by **
I. Steinwart**

**Preprint series:**
99-59, Reports on Analysis

**MSC:**- 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
- 68T05 Learning and adaptive systems

**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*

**Upload:** 1999-07-16

**Update:** 2002-04-19

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