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In optimal covariance cleaning theory, minimizing the Frobenius norm betweenthe true population covariance matrix and a rotational invariant estimator is akey step. This estimator can be obtained asymptotically for large covariancematrices, without knowledge of the true covariance matrix. In this study, wedemonstrate that this minimization problem is equivalent to minimizing the lossof information between the true population covariance and the rotationalinvariant estimator for normal multivariate variables. However, for Student's tdistributions, the minimal Frobenius norm does not necessarily minimize theinformation loss in finite-sized matrices. Nevertheless, such deviations vanishin the asymptotic regime of large matrices, which might extend theapplicability of random matrix theory results to Student's t distributions.These distributions are characterized by heavy tails and are frequentlyencountered in real-world applications such as finance, turbulence, or nuclearphysics. Therefore, our work establishes a connection between statisticalrandom matrix theory and estimation theory in physics, which is predominantlybased on information theory.

Optimal Covariance Cleaning for Heavy-Tailed Distributions: Insights from Information Theory