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A novel algorithm is proposed for CANDECOMP/PARAFAC tensor decomposition toexploit best rank-1 tensor approximation. Different from the existingalgorithms, our algorithm updates rank-1 tensors simultaneously in parallel. Inorder to achieve this, we develop new all-at-once algorithms for best rank-1tensor approximation based on the Levenberg-Marquardt method and the rotationalupdate. We show that the LM algorithm has the same complexity of first-orderoptimisation algorithms, while the rotational method leads to solving the bestrank-1 approximation of tensors of size $2 \times 2 \times \cdots \times 2$. Wederive a closed-form expression of the best rank-1 tensor of $2\times 2 \times2$ tensors and present an ALS algorithm which updates 3 component at a time forhigher order tensors. The proposed algorithm is illustrated in decomposition ofdifficult tensors which are associated with multiplication of two matrices.

Best Rank-One Tensor Approximation and Parallel Update Algorithm for CPD