  In this manuscript we consider denoising of large rectangular matrices: given
a noisy observation of a signal matrix, what is the best way of recovering the
signal matrix itself? For Gaussian noise and rotationally-invariant signal
priors, we completely characterize the optimal denoiser and its performance in
the high-dimensional limit, in which the size of the signal matrix goes to
infinity with fixed aspects ratio, and under the Bayes optimal setting, that is
when the statistician knows how the signal and the observations were generated.
Our results generalise previous works that considered only symmetric matrices
to the more general case of non-symmetric and rectangular ones. We explore
analytically and numerically a particular choice of factorized signal prior
that models cross-covariance matrices and the matrix factorization problem. As
a byproduct of our analysis, we provide an explicit asymptotic evaluation of
the rectangular Harish-Chandra-Itzykson-Zuber integral in a special case.
