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Archiv der Mathematik

, Volume 80, Issue 3, pp 310–318 | Cite as

Entropy numbers of convex hulls and an application to learning algorithms

  • I. Steinwart
Original paper

Abstract.

Given a positive sequence \( a = (a_n) \in \ell_{p,q} \), for 0 < p < 2 and \( 0 < q \leq \infty \), and a finite set \( A = \{x_1, \dots , x_m\} \subset \ell_2 \) with \( |x_i| \leq a \) for all \( i = 1, \dots , m \) we prove¶¶\( \|(e_{n}(\textrm{aco}A))\|_{p,q} \leq c_{p,q} \sqrt{\textrm{log}(m + 1)}\,\, \|a\|_{p,q}, \)¶¶where \( e_{n}(\textrm{aco}A \) is the n th dyadic entropy number of the absolutely convex hull acoA of A and c p,q > 0 is a suitable constant only depending on p and q. Moreover we show that this is asymptotically optimal in M for the most interesting case \( q = \infty \).¶As an application we give an upper bound for the so-called growth function which is of special interest in the theory of learning algorithms.

Mathematics Subject Classification (2000): 41A46, 68T05, 68Q32. 

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Copyright information

© Birkhäuser Verlag, Basel, 2003

Authors and Affiliations

  • I. Steinwart
    • 1
  1. 1.Mathematisches Institut, Friedrich-Schiller-Universität, D-07743 Jena, Germany,¶steinwart@minet.uni-jena.deDE

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