May 2022 Non-asymptotic properties of spectral decomposition of large Gram-type matrices and applications
Lyuou Zhang, Wen Zhou, Haonan Wang
Author Affiliations +
Bernoulli 28(2): 1224-1249 (May 2022). DOI: 10.3150/21-BEJ1384


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.


The authors thank the Editor, an Associate Editor, and a reviewer for many helpful and constructive comments. The work of Wen Zhou was partially supported by Department of Energy grant DE-SC0018344 and National Science Foundation grants IIS-1545994 and IOS-1922701. The research of Haonan Wang was partially supported by National Science Foundation grants DMS-1737795, DMS-1923142 and CNS-1932413.


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Lyuou Zhang. Wen Zhou. Haonan Wang. "Non-asymptotic properties of spectral decomposition of large Gram-type matrices and applications." Bernoulli 28 (2) 1224 - 1249, May 2022.


Received: 1 February 2020; Revised: 1 June 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388936
zbMATH: 07526582
Digital Object Identifier: 10.3150/21-BEJ1384

Keywords: approximate factor model , Gram-type matrices , high-dimensional time series , non-asymptotic analysis , Principal Component Analysis , spectral decomposition

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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