Tensor trees and tensor trains

In a new paper, we expose connections between tensor networks and recent recursive representations of high-dimensional tensors described by binary tensor trees and based on a sequence of skeleton (dyadic) decompositions for special unfolding matrices for a given tensor.     As the main result, we prove that a tensor decomposition by any binary tensor tree with certain restrictions on the distribution of spatial and auxiliary indices reduces to one and same for a particular case of tree. Since the latter tree is of simple predetermined shape, it becomes not needed at all in the construction of numerical algorithms. The tree input is replaced with a permutation of spatial indices (modes). The corresponding decomposition is given by the so-called tensor trains and known as TT decomposition.