  In this paper, we focus on the fixed TT-rank and precision problems of
finding an approximation of the tensor train (TT) decomposition of a tensor.
Note that the TT-SVD and TT-cross are two well-known algorithms for these two
problems. Firstly, by combining the random projection technique with the power
scheme, we obtain two types of randomized algorithms for the fixed TT-rank
problem. Secondly, by using the non-asymptotic theory of sub-random Gaussian
matrices, we derive the upper bounds of the proposed randomized algorithms.
Thirdly, we deduce a new deterministic strategy to estimate the desired TT-rank
with a given tolerance and another adaptive randomized algorithm that finds a
low TT-rank representation satisfying a given tolerance, and is beneficial when
the target TT-rank is not known in advance. We finally illustrate the accuracy
of the proposed algorithms via some test tensors from synthetic and real
databases. In particular, for the fixed TT-rank problem, the proposed
algorithms can be several times faster than the TT-SVD, and the accuracy of the
proposed algorithms and the TT-SVD are comparable for several test tensors.
