  In this article two new algorithms are presented that convert a given data
tensor train into either a Tucker decomposition with orthogonal matrix factors
or a multi-scale entanglement renormalization ansatz (MERA). The Tucker core
tensor is never explicitly computed but stored as a tensor train instead,
resulting in both computationally and storage efficient algorithms. Both the
multilinear Tucker-ranks as well as the MERA-ranks are automatically determined
by the algorithm for a given upper bound on the relative approximation error.
In addition, an iterative algorithm with low computational complexity based on
solving an orthogonal Procrustes problem is proposed for the first time to
retrieve optimal rank-lowering disentangler tensors, which are a crucial
component in the construction of a low-rank MERA. Numerical experiments
demonstrate the effectiveness of the proposed algorithms together with the
potential storage benefit of a low-rank MERA over a tensor train.
