In optimal covariance cleaning theory, minimizing the Frobenius norm between the true population covariance matrix and a rotational invariant estimator is a key step. This estimator can be obtained asymptotically for large covariance matrices, without knowledge of the true covariance matrix. In this study, we demonstrate that this minimization problem is equivalent to minimizing the loss of information between the true population covariance and the rotational invariant estimator for normal multivariate variables. However, for Student's t distributions, the minimal Frobenius norm does not necessarily minimize the information loss in finite-sized matrices. Nevertheless, such deviations vanish in the asymptotic regime of large matrices, which might extend the applicability of random matrix theory results to Student's t distributions. These distributions are characterized by heavy tails and are frequently encountered in real-world applications such as finance, turbulence, or nuclear physics. Therefore, our work establishes a connection between statistical random matrix theory and estimation theory in physics, which is predominantly based on information theory.

翻译：在最优协方差清理理论中，最小化真实总体协方差矩阵与旋转不变估计量之间的Frobenius范数是一个关键步骤。该估计量可在无需真实协方差矩阵先验知识的情况下，针对大型协方差矩阵渐近获得。在本研究中，我们证明，对于正态多元变量，该最小化问题等价于最小化真实总体协方差与旋转不变估计量之间的信息损失。然而，对于学生t分布，在有限尺寸矩阵中，最小化Frobenius范数未必能最小化信息损失。尽管如此，这种偏差在大型矩阵的渐近极限下会消失，这可能将随机矩阵理论结果的适用性扩展到学生t分布。这些分布以重尾为特征，常见于金融、湍流或核物理等实际应用场景。因此，我们的工作建立了统计随机矩阵理论与物理学中主要基于信息论的估计理论之间的联系。