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，用于在高维可分协方差结构噪声（即噪声在样本间存在颜色依赖）条件下进行矩阵去噪。该算法被命名为扩展最优收缩（eOptShrink），其依赖于含噪数据关联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粘性性质、弱信号非离群奇异向量以收敛速度进行的离域化，以及离群奇异值与向量的谱行为），我们构建了三种估计量，每种均具有独立研究价值。首先，我们设计了一种新颖的秩估计量，并基于此给出了纯噪声矩阵谱分布的估计量，进而得到名为eOptShrink的最优收缩器。该算法无需估计噪声的可分协方差结构，并提供了具有收敛速度的估计量理论保证。在应用方面，除开展一系列与多种最新最优收缩算法的数值模拟对比外，我们将eOptShrink应用于单通道腹壁母体心电图中母体与胎儿心电信号的提取。