Robust Max Entrywise Error Bounds for Tensor Estimation from Sparse Observations via Similarity Based Collaborative Filtering
This work addresses tensor estimation in sparse regimes, which is incremental as it builds on existing collaborative filtering methods but extends them to tensors with theoretical guarantees.
The paper tackles the problem of estimating a 3-order tensor from sparse, noisy observations by introducing a similarity-based collaborative filtering algorithm that achieves near-optimal sample complexity, with maximum entry-wise error and mean-squared-error decaying to zero under specific observation probabilities and robustness to bounded noise.
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 estimating a tensor from sparse observations 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 finite 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 $p = Ω(n^{-3/2 + κ})$ for any arbitrarily small $κ> 0$. More generally, we establish robustness of the estimator, showing that when arbitrary noise bounded by $\varepsilon \geq 0$ is added to each observation, the estimation error with respect to MEE and MSE degrades by $\text{poly}(\varepsilon)$. Consequently, even if the tensor may not have finite rank but can be approximated within $\varepsilon \geq 0$ by a finite rank tensor, then the estimation error converges to $\text{poly}(\varepsilon)$. 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.