Rectangular rotational invariant estimator for high-rank matrix estimation

We consider estimating a matrix from noisy observations coming from an arbitrary additive bi-rotational invariant perturbation. We propose an estimator, which we conjecture 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 asymptotic optimality of the proposed estimator and find an explicit expression for the minimum mean square error in terms of the limiting singular value distribution of the observation matrix. Moreover, we prove a formula linking the asymptotic mutual information under Gaussian noise to 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.