Lower bound on VC-dimension by local shattering
Neural Computation
Neural networks with local receptive fields and superlinear VC Dimension
Neural Computation
Radial Basis Function Neural Networks Have Superlinear VC Dimension
COLT '01/EuroCOLT '01 Proceedings of the 14th Annual Conference on Computational Learning Theory and and 5th European Conference on Computational Learning Theory
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We examine the relationship between the VC dimension and thenumber of parameters of a threshold smoothly parameterized functionclass. We show that the VC dimension of such a function class is atleast k if there exists a k-dimensionaldifferentiable manifold in the parameter space such that eachmember of the manifold corresponds to a different decisionboundary. Using this result, we are able to obtain lower bounds onthe VC dimension proportional to the number of parameters forseveral thresholded function classes including two-layer neuralnetworks with certain smooth activation functions and radial basisfunctions with a gaussian basis. These lower bounds hold even ifthe magnitudes of the parameters are restricted to be arbitrarilysmall. In Valiant's probably approximately correct learningframework, this implies that the number of examples necessary forlearning these function classes is at least linear in the number ofparameters.