Linear Shrinkage Estimation of Covariance Matrices Using Low-Complexity Cross-Validation

TL;DR

This paper introduces low-complexity, data-driven cross-validation methods for optimal linear shrinkage of covariance matrices, improving estimation accuracy especially in low-sample scenarios.

Contribution

It proposes analytically tractable LOOCV techniques for automatic shrinkage coefficient selection, applicable to various targets and array signal processing tasks.

Findings

LOOCV achieves near-oracle performance with sample covariance matrices

Methods are computationally efficient with small optimization problems

Applicable to multiple shrinkage targets and OLS-based covariance estimators

Abstract

Shrinkage can effectively improve the condition number and accuracy of covariance matrix estimation, especially for low-sample-support applications with the number of training samples smaller than the dimensionality. This paper investigates parameter choice for linear shrinkage estimators. We propose data-driven, leave-one-out cross-validation (LOOCV) methods for automatically choosing the shrinkage coefficients, aiming to minimize the Frobenius norm of the estimation error. A quadratic loss is used as the prediction error for LOOCV. The resulting solutions can be found analytically or by solving optimization problems of small sizes and thus have low complexities. Our proposed methods are compared with various existing techniques. We show that the LOOCV method achieves near-oracle performance for shrinkage designs using sample covariance matrix (SCM) and several typical shrinkage targets. Furthermore, the LOOCV method provides low-complexity solutions for estimators that use general shrinkage targets, multiple targets, and/or ordinary least squares (OLS)-based covariance matrix estimation. We also show applications of our proposed techniques to several different problems in array signal processing.