Open Access
April 2019 Cross: Efficient low-rank tensor completion
Anru Zhang
Ann. Statist. 47(2): 936-964 (April 2019). DOI: 10.1214/18-AOS1694


The completion of tensors, or high-order arrays, attracts significant attention in recent research. Current literature on tensor completion primarily focuses on recovery from a set of uniformly randomly measured entries, and the required number of measurements to achieve recovery is not guaranteed to be optimal. In addition, the implementation of some previous methods are NP-hard. In this article, we propose a framework for low-rank tensor completion via a novel tensor measurement scheme that we name Cross. The proposed procedure is efficient and easy to implement. In particular, we show that a third-order tensor of Tucker rank-$(r_{1},r_{2},r_{3})$ in $p_{1}$-by-$p_{2}$-by-$p_{3}$ dimensional space can be recovered from as few as $r_{1}r_{2}r_{3}+r_{1}(p_{1}-r_{1})+r_{2}(p_{2}-r_{2})+r_{3}(p_{3}-r_{3})$ noiseless measurements, which matches the sample complexity lower bound. In the case of noisy measurements, we also develop a theoretical upper bound and the matching minimax lower bound for recovery error over certain classes of low-rank tensors for the proposed procedure. The results can be further extended to fourth or higher-order tensors. Simulation studies show that the method performs well under a variety of settings. Finally, the procedure is illustrated through a real dataset in neuroimaging.


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Anru Zhang. "Cross: Efficient low-rank tensor completion." Ann. Statist. 47 (2) 936 - 964, April 2019.


Received: 1 November 2016; Revised: 1 November 2017; Published: April 2019
First available in Project Euclid: 11 January 2019

zbMATH: 07033157
MathSciNet: MR3909956
Digital Object Identifier: 10.1214/18-AOS1694

Primary: 62H12
Secondary: 62C20

Keywords: Cross tensor measurement , Denoising , minimax rate-optimal , neuroimaging , tensor completion

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 2 • April 2019
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