Tensor Estimation with Nearly Linear Samples Given Weak Side Information

• URL: http://arxiv.org/abs/2007.00736v3
• Date: Sat, 19 Oct 2024 13:34:40 GMT
• Title: Tensor Estimation with Nearly Linear Samples Given Weak Side Information
• Authors: Christina Lee Yu, Xumei Xi,
• Abstract summary: We show that weak side information is sufficient to reduce the sample to $O(n)$. We provide an algorithm that utilizes this side information to produce a consistent estimator with $O(n1+kappa)$ samples for any small constant $kappa > 0$.
• Score: 5.69361786082969
• License:
• 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.

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