Tractability of the Approximation of High-Dimensional Rank One Tensors

Abstract

We study the approximation of high-dimensional rank one tensors using point evaluations and consider deterministic as well as randomized algorithms. We prove that for certain parameters (smoothness and norm of the $$r$$ th derivative), this problem is intractable, while for other parameters, the problem is tractable and the complexity is only polynomial in the dimension for every fixed $$\varepsilon >0$$ . For randomized algorithms, we completely characterize the set of parameters that lead to easy or difficult problems, respectively. In the “difficult” case, we modify the class to obtain a tractable problem: The problem gets tractable with a polynomial (in the dimension) complexity if the support of the function is not too small.