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Tensor completion exhibits an interesting computational-statistical gap interms of the number of samples needed to perform tensor estimation. While thereare only $\Theta(tn)$ degrees of freedom in a $t$-order tensor with $n^t$entries, the best known polynomial time algorithm requires $O(n^{t/2})$ samplesin order to guarantee consistent estimation. In this paper, we show that weakside information is sufficient to reduce the sample complexity to $O(n)$. Theside information consists of a weight vector for each of the modes which is notorthogonal to any of the latent factors along that mode; this is significantlyweaker than assuming noisy knowledge of the subspaces. We provide an algorithmthat utilizes this side information to produce a consistent estimator with$O(n^{1+\kappa})$ samples for any small constant $\kappa > 0$.

Tensor Estimation with Nearly Linear Samples Given Weak Side Information