Best Rank-One Tensor Approximation and Parallel Update Algorithm for CPD
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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 × 2 ×...× 2. We derive a closed-form expression of the best rank-1 tensor of 2× 2 × 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.
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