Sequential Algorithms for Matrix and Tensor Completion

We study low rank matrix and tensor completion and propose novel algorithms with the best known sample complexity guarantees for these problems. Our algorithms are active in that they interact with the sampling mechanism to obtain informative measurements. They are also sequential, processing the columns (sub-tensors) one at a time, and can easily be implemented in a streaming setting. For matrix completion, we show that one can exactly recover an $n \times n$ matrix of rank $r$ using $O(r^2n \log(r))$ observations and for tensor completion, one can recover an order-$T$ tensor using $O(r^{2(T-1)} T^2 n\log(r))$ observations. We also establish a necessary condition for exact tensor completion from random observations. We complement our study with simulations that verify our theoretical guarantees and demonstrate the scalability of our algorithms.