使用Nyström方法估计协方差阵的主成分
Estimating principal components of covariance matrices using the Nyström method
摘要：
协方差阵估计是许多信号处理算法的重要组成部分，通常用于通过谱分解确定低维主子空间。然而，精确的特征分析是计算上棘手的足够高的维度矩阵，并在小的样本量的情况下，样本特征值和特征向量是已知的穷人估计他们的人口同行。为了解决这些问题，我们提出了一个协方差估计，这是计算效率，同时也执行收缩的样本特征值。我们的方法是基于Nyström方法，它使用一个依赖于数据的正交投影，以获得一个快速的低秩近似的大型半正定矩阵。我们提供了一个理论分析的误差特性，我们的估计以及实证结果，包括其应用程序的例子，自适应波束形成和图像去噪。
英文摘要：
Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a low-dimensional principal subspace via their spectral decomposition. However, exact eigenanalysis is computationally intractable for sufficiently high-dimensional matrices, and in the case of small sample sizes, sample eigenvalues and eigenvectors are known to be poor estimators of their population counterparts. To address these issues, we propose a covariance estimator that is computationally efficient while also performing shrinkage on the sample eigenvalues. Our approach is based on the Nyström method, which uses a data-dependent orthogonal projection to obtain a fast low-rank approximation of a large positive semidefinite matrix. We provide a theoretical analysis of the error properties of our estimator as well as empirical results, including examples of its application to adaptive beamforming and image denoising.
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