A Weak Condition on Linear Independence of Unscaled Shifts of a Function and Finite Mappings by Neural Networks

Authors:
Yoshifusa Ito
Affiliations:
-
Venue:
ICANN '02 Proceedings of the International Conference on Artificial Neural Networks
Year:
2002

Citing 5
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Activation functions defined on higher-dimensional spaces for approximation on compact sets with and without scaling

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Abstract

Let 1 驴 c 驴 d and let g be a slowly increasing function defined on Rc. Suppose that the support of the Fourier transform Fcg of g includes a converging sequence of distinct points yk which sufficiently rapidly come close to a line as k驴驴. Then, any mapping of any number, say n, of any points x1, . . ., xn in Rd onto R can be implemented by a linear sum of the form 驴jn=1 ajg(Wxi + zj). Here, W is a d 脳 c matrix having orthonormal row vectors, implying that g is used without scaling, and that the sigmoid function defined on R and the radial basis function defined on Rd are treated on a common basis.