Some estimates for the entropy numbers of convex hulls with

Preprint series: 99-59, Reports on Analysis

MSC:
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy

Abstract: Given a positive sequence $a=(a_i) \in \ell_{p,q}$ for $0<p<2$
and $0<q\leq \infty$ and a finite set $A=\{x_1,\dots,x_m\}\subset \ell_2$
with $|x_i| \leq a$ we proof
$$\ynorm {(e_n(\co A))}{p,q} \Leq c_{p,q} \ \sqrt{\log (m+1)} \ \ynorm a {p,q}$$
for some constant $c_{p,q}>0$ only depending on $p$ and $q$. Moreover we show
that this is asymptotically optimal for $q=\infty$.\\
As an application we give an upper bound for the so-called growth function
which is of interest in the theory of learning machines of
support vector type.

Keywords: entropy numbers, convex hulls, support vector machines