Iterative Collaborative Filtering for Sparse Noisy Tensor Estimation

Consider the task of tensor estimation, i.e. estimating a low-rank 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a generalization of the 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. 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 the tensor with maximum entry-wise error 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 $\kappa> 0$. 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.