Paper: arXiv 2304.14098

Authors: Christian Bongiorno, Marco Berritta

Abstract

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical bridge between Random Matrix Theory and Information Theory for heavy-tailed distributions using advanced mathematical formulations. However, it lacks empirical validation and backtesting on real-world market datasets, remaining primarily a theoretical contribution.

Research Flowchart

  flowchart TD
    A["Research Goal: Link Covariance Cleaning<br>to Information Theory for Heavy-Tailed Data"] --> B["Input Data: Normal & Student's t<br>Multivariate Distributions"]
    B --> C["Method: Minimize Frobenius Norm vs.<br>Information Loss (Kullback-Leibler)"]
    C --> D["Computational Process: Evaluate Estimators<br>in Finite vs. Asymptotic Regimes"]
    D --> E{"Are Regimes Asymptotic<br>(Large Matrices)?"}
    E -- "Normal Distribution" --> F["Outcomes: Frobenius Minimization<br>Equivalents Information Loss"]
    E -- "Student's t (Finite)" --> G["Outcomes: Frobenius Deviates<br>from Optimal Information Loss"]
    E -- "Student's t (Asymptotic)" --> F