Robust Max Entrywise Error Bounds for Sparse Tensor Estimation via Similarity Based Collaborative Filtering

Consider the task of estimating a 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a similarity based collaborative filtering algorithm for sparse tensor estimation and argue that it achieves sample complexity that nearly matches the conjectured computationally efficient lower bound on the sample complexity for the setting of low-rank tensors. Our algorithm uses the matrix obtained from the flattened tensor to compute similarity, and estimates the tensor entries using a nearest neighbor estimator. We prove that the algorithm recovers a low rank tensor with maximum entry-wise error (MEE) and mean-squared-error (MSE) decaying to $0$ as long as each entry is observed independently with probability $p = \Omega(n^{-3/2 + \kappa})$ for any arbitrarily small $\kappa > 0$. % as long as tensor has finite rank $r = \Theta(1)$. More generally, we establish robustness of the estimator, showing that when arbitrary noise bounded by $\epsilon \geq 0$ is added to each observation, the estimation error with respect to MEE and MSE degrades by ${\sf poly}(\epsilon)$. Consequently, even if the tensor may not have finite rank but can be approximated within $\epsilon \geq 0$ by a finite rank tensor, then the estimation error converges to ${\sf poly}(\epsilon)$. Our analysis sheds insight into the conjectured sample complexity lower bound, showing that it matches the connectivity threshold of the graph used by our algorithm for estimating similarity between coordinates.