  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 $p = \Omega(n^{-3/2 + \kappa})$ for any
arbitrarily small $\kappa > 0$. 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.
