Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the worst-case error of least squares algorithms for
  $L_2$-approximation with high probability   It was recently shown in [4] that, for $L_2$-approximation of functions from
a Hilbert space, function values are almost as powerful as arbitrary linear
information, if the approximation numbers are square-summable. That is, we
showed that \[
  e_n \,\lesssim\, \sqrt{\frac{1}{k_n} \sum_{j\geq k_n} a_j^2} \qquad \text{
with }\quad k_n \asymp \frac{n}{\ln(n)}, \] where $e_n$ are the sampling
numbers and $a_k$ are the approximation numbers. In particular, if
$(a_k)\in\ell_2$, then $e_n$ and $a_n$ are of the same polynomial order. For
this, we presented an explicit (weighted least squares) algorithm based on
i.i.d. random points and proved that this works with positive probability. This
implies the existence of a good deterministic sampling algorithm.
  Here, we present a modification of the proof in [4] that shows that the same
algorithm works with probability at least $1-{n^{-c}}$ for all $c>0$.

9
The categories of this paper are: nlin
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg