Open Access
April 2021 Subspace estimation from unbalanced and incomplete data matrices: 2, statistical guarantees
Changxiao Cai, Gen Li, Yuejie Chi, H. Vincent Poor, Yuxin Chen
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Ann. Statist. 49(2): 944-967 (April 2021). DOI: 10.1214/20-AOS1986


This paper is concerned with estimating the column space of an unknown low-rank matrix ARd1×d2, given noisy and partial observations of its entries. There is no shortage of scenarios where the observations—while being too noisy to support faithful recovery of the entire matrix—still convey sufficient information to enable reliable estimation of the column space of interest. This is particularly evident and crucial for the highly unbalanced case where the column dimension d2 far exceeds the row dimension d1, which is the focal point of the current paper.

We investigate an efficient spectral method, which operates upon the sample Gram matrix with diagonal deletion. While this algorithmic idea has been studied before, we establish new statistical guarantees for this method in terms of both 2 and 2, estimation accuracy, which improve upon prior results if d2 is substantially larger than d1. To illustrate the effectiveness of our findings, we derive matching minimax lower bounds with respect to the noise levels, and develop consequences of our general theory for three applications of practical importance: (1) tensor completion from noisy data, (2) covariance estimation/principal component analysis with missing data and (3) community recovery in bipartite graphs. Our theory leads to improved performance guarantees for all three cases.


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Changxiao Cai. Gen Li. Yuejie Chi. H. Vincent Poor. Yuxin Chen. "Subspace estimation from unbalanced and incomplete data matrices: 2, statistical guarantees." Ann. Statist. 49 (2) 944 - 967, April 2021.


Received: 1 October 2019; Revised: 1 March 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1986

Primary: 62F10
Secondary: 62H25

Keywords: Covariance estimation , leave-one-out analysis , missing data , Principal Component Analysis , spectral clustering , Spectral method , tensor completion

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 2 • April 2021
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