A Practical Randomized CP Tensor Decomposition

The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multi-way data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined least squares problems. We show that, by extending randomized least squares ("sketching") methods to tensors, the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP iteration) produces conditions favorable for direct sampling without any preprocessing. In numerical results, we see improvements in speed, storage, and robustness.