High-dimensional integration on $\mathbb{R}^d$, weighted Hermite spaces, and orthogonal transforms

TL;DR

This paper investigates how orthogonal transforms affect high-dimensional integration in weighted Hermite spaces, explaining the empirical success of quasi-Monte Carlo methods through the impact on function structure.

Contribution

It introduces the concept of Hermite spaces and demonstrates how orthogonal transforms can significantly improve the tractability of high-dimensional integration.

Findings

Orthogonal transforms can dramatically reduce weighted norms of functions.

Certain Hermite spaces admit tractability of integration.

Transformations explain the efficiency of quasi-Monte Carlo methods in high dimensions.

Abstract

It has been found empirically that quasi-Monte Carlo methods are often efficient for very high-dimensional problems, that is, with dimension in the hundreds or even thousands. The common explanation for this surprising fact is that those functions for which this holds true behave rather like low-dimensional functions in that only few of the coordinates have a size- able influence on its value. However, this statement may be true only after applying a suitable orthogonal transform to the input data, like utilizing the Brownian bridge construction or principal component analysis construction. We study the effect of general orthogonal transforms on functions on $\mathbb{R}^d$ which are ele- ments of certain weighted reproducing kernel Hilbert spaces. The notion of Hermite spaces is defined and it is shown that there are examples which admit tractability of integration. We translate the action of the orthogonal transform of $\mathbb{R}^d$ into an action on the Hermite coefficients and we give examples where orthogonal transforms have a dramatic effect on the weighted norm, thus providing an explanation for the efficiency of using suitable orthogonal transforms.