Tractability of the approximation of high-dimensional rank one tensors

Erich Novak; Daniel Rudolf

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arXiv · 1402.5011

Tractability of the approximation of high-dimensional rank one tensors

Erich Novak·Daniel Rudolf

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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.

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Erich Novak, Daniel Rudolf. 2014-02-20. Tractability of the approximation of high-dimensional rank one tensors. https://arxiv.org/abs/1402.5011

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