SVD-based algorithms for fully-connected tensor network decomposition

The popular fully-connected tensor network (FCTN) decomposition has achieved successful applications in many fields. A standard method to this decomposition is the alternating least squares. However, it often converges slowly and suffers from issues of numerical stability. In this work, we investigate the SVD-based algorithms for FCTN decomposition to tackle the aforementioned deficiencies. On the basis of a result about FCTN-ranks, a deterministic algorithm, namely FCTN-SVD, is first proposed, which can approximate the FCTN decomposition under a fixed accuracy. Then, we present the randomized version of the algorithm. Both synthetic and real data are used to test our algorithms. Numerical results show that they perform much better than the existing methods, and the randomized algorithm can indeed yield acceleration on FCTN-SVD. Moreover, we also apply our algorithms to tensor-on-vector regression and achieve quite decent performance.