  This paper considers the regularized estimation of covariance matrices (CM)
of high-dimensional (compound) Gaussian data for minimum variance
distortionless response (MVDR) beamforming. Linear shrinkage is applied to
improve the accuracy and condition number of the CM estimate for
low-sample-support cases. We focus on data-driven techniques that automatically
choose the linear shrinkage factors for shrinkage sample covariance matrix
($\text{S}^2$CM) and shrinkage Tyler's estimator (STE) by exploiting cross
validation (CV). We propose leave-one-out cross-validation (LOOCV) choices for
the shrinkage factors to optimize the beamforming performance, referred to as
$\text{S}^2$CM-CV and STE-CV. The (weighted) out-of-sample output power of the
beamfomer is chosen as a proxy of the beamformer performance and concise
expressions of the LOOCV cost function are derived to allow fast optimization.
For the large system regime, asymptotic approximations of the LOOCV cost
functions are derived, yielding the $\text{S}^2$CM-AE and STE-AE. In general,
the proposed algorithms are able to achieve near-oracle performance in choosing
the linear shrinkage factors for MVDR beamforming. Simulation results are
provided for validating the proposed methods.
