In this work, we examine the prevalent use of Frobenius error minimization in covariance matrix cleaning. Currently, minimizing the Frobenius error offers a limited interpretation within information theory. To better understand this relationship, we focus on the Kullback-Leibler divergence as a measure of the information lost by the optimal estimators. Our analysis centers on rotationally invariant estimators for data following an inverse Wishart population covariance matrix, and we derive an analytical expression for their Kullback-Leibler divergence. Due to the intricate nature of the calculations, we use genetic programming regressors paired with human intuition. Ultimately, we establish a more defined link between the Frobenius error and information theory, showing that the former corresponds to a first-order expansion term of the Kullback-Leibler divergence.

翻译：本研究探讨了协方差矩阵清洗中广泛使用的Frobenius误差最小化方法。当前，Frobenius误差最小化在信息论框架下的解释较为有限。为深入理解二者关系，我们聚焦于Kullback-Leibler散度作为最优估计器信息损失的度量标准。针对服从逆Wishart总体协方差矩阵的数据，我们重点分析了旋转不变估计器，并推导出其Kullback-Leibler散度的解析表达式。考虑到计算的复杂性，我们采用遗传规划回归器结合人类直觉进行分析。最终，我们建立了Frobenius误差与信息论之间更明确的联系，证明前者对应Kullback-Leibler散度的一阶展开项。