Non-asymptotic properties of spectral decomposition of large Gram-type matrices and applications
Non-asymptotic properties of spectral decomposition of large Gram-type matrices and applications
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DOI：
10.3150/21-bej1384
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发表时间：
2022-05
期刊：
影响因子：
1.5
通讯作者：
Lyuou Zhang;Wen Zhou;Haonan Wang
中科院分区：
文献类型：
--
作者：
Lyuou Zhang;Wen Zhou;Haonan Wang
Gram-type矩阵及其谱分解对于统计学、应用数学、物理学和机器学习中的许多问题至关重要。本文研究了数据不一定独立时大型Gram型矩阵谱分解的非渐近性质。具体地说，我们推导出右Gram矩阵的特征向量与其总体对应特征向量之间的偏差的指数尾界以及这些偏差的Berry-Esseen型界。我们还得到了左格拉姆矩阵，即样本协方差矩阵的特征值之间的比率的非渐近尾界，和他们的人口对应的数据矩阵的大小无关。这些非渐近性质在一系列的应用中得到了进一步的证明，包括因子模型中潜在因子个数估计的非渐近性质和相关的机器学习问题，高维时间序列的估计和预测，大样本协方差矩阵的谱性质如扰动界和谱投影的推断，以及使用相关数据的低秩矩阵去噪。
Gram-type matrices and their spectral decomposition are of central importance for numerous problems in statistics, applied mathematics, physics, and machine learning. In this paper, we carefully study the non-asymptotic properties of spectral decomposition of large Gram-type matrices when data are not necessarily independent. Specifically, we derive the exponential tail bounds for the deviation between eigenvectors of the right Gram matrix to their population counterparts as well as the Berry-Esseen type bound for these deviations. We also obtain the non-asymptotic tail bound of the ratio between eigenvalues of the left Gram matrix, namely the sample covariance matrix, and their population counterparts regardless of the size of the data matrix. The documented non-asymptotic properties are further demonstrated in a suite of applications, including the non-asymptotic characterization of the estimated number of latent factors in factor models and relate machine learning problems, the estimation and forecasting of high-dimensional time series, the spectral properties of large sample covariance matrix such as perturbation bounds and inference on the spectral projectors, and low-rank matrix denoising using dependent data.