High-Dimensional Covariance Denoising and Geometric Robustness: A Statistical Framework for Portfolio Stability

Abstract

In high-dimensional financial settings, covariance matrix estimation suffers from the curse of dimensionality and parameter instability, which substantially limit the practical effectiveness of the Markowitz mean–variance framework. To address this issue, this paper constructs a "covariance estimation - portfolio strategy" collaborative framework, systematically examining the performance differences of four classic methods - sample covariance, shrinkage estimation, MCD robust estimation, and Tyler M-estimation in global minimum variance portfolios, mean-variance portfolios, and risk parity portfolios, to enhance the stability of investment portfolios in high-dimensional, noisy, and heavy-tailed environments. The empirical analysis covers three scenarios: ideal Gaussian distribution, contaminated data scenario, and resampling of the S&P 500 real market data, and uses estimation error, Sharpe ratio, maximum drawdown, Sortino ratio, and the time series stability of portfolio weights as evaluation metrics. Simulation experiments show that in the contaminated data environment, the estimation error of MCD and Tyler estimators is significantly lower than that of sample covariance and shrinkage estimation, demonstrating stronger robustness against outliers; real market backtesting further reveals that embedding robust covariance estimation into low-sensitivity optimization models such as risk parity and global minimum variance substantially reduces the time series fluctuations of portfolio weights and improves risk-adjusted returns, where the Risk Parity_Shrinkage and Risk Parity_Robust strategies outperform traditional benchmark portfolios in Sharpe ratio, maximum drawdown, and stability indicators. The research shows that the "estimator- strategy" adaptation framework is an effective paradigm for eliminating high-dimensional estimation errors and enhancing strategy resilience. This framework offers practical value for high-dimensional asset allocation, quantitative strategy development, and systemic risk prevention and mitigation.