An improved VC dimension bound for sparse polynomials
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We show that the function class consisting of $k$-sparse polynomials in $n$ variables has Vapnik-Chervonenkis (VC) dimension at least $nk+1$. This result supersedes the previously known lower bound via $k$-term monotone disjunctive normal form (DNF) formulas obtained by Littlestone (1988). Moreover, it implies that the VC dimension for $k$-sparse polynomials is strictly larger than the VC dimension for $k$-term monotone DNF. The new bound is achieved by introducing an exponential approach that employs Gaussian radial basis function (RBF) neural networks for obtaining classifications of points in terms of sparse polynomials.
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