On the Vapnik-Chervonenkis dimension of computer programs which use transcendental elementary operations

Article

Abstract

We exhibit upper bounds for the Vapnik-Chervonenkis (VC) dimension of a wide family of concept classes that are defined by algorithms using analytic Pfaffian functions. We give upper bounds on the VC dimension of concept classes in which the membership test for whether an input belongs to a concept in the class can be performed either by a computation tree or by a circuit with sign gates containing Pfaffian functions as operators. These new bounds are polynomial both in the height of the tree and in the depth of the circuit. As consequence we obtain polynomial VC dimension not also for classes of concepts whose membership test can be defined by polynomial time algorithms but also for those defined by well-parallelizable sequential exponential time algorithms.

Keywords

Concept learning Vapnik–Chervonenkis dimension Khovanskii bounds Pfaffian functions Parallel computation Formula size 

Mathematics Subject Classifications (2000)

68T05 58A17 

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Depto. de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain

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