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
Complexity lower bounds for computation trees with elementary transcendental function gates
Theoretical Computer Science - Special issue on complexity theory and the theory of algorithms as developed in the CIS
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
Combinatorial Hardness Proofs for Polynomial Evaluation
MFCS '98 Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
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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.