  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.
