Faster Tensor Train Decomposition for Sparse Data

In recent years, the application of tensors has become more widespread in fields that involve data analytics and numerical computation. Due to the explosive growth of data, low-rank tensor decompositions have become a powerful tool to harness the notorious curse of dimensionality. The main forms of tensor decomposition include CP decomposition, Tucker decomposition, tensor train (TT) decomposition, etc. Each of the existing TT decomposition algorithms, including the TT-SVD and randomized TT-SVD, is successful in the field, but neither can both accurately and efficiently decompose large-scale sparse tensors. Based on previous research, this paper proposes a new quasi-best fast TT decomposition algorithm for large-scale sparse tensors with proven correctness and the upper bound of its complexity is derived. In numerical experiments, we verify that the proposed algorithm can decompose sparse tensors faster than the TT-SVD, and have more speed, precision and versatility than randomized TT-SVD, and it can be used to decomposes arbitrary high-dimensional tensor without losing efficiency when the number of non-zero elements is limited. The new algorithm implements a large-scale sparse matrix TT decomposition that was previously unachievable, enabling tensor decomposition based algorithms to be applied in larger-scale scenarios.