On Polynomial Time Methods for Exact Low-Rank Tensor Completion

In this paper, we investigate the sample size requirement for exact recovery of a high-order tensor of low rank from a subset of its entries. We show that a gradient descent algorithm with initial value obtained from a spectral method can, in particular, reconstruct a d×d×d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d\times d\times d}$$\end{document} tensor of multilinear ranks (r, r, r) with high probability from as few as O(r7/2d3/2log7/2d+r7dlog6d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(r^{7/2}d^{3/2}\log ^{7/2}d+r^7d\log ^6d)$$\end{document} entries. In the case when the ranks r=O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=O(1)$$\end{document}, our sample size requirement matches those for nuclear norm minimization (Yuan and Zhang in Found Comput Math 1031–1068, 2016), or alternating least squares assuming orthogonal decomposability (Jain and Oh in Advances in Neural Information Processing Systems, pp 1431–1439, 2014). Unlike these earlier approaches, however, our method is efficient to compute, is easy to implement, and does not impose extra structures on the tensor. Numerical results are presented to further demonstrate the merits of the proposed approach.