具有可分离协方差结构的高维噪声下奇异值的数据驱动最优收缩及应用
Data-Driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application
AI总结：
本文提出无需估计噪声可分离协方差结构的数据驱动 eOptShrink 矩阵去噪算法，建立秩、谱分布和最优收缩估计的理论保证，并通过模拟及胎儿心电提取验证其效果。
摘要：
我们开发了一种数据驱动的最优收缩算法，用于在存在具有可分离协方差结构的高维噪声时进行矩阵去噪；也就是说，噪声是有色的，并且在样本之间相互依赖。该算法被命名为 extended OptShrink（eOptShrink），它依赖于与含噪数据相关的随机矩阵的奇异值和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附特性、与弱信号相关的非离群奇异向量在一定收敛速率下的离域性，以及离群奇异值和奇异向量的谱行为，我们开发了三个估计器，每个都有其自身的意义。首先，我们设计了一种新颖的秩估计器，并在此基础上给出了纯噪声矩阵谱分布的估计器，进而得到称为 eOptShrink 的最优收缩器。在该算法中，我们不需要估计噪声的可分离协方差结构。文中给出了这些估计器在一定收敛速率下的理论保证。在应用方面，除了进行一系列数值模拟并与多种最先进的最优收缩算法比较外，我们还应用 eOptShrink 从单通道经腹壁母体心电图中提取母体和胎儿心电图。
英文摘要：
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.
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