We develop a data-driven optimal shrinkage algorithm for matrix denoising in the 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 behavior of singular values and singular vectors of the random matrix associated with the noisy data. Based on the developed theory, including the sticking property of non-outlier singular values and delocalization of the non-outlier singular vectors associated with weak signals with a convergence rate, and the spectral behavior 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 the pure noise matrix, and hence the optimal shrinker called eOptShrink. In this algorithm we do not need to estimate the separable covariance structure of the noise. A theoretical guarantee of these estimators with a convergence rate is given. 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 maternal and fetal electrocardiograms from the single channel trans-abdominal maternal electrocardiogram.

翻译：我们提出了一种数据驱动的奇异值最优收缩算法，用于在高维噪声具有可分离协方差结构（即噪声在样本间存在色彩化和依赖关系）的情况下进行矩阵去噪。该算法被命名为**扩展最优收缩（eOptShrink）**，其依赖于含噪数据随机矩阵的奇异值与奇异向量的渐近行为。基于所发展的理论（包括非离群奇异值的黏滞性质、与弱信号关联的非离群奇异向量的离域化及其收敛速率，以及离群奇异值与向量的谱行为），我们构建了三种估计量，每种估计量均具有独立的理论价值。首先，设计了一种新型秩估计量，并基于此提出纯噪声矩阵谱分布的估计方法，进而获得称为eOptShrink的最优收缩器。该算法无需估计噪声的可分协方差结构。我们给出了这些估计量的理论保证及其收敛速率。在应用方面，除了一系列与当前最优收缩算法的数值模拟对比外，我们将eOptShrink应用于从单通道经腹母体心电图中提取母体与胎儿心电信号。