Asymptotic density of the SOM units.

The convergence of the process can be forced by decreasing the learning rate gradually towards zero following a schedule that allows enough learning and controls the balance between the effect of successive adaptations. Stochastic approximation [Robbins and Monro, 1951] can be used to reach a suitable stable state for the SOM and for a M=D=1 SOM it can be shown [Ritter and Schulten, 1988] that the following conditions are necessary and sufficient to reach the existing stable state from all sufficiently close initial states with a pre-specified neighborhood function using any small positive function $\alpha(t)\:$:

$$\lim_{T \rightarrow \infty} \int_0^T \alpha(t) dt = \infty \:,$$ (7)

$$\lim_{T \rightarrow \infty} \alpha(T) = 0\:.$$ (8)

As shown in [Ritter, 1989] for M=D=1 SOMs, the width of the neighborhood function h_ci affects strongly the asymptotic point density of the SOM units. The asymptotic point density is the number of units in the small unit area of input space and is given proportional to the probability density of the input samples p(x). This relation of the densities is also known as the magnification factor [Kohonen, 1995,Bauer et al., 1996] of the SOM. If no neighbors are adjusted, the magnification factor is proportional to p(x)^1/3 and as the number of adjusted neighbors is increased it will increase towards p(x)^2/3 [Ritter and Schulten, 1986]. If the number of adjusted neighbors is constant through the learning process the asymptotic value of the magnification factor can be determined exactly [Ritter, 1989,Ritter, 1991].

For multidimensional input the analysis of the magnification factor of the SOM is very difficult. For the classical vector quantization (VQ) with no topology (r=0) the exponent of the magnification factor proportionality discussed above will be quite close to one, if the dimension is high. It can be shown that the density of the best random quantizer is proportional to $p(\mbox{\boldmath$x$})^{\frac{D}{D+2}}$, where D is the dimension of the input vectors [Zador, 1982,Cottrell et al., 1997].

Depending on the shape of the neighborhood function, there may exist stationary states other than the ordered state (4) as can be shown for M=D=1 SOMs [Erwin et al., 1992b]. Due to the meta-stable states the ordering time may increase by orders of magnitude, if too narrow pre-specified neighborhood function is used throughout the process. According to empirical experience with different SOMs and different input the organization is most conveniently achieved by applying wide neighborhood at first, then decreasing it gradually and finally using very slow decrease, when the number of affected neighbors approach one [Kohonen, 1990b,Erwin et al., 1992b,Kohonen, 1995].