  We consider estimating a matrix from noisy observations coming from an
arbitrary additive bi-rotational invariant perturbation. We propose an
estimator which is optimal among the class of rectangular rotational invariant
estimators and can be applied irrespective of the prior on the signal. For the
particular case of Gaussian noise, we prove the optimality of the proposed
estimator, and we find an explicit expression for the MMSE in terms of the
limiting singular value distribution of the observation matrix. Moreover, we
prove a formula linking the asymptotic mutual information and the limit of a
log-spherical integral of rectangular matrices. We also provide numerical
checks for our results for general bi-rotational invariant noise, as well as
Gaussian noise, which match our theoretical predictions.
