Communications of the ACM
Parallel arithmetic computations: a survey
Proceedings of the 12th symposium on Mathematical foundations of computer science 1986
Learnability and the Vapnik-Chervonenkis dimension
Journal of the ACM (JACM)
Bounding the Vapnik-Chervonenkis Dimension of Concept Classes Parameterized by Real Numbers
Machine Learning - Special issue on COLT '93
Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks
Journal of Computer and System Sciences - Special issue: dedicated to the memory of Paris Kanellakis
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Vapnik-Chervonenkis Dimension of Parallel Arithmetic Computations
ALT '07 Proceedings of the 18th international conference on Algorithmic Learning Theory
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We study the Vapnik-Chervonenkis dimension of concept classes that are defined by computer programs using analytic algebraic functionals (Nash operators) as primitives. Such bounds are of interest in learning theory because of the fundamental role the Vapnik-Chervonenkis dimension plays in characterizing the sample complexity required to learn concept classes. We strengthen previous results by Goldberg and Jerrum giving 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 an algebraic computation tree or by an algebraic circuit containing analytic algebraic gates. These new bounds are polynomial both in the height of the tree and in the depth of the circuit. This means in particular that VC dimension of computer programs using Nash operators is polynomial not only in the sequential complexity but also in the parallel complexity what ensures polynomial VC dimension for classes of concepts whose membership test can be defined by well-parallelizable sequential exponential time algorithms using analytic algebraic operators.