  Low-rank tensor decomposition generalizes low-rank matrix approximation and
is a powerful technique for discovering low-dimensional structure in
high-dimensional data. In this paper, we study Tucker decompositions and use
tools from randomized numerical linear algebra called ridge leverage scores to
accelerate the core tensor update step in the widely-used alternating least
squares (ALS) algorithm. Updating the core tensor, a severe bottleneck in ALS,
is a highly-structured ridge regression problem where the design matrix is a
Kronecker product of the factor matrices. We show how to use approximate ridge
leverage scores to construct a sketched instance for any ridge regression
problem such that the solution vector for the sketched problem is a
$(1+\varepsilon)$-approximation to the original instance. Moreover, we show
that classical leverage scores suffice as an approximation, which then allows
us to exploit the Kronecker structure and update the core tensor in time that
depends predominantly on the rank and the sketching parameters (i.e., sublinear
in the size of the input tensor). We also give upper bounds for ridge leverage
scores as rows are removed from the design matrix (e.g., if the tensor has
missing entries), and we demonstrate the effectiveness of our approximate ridge
regressioni algorithm for large, low-rank Tucker decompositions on both
synthetic and real-world data.
