AFSR: An Adaptive Factor Score Regression Framework for High-Dimensional Correlation Matrix Estimation

The accurate estimation of correlation matrices is a foundational challenge in high-dimensional statistics. The sample correlation matrix, while unbiased, suffers from high variance when the number of variables p is large relative to the sample size n, leading to unstable estimates and poor performance in downstream applications. This paper introduces the adaptive factor-informed shrinkage regression (AFSR) framework, a novel methodology that moves beyond the traditional binary choice between the noisy sample matrix and potentially biased structured estimators. The AFSR estimator is defined as a convex combination of the sample correlation matrix and a low-rank factor-informed target, forming a continuum of estimators that accommodates both exact and approximate factor model assumptions. The key innovation is a cross-validation procedure to select the optimal shrinkage intensity, effectively approximating the oracle estimator that minimizes the mean squared error. Connections to Ridge-type shrinkage, Ledoit–Wolf methods, POET, and nuclear-norm regularization are discussed, positioning AFSR within the broader landscape of regularization approaches. We demonstrate that this adaptive approach consistently outperforms established benchmarks, including the sample matrix, the model-implied factor analysis matrix, Ledoit–Wolf shrinkage, POET, and graphical lasso, across a wide range of high-dimensional simulation scenarios. The practical utility of the AFSR framework is further validated through applications to real-world datasets from genomics and finance, showcasing its robustness to non-normality and automatic adaptability to underlying data structures.