Tensor Completion with Nearly Linear Samples Given Weak Side Information

Tensor completion exhibits an interesting computational-statistical gap in terms of the number of samples needed to perform tensor estimation. While there are only Θ(tn) degrees of freedom in a t-order tensor with nt entries, the best known polynomial time algorithm requires O(nt/2) samples in order to guarantee consistent estimation. In this paper, we show that weak side information is sufficient to reduce the sample complexity to O(n). The side information consists of a weight vector for each of the modes which is not orthogonal to any of the latent factors along that mode; this is significantly weaker than assuming noisy knowledge of the subspaces. We provide an algorithm that utilizes this side information to produce a consistent estimator with O(n1+κ) samples for any small constant κ > 0. We also provide experiments on both synthetic and real-world datasets that validate our theoretical insights.