Optimal shrinkage of singular values under high-dimensional noise with separable covariance structure

We consider an optimal shrinkage algorithm that depends on an effective rank estimation and imputation, coined optimal shrinkage with imputation and rank estimation (OSIR), for matrix denoising in the presence of high-dimensional noise with the separable covariance structure (colored and dependent noise). The algorithm does not depend on estimating separable covariance structure of the noise. On the theoretical side, we study the asymptotic behavior of outlier singular values and singular vectors and prove the delocalization of the non-outlier singular vectors of the associated random matrix with a convergence rate, and apply these results to analyze OSIR over different signal strengths and sizes of data matrices. On the application side, we carry out simulations to demonstrate the effectiveness of OSIR, and apply it to study the fetal electrocardiogram signal processing challenge and the two-dimensional random tomography problem.