Data-Driven Optimal Shrinkage of Singular Values under High-Dimensional Noise with Separable Covariance Structure

We develop a data-driven optimal shrinkage algorithm for matrix denoising in the presence of high-dimensional noise with separable covariance structure; that is, the nose is colored and dependent. The algorithm, coined extended OptShrink (eOptShrink), involves a new imputation and rank estimation and we do not need to estimate the separable covariance structure of the noise. On the theoretical side, we study the asymptotic behavior of singular values and singular vectors of the random matrix associated with the noisy data, including the sticking property of non-outlier singular values and delocalization of the non-outlier singular vectors with a convergence rate. We apply these results to establish the guarantee of the imputation, rank estimation and eOptShrink algorithm with a convergence rate. On the application side, in addition to a series of numerical simulations with a comparison with various state-of-the-art optimal shrinkage algorithms, we apply eOptShrink to extract fetal electrocardiogram from the single channel trans-abdominal maternal electrocardiogram.