A note on tensor chain approximation

This paper deals with the approximation of ddimensional tensors, as discrete representations of arbitrary functions f(x(1),..., x(d)) on [0, 1](d), in the so-called tensor chain format. The main goal of this paper is to show that the construction of a tensor chain approximation is possible using skeleton/ cross approximation type methods. The complete algorithm is described, computational issues are discussed in detail and the complexity of the algorithm is shown to be linear in d. Some numerical examples are given to validate the theoretical results.