Title: Best Rank-One Tensor Approximation and Parallel Update Algorithm for CPD,
Abstract: A novel algorithm is proposed for CANDECOMP/PARAFAC tensor decomposition to
exploit best rank-1 tensor approximation. Different from the existing
algorithms, our algorithm updates rank-1 tensors simultaneously in parallel. In
order to achieve this, we develop new all-at-once algorithms for best rank-1
tensor approximation based on the Levenberg-Marquardt method and the rotational
update. We show that the LM algorithm has the same complexity of first-order
optimisation algorithms, while the rotational method leads to solving the best
rank-1 approximation of tensors of size $2 \times 2 \times \cdots \times 2$. We
derive a closed-form expression of the best rank-1 tensor of $2\times 2 \times
2$ tensors and present an ALS algorithm which updates 3 component at a time for
higher order tensors. The proposed algorithm is illustrated in decomposition of
difficult tensors which are associated with multiplication of two matrices.