Fast and accurate randomized algorithms for low-rank tensor decompositions

Low-rank Tucker and CP tensor decompositions are powerful tools in data analytics. The widely used alternating least squares (ALS) method, which solves a sequence of over-determined least squares subproblems, is costly for large and sparse tensors. We propose a fast and accurate sketched ALS algorithm for Tucker decomposition, which solves a sequence of sketched rank-constrained linear least squares subproblems. Theoretical sketch size upper bounds are provided to achieve $O(\epsilon)$ relative error for each subproblem with two sketching techniques, TensorSketch and leverage score sampling. Experimental results show that this new ALS algorithm, combined with a new initialization scheme based on the randomized range finder, yields decomposition accuracy comparable to the standard higher-order orthogonal iteration (HOOI) algorithm. The new algorithm achieves up to $22.0\%$ relative decomposition residual improvement compared to the state-of-the-art sketched randomized algorithm for Tucker decomposition of various synthetic and real datasets. This Tucker-ALS algorithm is further used to accelerate CP decomposition, by using randomized Tucker compression followed by CP decomposition of the Tucker core tensor. Experimental results show that this algorithm not only converges faster, but also yields more accurate CP decompositions.