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

We study the problem of approximating functions of $d$ variables in theaverage case setting for the $L_2$ space $L_{2,d}$ with the standard Gaussianweight equipped with a zero-mean Gaussian measure. The covariance kernel ofthis Gaussian measure takes the form of a Gaussian kernel with non-increasingpositive shape parameters $\gamma_j^2$ for $j = 1, 2, \dots, d$. The error ofapproximation is defined in the norm of $L_{2,d}$. We study the average caseerror of algorithms that use at most $n$ arbitrary continuous linearfunctionals. The information complexity $n(\varepsilon, d)$ is defined as theminimal number of linear functionals which are needed to find an algorithmwhose average case error is at most $\varepsilon$. We study different notionsof tractability or exponentially-convergent tractability (EC-tractability)which the information complexity $n(\varepsilon, d)$ describe how behaves as afunction of $d$ and $\varepsilon^{-1}$ or as one of $d$ and$(1+\ln\varepsilon^{-1})$.  We find necessary and sufficient conditions on various notions oftractability and EC-tractability in terms of shape parameters. In particular,for any positive $s>0$ and $t\in(0,1)$ we obtain that the sufficient andnecessary condition on $\gamma^2_ j$ for which$$\lim_{d+\varepsilon^{-1}\to\infty}\frac{n(\varepsilon,d)}{\varepsilon^{-s}+d^t}=0$$holds is $$ \lim_{j\to \infty}j^{1-t}\gamma_j^2\,\ln^+ \gamma_j^{-2}=0,$$where$\ln^+ x=\max(1,\ln x)$.

Average Case tractability of multivariate approximation with Gaussian kernels