Pattern Recognition Techniques - Biomedical Signal Analysis: Contemporary Methods and Applications - page 181

x 1 x x n Input layer

2

Radial basis
functions

h 1 h 2 h n

c 1 c 2 c n

+

Output layer

F

Figure 6.10
Approximation network.

rons represents a universal approximator based on the Stone-Weierstrass
theorem [209]. In essence, every multivariate, nonlinear, and continuous
function can be approximated.
2. The interpolation network with radial-basis functions has the best ap-
proximation property compared to other neural networks, such as the
three-layer perceptron. The sigmoid function does not represent a trans-
lation and rotation-invariant function, as the radial-basis function does.
Thus, every unknown nonlinear function f is better approximated by a
choice of coecients than any other choice.
3. The interpolation problem can be solved even more simply by choosing
radial-basis functions of the same width σ i = σ , as shown in [197]:

c i g ||

N

x − m i ||

F ( x )= (6.22)

σ

i=1

In other words, Gaussian functions of the same width can approximate
any given function.

Data processing in radial-basis function networks

Radial-basis neural networks implement a hybrid learning algorithm.
They have a combined learning scheme of supervised learning for the out-
put weights and unsupervised learning for radial-basis neurons. The ac-