Tensor Completion with Nearly Linear Samples Given Weak Side Information

TL;DR

This paper demonstrates that weak side information can significantly reduce the sample complexity in tensor completion, achieving near-linear sample requirements with a new algorithm.

Contribution

It introduces a novel tensor completion algorithm that leverages weak side information to reduce sample complexity to nearly linear in tensor dimensions.

Findings

Achieves tensor estimation with $O(n^{1+\

experiments validate theoretical results on synthetic and real datasets.

Abstract

Tensor completion exhibits an interesting computational-statistical gap in terms of the number of samples needed to perform tensor estimation. While there are only $\Theta(tn)$ degrees of freedom in a $t$-order tensor with $n^t$ entries, the best known polynomial time algorithm requires $O(n^{t/2})$ 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 $O(n^{1+\kappa})$ samples for any small constant $\kappa > 0$. We also provide experiments on both synthetic and real-world datasets that validate our theoretical insights.