Comments on "Stochastic choice of basis functions in adaptive function approximation and the functional-link net"...

Area of Science:

• Neural Networks
• Machine Learning Theory
• Computational Neuroscience

Background:

• The paper addresses comments and amendments to a previous publication by Igelnik et al. (1995).
• Theorem 1 from the original paper is the focus of revision.
• The core issue involves the space of thresholds in the hidden layer of neural networks.

Purpose of the Study:

• To provide amendments and comments on the paper by Igelnik et al. (1995).
• To revise Theorem 1 concerning the thresholds in the hidden layer.
• To clarify the mathematical space for these thresholds.

Main Methods:

• Mathematical analysis and revision of Theorem 1.
• Inclusion of specific mathematical expressions for thresholds: b(0) = -w(0).y(0)-u(0).
• Definition of the distribution space for u(0) as V(d)=[0; Omega]x[-Omega; Omega](d-1), I(d), and [-2dOmega, 2dOmega].

Main Results:

• A revised statement for Theorem 1 is presented, specifying the thresholds in the hidden layer.
• The revised theorem defines the thresholds based on weight vectors and uniformly distributed random variables.
• Igelnik et al. acknowledged a minor omission in a trigonometric identity but confirmed the essential validity of Theorem 1.

Conclusions:

• The revision clarifies the mathematical underpinnings of Theorem 1.
• Despite a minor correction, the fundamental conclusions of the original Theorem 1 remain valid.
• This work contributes to the theoretical understanding of neural network hidden layer thresholds.