English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It has been found empirically that quasi-Monte Carlo methods are oftenefficient for very high-dimensional problems, that is, with dimension in thehundreds or even thousands. The common explanation for this surprising fact isthat those functions for which this holds true behave rather likelow-dimensional functions in that only few of the coordinates have a size- ableinfluence on its value. However, this statement may be true only after applyinga suitable orthogonal transform to the input data, like utilizing the Brownianbridge construction or principal component analysis construction. We study theeffect of general orthogonal transforms on functions on $\mathbb{R}^d$ whichare ele- ments of certain weighted reproducing kernel Hilbert spaces. Thenotion of Hermite spaces is defined and it is shown that there are exampleswhich admit tractability of integration. We translate the action of theorthogonal transform of $\mathbb{R}^d$ into an action on the Hermitecoefficients and we give examples where orthogonal transforms have a dramaticeffect on the weighted norm, thus providing an explanation for the efficiencyof using suitable orthogonal transforms.

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