Optimal Data Splitting for Holdout Cross-Validation in Large Covariance Matrix Estimation

ArXiv ID: 2503.15186 “View on arXiv”

Authors: Unknown

Abstract

Cross-validation is a statistical tool that can be used to improve large covariance matrix estimation. Although its efficiency is observed in practical applications and a convergence result towards the error of the non linear shrinkage is available in the high-dimensional regime, formal proofs that take into account the finite sample size effects are currently lacking. To carry on analytical analysis, we focus on the holdout method, a single iteration of cross-validation, rather than the traditional $k$-fold approach. We derive a closed-form expression for the expected estimation error when the population matrix follows a white inverse Wishart distribution, and we observe the optimal train-test split scales as the square root of the matrix dimension. For general population matrices, we connected the error to the variance of eigenvalues distribution, but approximations are necessary. In this framework and in the high-dimensional asymptotic regime, both the holdout and $k$-fold cross-validation methods converge to the optimal estimator when the train-test ratio scales with the square root of the matrix dimension which is coherent with the existing theory.

Keywords: Covariance Matrix Estimation, Cross-Validation, High-Dimensional Statistics, Nonlinear Shrinkage, Eigenvalue Distribution, Multi-Asset (Risk Management)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily mathematical, deriving closed-form error expressions under specific distributions (white inverse Wishart) and analyzing high-dimensional asymptotics with random matrix theory. Empirically, it is purely theoretical, focusing on analytical proofs and derivations with no backtests, data sets, or implementation details provided in the excerpt.

  flowchart TD
    A["Research Goal:<br>Formalize Cross-Validation<br>in Covariance Estimation"] --> B["Methodology:<br>Analyze Holdout Method &<br>High-Dim Asymptotics"]
    B --> C{"Data/Input:<br>White Inverse Wishart<br>Population Matrix"}
    C --> D["Computational Process:<br>Derive Closed-Form<br>Expected Estimation Error"]
    D --> E["Key Finding:<br>Optimal Split Ratio ~ sqrt(d)"]
    D --> F["Key Finding:<br>Convergence to Nonlinear<br>Shrinkage Estimator"]
    E --> G["Outcome:<br>Finite-sample proof for<br>High-dimensional CV efficiency"]
    F --> G