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-of-squares (sos) semidefinite programming hierarchy (Barak and Moitra, 2014). However, the sos approach does not scale well to large problem instances. By contrast, spectral methods — based 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 unfolding-based method, 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 rd3{2 revealed entries), subject to a worse rank condition (r d3{4 rather than r d3{2). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete (r ě d) regime. Under a random model, this second approach succeeds in estimating tensors of rank d ď r d3{2 from about rd3{2 revealed entries.