  The problem of infering the top component of a noisy sample covariance matrix
with prior information about the distribution of its entries is considered, in
the framework of the spiked covariance model. Using the replica method of
statistical physics the computation of the overlap between the top components
of the sample and population covariance matrices is formulated as an explicit
optimization problem for any kind of entry-wise prior information. The approach
is illustrated on the case of top components including large entries, and the
corresponding phase diagram is shown. The calculation predicts that the maximal
sampling noise level at which the recovery of the top population component
remains possible is higher than its counterpart in the spiked covariance model
with no prior information.
