We propose a rectangular rotational invariant estimator to recover a real matrix from noisy matrix observations coming from an arbitrary additive rotational invariant perturbation, in the large dimension limit. Using the Bayes-optimality of this estimator, we derive the asymptotic minimum mean squared error (MMSE). For the particular case of Gaussian noise, 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 log-spherical integral of rectangular matrices. We also provide numerical checks for our results, which match our theoretical predictions and known Bayesian inference results.

翻译：我们提出了一种矩形旋转不变估计器，用于从任意加性旋转不变扰动产生的含噪矩阵观测中恢复实矩阵，在大维度极限下。利用该估计器的贝叶斯最优性，我们推导出了渐近最小均方误差（MMSE）。对于高斯噪声的特殊情况，我们得到了MMSE关于观测矩阵奇异值分布极限的显式表达式。此外，我们证明了一个公式，将渐近互信息与矩形矩阵的对数球体积分的极限联系起来。我们还对我们的结果进行了数值验证，这些结果与理论预测及已知的贝叶斯推断结果吻合。