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The Stone-Weierstrass theorem and its application to neural networks
1Dept. of Electr. Eng., Utah Univ., Salt Lake City, UT.
IEEE Transactions on Neural Networks
|January 1, 1990
Summary
The Stone-Weierstrass theorem enables neural networks to approximate any bounded measurable function using various building blocks. A novel modified logistic network is proposed for enhanced function approximation capabilities.
Area of Science:
- Mathematical analysis
- Artificial intelligence
- Neural networks
Background:
- The Stone-Weierstrass theorem is a fundamental result in real analysis.
- Neural networks are widely used for function approximation.
- Commonly used activation functions in neural networks have limitations.
Purpose of the Study:
- To review the Stone-Weierstrass theorem and its relevance to neural networks.
- To present neural network architectures inspired by the theorem.
- To propose a novel modified logistic network architecture.
Main Methods:
- Review of the Stone-Weierstrass theorem and its mathematical underpinnings.
- Design of neural network architectures utilizing exponential functions, polynomials, partial fractions, and Boolean functions.
- Development of a modified logistic network architecture.
Main Results:
- Demonstration that networks built with specified functions can approximate arbitrary bounded measurable functions.
- Validation of the proposed network architectures' theoretical capabilities.
- Introduction of a modified logistic network as a viable alternative to standard logistic networks.
Conclusions:
- The Stone-Weierstrass theorem provides a theoretical foundation for universal approximation in neural networks.
- Diverse function sets can be employed to construct powerful neural network approximators.
- The proposed modified logistic network offers a promising direction for advanced function approximation.
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