Data-driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application

We develop a data-driven optimal shrinkage algorithm, named extended OptShrink (eOptShrink), for matrix denoising with high-dimensional noise and a separable covariance structure. This noise is colored and dependent across samples. The algorithm leverages the asymptotic behavior of singular values and vectors of the noisy data's random matrix. Our theory includes the sticking property of non-outlier singular values, delocalization of weak signal singular vectors, and the spectral behavior of outlier singular values and vectors. We introduce three estimators: a novel rank estimator, an estimator for the spectral distribution of the pure noise matrix, and the optimal shrinker eOptShrink. Notably, eOptShrink does not require estimating the noise's separable covariance structure. We provide a theoretical guarantee for these estimators with a convergence rate. Through numerical simulations and comparisons with state-of-the-art optimal shrinkage algorithms, we demonstrate eOptShrink's application in extracting maternal and fetal electrocardiograms from single-channel trans-abdominal maternal electrocardiograms.