|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
A new upper bound for sampling numbers | | Nicolas Nagel
; Martin Schäfer
; Tino Ullrich
; | | Date: |
30 Sep 2020 | | Abstract: | We provide a new upper bound for sampling numbers $(g_n)_{nin mathbb{N}}$
associated to the compact embedding of a separable reproducing kernel Hilbert
space into the space of square integrable functions. There are universal
constants $C,c>0$ (which are specified in the paper) such that $$
g^2_n leq frac{Clog(n)}{n}sumlimits_{kgeq lfloor cn
floor}
sigma_k^2quad,quad ngeq 2,, $$ where $(sigma_k)_{kin mathbb{N}}$ is the
sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt
embedding $ ext{Id}:H(K) o L_2(D,varrho_D)$. The algorithm which realizes
the bound is a least squares algorithm based on a specific set of sampling
nodes. These are constructed out of a random draw in combination with a
down-sampling procedure coming from the celebrated proof of Weaver’s
conjecture, which was shown to be equivalent to the Kadison-Singer problem. Our
result is non-constructive since we only show the existence of a linear
sampling operator realizing the above bound. The general result can for
instance be applied to the well-known situation of
$H^s_{ ext{mix}}(mathbb{T}^d)$ in $L_2(mathbb{T}^d)$ with $s>1/2$. We obtain
the asymptotic bound $$
g_n leq C_{s,d}n^{-s}log(n)^{(d-1)s+1/2},, $$ which improves on very
recent results by shortening the gap between upper and lower bound to
$sqrt{log(n)}$. | | Source: | arXiv, 2010.00327 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER