Dependence on the Dimension for Complexity of Approximation of Random Fields (in Russian)

TL;DR

This paper investigates how the complexity of approximating high-dimensional tensor-type random fields grows with dimension, providing precise asymptotic formulas that clarify the curse of dimensionality in the average case.

Contribution

The paper develops a new technique to derive sharp asymptotic expressions for the approximation complexity of tensor-type random fields as dimension increases.

Findings

Complexity grows exponentially with dimension for fixed accuracy.

Sharp asymptotic formulas for approximation complexity are obtained.

The results clarify the nature of the curse of dimensionality.

Abstract

In the present paper a behavior of the "average case" approximation complexity for d-parametric random fields of tensor-type is studied. It was shown in [Lifshits and Tulyakova, 2006] that for a given approximation accuracy level the complexity of approximation increases exponentially, as d tends to infinity; that is the curse of dimensionality is observed. In this paper a technique allowing obtaining sharp asymptotic expressions for the approximation complexity is developed and such an expression is obtained.