Abstract
Devices such as neural networks typically approximate the elements of some function space X by elements of a nontrivial finite union M of finite-dimensional spaces. It is shown that if X=L p(Ω) (1<p<∞ and Ω⊂R d), then for any positive constant Γ and any continuous function φ from X to M, ‖f−φ(f)‖>‖f−M‖+Γ for some f in X. Thus, no continuous finite neural network approximation can be within any positive constant of a best approximation in the L p-norm.
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Kainen, P.C., Kůrková, V. & Vogt, A. Continuity of Approximation by Neural Networks in Lp Spaces. Annals of Operations Research 101, 143–147 (2001). https://doi.org/10.1023/A:1010916406274
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DOI: https://doi.org/10.1023/A:1010916406274