Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Average Case tractability of multivariate approximation with Gaussian
  kernels   We study the problem of approximating functions of $d$ variables in the
average case setting for the $L_2$ space $L_{2,d}$ with the standard Gaussian
weight equipped with a zero-mean Gaussian measure. The covariance kernel of
this Gaussian measure takes the form of a Gaussian kernel with non-increasing
positive shape parameters $\gamma_j^2$ for $j = 1, 2, \dots, d$. The error of
approximation is defined in the norm of $L_{2,d}$. We study the average case
error of algorithms that use at most $n$ arbitrary continuous linear
functionals. The information complexity $n(\varepsilon, d)$ is defined as the
minimal number of linear functionals which are needed to find an algorithm
whose average case error is at most $\varepsilon$. We study different notions
of tractability or exponentially-convergent tractability (EC-tractability)
which the information complexity $n(\varepsilon, d)$ describe how behaves as a
function of $d$ and $\varepsilon^{-1}$ or as one of $d$ and
$(1+\ln\varepsilon^{-1})$.
  We find necessary and sufficient conditions on various notions of
tractability 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 and
necessary 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
$\ln^+ x=\max(1,\ln x)$.

35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex