Spectral algorithms for tensor completion

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 (rll d3/4 rather than rll d3/2). 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≤ r ll d3/2 from about rd3/2 revealed entries.