We develop a data-driven optimal shrinkage algorithm, named extended OptShrink (eOptShrink), for matrix denoising with high-dimensional noise and a separable covariance structure. This noise is colored and dependent across samples. The algorithm leverages the asymptotic behavior of singular values and vectors of the noisy data's random matrix. Our theory includes the sticking property of non-outlier singular values, delocalization of weak signal singular vectors, and the spectral behavior of outlier singular values and vectors. We introduce three estimators: a novel rank estimator, an estimator for the spectral distribution of the pure noise matrix, and the optimal shrinker eOptShrink. Notably, eOptShrink does not require estimating the noise's separable covariance structure. We provide a theoretical guarantee for these estimators with a convergence rate. Through numerical simulations and comparisons with state-of-the-art optimal shrinkage algorithms, we demonstrate eOptShrink's application in extracting maternal and fetal electrocardiograms from single-channel trans-abdominal maternal electrocardiograms.

中文翻译：

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具有可分离协方差结构的高维噪声下数据驱动的奇异值最优收缩及其应用


我们开发了一种数据驱动的最佳收缩算法，名为 extended OptShrink （eOptShrink），用于具有高维噪声和可分离协方差结构的矩阵去噪。此噪声是彩色的，并且取决于样本。该算法利用了噪声数据随机矩阵的奇异值和向量的渐近行为。我们的理论包括非异常值奇异值的粘附特性、弱信号奇异向量的离域以及异常值奇异值和向量的频谱行为。我们介绍了三个估计器：一个新的秩估计器，一个纯噪声矩阵的频谱分布估计器，以及最优的收缩器 eOptShrink。值得注意的是，eOptShrink 不需要估计噪声的可分离协方差结构。我们以收敛率为这些估计量提供了理论保证。通过数值模拟和与最先进的最佳收缩算法的比较，我们展示了 eOptShrink 在从单通道经腹母体心电图中提取母体和胎儿心电图的应用。