🤖 AI Summary
Tensor completion suffers from a computational-statistical gap: existing polynomial-time algorithms require $O(n^{t/2})$ samples, whereas the information-theoretic lower bound is merely $O(n)$. This work bridges the gap by demonstrating—*for the first time*—that near-linear sample complexity $O(n^{1+kappa})$ is achievable using only weak side information per mode that is non-orthogonal to the latent factors, *without requiring subspace observations*. We propose a weighted higher-order SVD framework integrated with mode-specific side information, coupled with an iterative projection algorithm and a robust initialization scheme. We establish theoretical guarantees of uniform convergence. Experiments on recommendation and neuroimaging datasets show that, even when sample size is reduced by over 90%, our method achieves 35–52% lower RMSE compared to state-of-the-art approaches, significantly outperforming existing methods.
📝 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 Θ(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+κ ) samples for any small constant κ > 0. We also provide experiments on both synthetic and real-world datasets that validate our theoretical insights.
