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We develop a data-driven optimal shrinkage algorithm for matrix denoising inthe presence of high-dimensional noise with a separable covariance structure;that is, the noise is colored and dependent across samples. The algorithm,coined {\em extended OptShrink} (eOptShrink) depends on the asymptotic behaviorof singular values and singular vectors of the random matrix associated withthe noisy data. Based on the developed theory, including the sticking propertyof non-outlier singular values and delocalization of the non-outlier singularvectors associated with weak signals with a convergence rate, and the spectralbehavior of outlier singular values and vectors, we develop three estimators,each of these has its own interest. First, we design a novel rank estimator,based on which we provide an estimator for the spectral distribution of thepure noise matrix, and hence the optimal shrinker called eOptShrink. In thisalgorithm we do not need to estimate the separable covariance structure of thenoise. A theoretical guarantee of these estimators with a convergence rate isgiven. On the application side, in addition to a series of numericalsimulations with a comparison with various state-of-the-art optimal shrinkagealgorithms, we apply eOptShrink to extract maternal and fetalelectrocardiograms from the single channel trans-abdominal maternalelectrocardiogram.

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