Arbitrary Nonlinearity Is Sufficient to Represent All Functions by Neural Networks: A Theorem. Kreinovich, V. Y. 4(3):381–383.
Arbitrary Nonlinearity Is Sufficient to Represent All Functions by Neural Networks: A Theorem [link]Paper  doi  abstract   bibtex   
It is proved that if we have neurons implementing arbitrary linear functions and a neuron implementing one (arbitrary but smooth) nonlinear function g(x), then for every continuous function f(x1, …, xm) of arbitrarily many variables and for arbitrary e $>$ 0 we can construct a network that consists of g-neurons and linear neurons and computes f with precision e.
@article{kreinovichArbitraryNonlinearitySufficient1991,
  title = {Arbitrary Nonlinearity Is Sufficient to Represent All Functions by Neural Networks: A Theorem},
  author = {Kreinovich, Vladik Y.},
  date = {1991-01},
  journaltitle = {Neural Networks},
  volume = {4},
  pages = {381--383},
  issn = {0893-6080},
  doi = {10.1016/0893-6080(91)90074-f},
  url = {https://doi.org/10.1016/0893-6080(91)90074-f},
  abstract = {It is proved that if we have neurons implementing arbitrary linear functions and a neuron implementing one (arbitrary but smooth) nonlinear function g(x), then for every continuous function f(x1, …, xm) of arbitrarily many variables and for arbitrary e {$>$} 0 we can construct a network that consists of g-neurons and linear neurons and computes f with precision e.},
  keywords = {*imported-from-citeulike-INRMM,~INRMM-MiD:c-10833268,artificial-neural-networks,back-propagation-networks,mapping-networks,network-representation-capability,nonideal-neurons,universal-approximation},
  number = {3}
}
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