entropy methods
in approximating such calculations. We prove the equivalence between mean value
constraints and sample expectations in the big data limit, that Covariance
matrix eigenvalue distributions can be completely defined by moment information
and that the reduction of the self entropy of a maximum entropy proposal
distribution, achieved by adding more moments reduces the KL divergence between
the proposal and true eigenvalue distribution. We empirically verify our
results on a variety of SparseSuite matrices and establish best practices.
