Original Contribution: Approximation of continuous functions on Rd by linear combinations of shifted rotations of a sigmoid function with and without scaling

Authors:
Yoshifusa Ito
Affiliations:
-
Venue:
Neural Networks
Year:
1992

Citing 4
Cited 12

Multilayer feedforward networks are universal approximators

Neural Networks
On the approximate realization of continuous mappings by neural networks

Neural Networks
Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory

Neural Networks
Approximation of functions on a compact set by finite sums of a sigmoid function without scaling

Neural Networks

Intelligent locomotion control on sloping surfaces

Information Sciences—Informatics and Computer Science: An International Journal
A Weak Condition on Linear Independence of Unscaled Shifts of a Function and Finite Mappings by Neural Networks

ICANN '02 Proceedings of the International Conference on Artificial Neural Networks
Activation functions defined on higher-dimensional spaces for approximation on compact sets with and without scaling

Neural Computation
Approximation capability of layered neural networks with sigmoid units on two layers

Neural Computation
Simultaneous approximations of polynomials and derivatives and their applications to neural networks

Neural Computation
Systemical convergence rate analysis of convex incremental feedforward neural networks

Neurocomputing
GPS orbit approximation using radial basis function networks

Computers & Geosciences
Equivalent Relationship of Feedforward Neural Networks and Real-Time Face Detection System

Proceedings of the FIRA RoboWorld Congress 2009 on Advances in Robotics
Some new results on neural network approximation

Neural Networks
Effects of using different neural network structures and cost functions in locomotion control

ICNC'06 Proceedings of the Second international conference on Advances in Natural Computation - Volume Part I
Approximation bound for fuzzy-neural networks with bell membership function

FSKD'05 Proceedings of the Second international conference on Fuzzy Systems and Knowledge Discovery - Volume Part I
Improvement on the approximation bound for fuzzy-neural networks clustering method with gaussian membership function

ADMA'05 Proceedings of the First international conference on Advanced Data Mining and Applications

Quantified Score

Hi-index	0.00

Visualization

Abstract

Let f be an arbitrary continuous function defined on R^d and h be any sigmoid function. Then, there is a linear combination of scaled shifted rotations of h, which approximates f uniformly on the whole space R^d, if a certain such linear combination approximates f uniformly in a neighbourhood of the infinite point. From this result, derived is that any function continuous on R@?^d (the one-point compactification of R^d) can be likewise approximated. Further, obtained is a necessary and sufficient condition on h, under which the uniform approximation can be implemented without scaling of h. The Heaviside function, the logistic function, the Gaussian distribution function, and the arctangent sigmoid function satisfy this condition. Any sigmoid function that increases only in a finite interval satisfies it. Examples of sigmoid functions incapable of implementing the uniform approximation without scaling are also illustrated.