Fast Monte Carlo Algorithms for Tensor Operations

We consider scalable randomized algorithms for many common tensor operations, including tensor multiplication, low-rank decomposition and nuclear norm minimization. Such operations arise in a number of modern day applications of tensors in machine learning and signal processing. Specifically, we introduce polynomial time algorithms that employ a small number of lateral and/or horizontal slices of the underlying 3-rd order tensor, that offer relative error guarantees for the quality of the solutions. All results can easily be extended to higher order tensors. The proposed methods are illustrated on selected real data sets.