Best Rank-One Tensor Approximation and Parallel Update Algorithm for CPD
This work addresses the bottleneck of slow sequential updates in tensor decomposition, offering a parallel approach that may accelerate CPD for practitioners.
The paper proposes a new algorithm for CANDECOMP/PARAFAC tensor decomposition that updates rank-1 tensors in parallel, achieving faster convergence. The method uses Levenberg-Marquardt and rotational updates, with closed-form solutions for 2x2x2 tensors and an ALS algorithm for higher orders.
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.