Tensor Trains Approximation Estimates in the Chebyshev Norm

A new elementwise bound on the cross approximation error used for approximating multi-index arrays (tensors) in the format of a tensor train is obtained. The new bound is the first known error bound that differs from the best bound by a factor that depends only on the rank of the approximation and on the dimensionality of the tensor , and the dependence on the dimensionality at a fixed rank has only the order rather than const(d). Thus, this bound justifies the use of the cross method even for high dimensional tensors.