  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 inefficient 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 randomized range finder, yields up to $22.0\%$ relative
decomposition residual improvement compared to the state-of-the-art sketched
randomized algorithm for Tucker decomposition of various synthetic 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.
