Spectral Algorithms for Tensor Completion

Abstract

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-ofsquares (SOS) semidefinite programming hierarchy. However, the SOS approach does not scale well to large problem instances. By contrast, spectral methodsbased on unfolding or matricizing the tensor-are attractive for their low complexity, but have been believed to require a much larger sample size.This paper presents two main contributions. First, we propose a new method, based on unfolding, which outperforms naive ones for symmetric k th -order tensors of rank r. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about rd 3=2 revealed entries), subject to a worse rank condition (r d 3=4 rather than r d 3=2 ). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete (r d ) regime. Under a random model, this second approach succeeds in estimating tensors of rank d Ä r d 3=2 from about rd 3=2 revealed entries.

On Polynomial Time Methods for Exact Low-Rank Tensor Completion

“…In contrast, our approach is essentially based on the spectral decomposition of a d × d matrix and can be computed fairly efficiently. Very recently, in independent work and under further restrictions on the tensor ranks, Montanari and Sun (2016) showed that a spectral method different from ours can also achieve consistency with O(d 3/2 polylog(r, log d)) observed entries. The rate of concentration for their estimate, however, is slower than ours and as a result, it is unclear if it provides a sufficiently accurate initial value for the exact recovery with the said sample size.…”

The Landscape of the Spiked Tensor Model

“…A large gap between known polynomial-time algorithms and statistical limits arises in the tensor completion problem, which shares many similarities with the spiked tensor model [GRY11,YZ15,MS16]. In the setting of tensor completion, hardness under Feige's hypothesis was proven in [BM16] for a certain regime of the number of observed entries.…”

Tensor Completion Algorithms in Big Data Analytics

“…Instead of exact recovery, their main result shows that O(I 3 2 r 2 lo 4 I ) observations could guarantee an approximation with an explicit upper bound on error. However, this sum-of-squares approach is point out not to be scale well to large tensors and is substituted by a spectral approach proposed by Montanari et al [137] with matching statistical guarantee (i.e., the required sample size). Recently, Yuan and Zhang [217] explore the correlations between tensor norms and di erent coherence conditions.…”