Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Breaking the Curse for Uniform Approximation in Hilbert Spaces via Monte
  Carlo Methods   We study the $L_{\infty}$-approximation of $d$-variate functions from Hilbert
spaces via linear functionals as information. It is a common phenomenon in
tractability studies that unweighted problems (with each dimension being
equally important) suffer from the curse of dimensionality in the deterministic
setting, that is, the number $n(\varepsilon,d)$ of information needed in order
to solve a problem to within a given accuracy $\varepsilon > 0$ grows
exponentially in $d$. We show that for certain approximation problems in
periodic tensor product spaces, in particular Korobov spaces with smoothness $r
> 1/2$, switching to the randomized setting can break the curse of
dimensionality, now having polynomial tractability, namely $n(\varepsilon,d)
\preceq \varepsilon^{-2} \, d \, (1 + \log d)$. Similar benefits of Monte Carlo
methods in terms of tractability have only been known for integration problems
so far.

7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an