  We study low-rank matrix estimation for a generic inhomogeneous output
channel through which the matrix is observed. This generalizes the commonly
considered spiked matrix model with homogeneous noise to include for instance
the dense degree-corrected stochastic block model. We adapt techniques used to
study multispecies spin glasses to derive and rigorously prove an expression
for the free energy of the problem in the large size limit, providing a
framework to study the signal detection thresholds. We discuss an application
of this framework to the degree corrected stochastic block models.
