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.

翻译：本文考虑从带有任意加性双旋转不变扰动的噪声观测中估计矩阵的问题。我们提出了一种在矩形旋转不变估计器类中最优的估计器，该估计器可应用于任何信号先验。针对高斯噪声的特殊情况，我们证明了所提估计器的最优性，并基于观测矩阵的极限奇异值分布得到了MMSE的显式表达式。此外，我们证明了一个将渐近互信息与矩形矩阵的对数球体积积分极限联系起来的公式。我们还对一般双旋转不变噪声及高斯噪声情况下的结果进行了数值验证，数值结果与理论预测一致。