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February 2021 Statistically optimal and computationally efficient low rank tensor completion from noisy entries
Dong Xia, Ming Yuan, Cun-Hui Zhang
Ann. Statist. 49(1): 76-99 (February 2021). DOI: 10.1214/20-AOS1942


In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell _{p}$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.


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Dong Xia. Ming Yuan. Cun-Hui Zhang. "Statistically optimal and computationally efficient low rank tensor completion from noisy entries." Ann. Statist. 49 (1) 76 - 99, February 2021.


Received: 1 November 2017; Revised: 1 August 2019; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1942

Primary: 60K35

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • February 2021
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