  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.
