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Neural Computation

February 15, 1997, Vol. 9, No. 2, Pages 337-348
(doi: 10.1162/neco.1997.9.2.337)
© 1997 Massachusetts Institute of Technology
Shattering All Sets of ‘k’ Points in “General Position” Requires (k — 1)/2 Parameters
Article PDF (153.25 KB)
Eduardo D. Sontag
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 USA
Abstract

For classes of concepts defined by certain classes of analytic functions depending on n parameters, there are nonempty open sets of samples of length 2n + 2 that cannot be shattered. A slighly weaker result is also proved for piecewise-analytic functions. The special case of neural networks is discussed.