论文标题
在高维噪声下具有可分离的协方差结构，数据驱动的奇异值的最佳收缩
Data-Driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application
论文作者
论文摘要
我们开发了一种具有可分离的协方差结构的高维噪声的数据驱动的最佳收缩算法，用于矩阵deno的基质。也就是说，噪声是有色的，并依赖于样品。算法创建的{\ em扩展的OptShrink}（eOptshrink）取决于与嘈杂数据相关的随机矩阵的奇异值的渐近行为。基于发达的理论，包括非外观奇异值的粘性特性和与弱信号相关的非外部奇异向量的定位，以及逆转率的弱信号，以及我们开发三个估计值的异常值奇异值的光谱行为，每个估计器都有其自身的利益。首先，我们设计了一种新型的等级估计量，基于我们为纯噪声矩阵的光谱分布提供了一个估计器，因此为最佳的收缩器（称为eOptshrink）提供了估计值。在该算法中，我们不需要估计噪声的可分离协方差结构。给出了具有收敛速率的这些估计器的理论保证。在应用方面，除了一系列与各种最佳最佳收缩算法进行比较的数值模拟外，我们还使用EOPTSHRINK从单个通道跨腹膜母体母体电脑图中提取母体和胎儿心电图。
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.
