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Neural Gas

The neural gas algorithm (Martinetz and Schulten, 1991) sorts for each input signal tex2html_wrap_inline4681 the units of the network according to the distance of their reference vectors to tex2html_wrap_inline4683. Based on this ``rank order'' a certain number of units is adapted. Both the number of adapted units and the adaptation strength are decreased according to a fixed schedule. The complete neural gas algorithm is the following:
Initialize the set tex2html_wrap_inline4685 to contain N units tex2html_wrap_inline4689
with reference vectors tex2html_wrap_inline4691 chosen randomly according to tex2html_wrap_inline4693.

Initialize the time parameter t:

Generate at random an input signal tex2html_wrap_inline4697 according to tex2html_wrap_inline4699.
Order all elements of tex2html_wrap_inline4701 according to their distance to tex2html_wrap_inline4703, i.e., find the sequence of indices tex2html_wrap_inline4705 such that tex2html_wrap_inline4707 is the reference vector closest to tex2html_wrap_inline4709, tex2html_wrap_inline4711 is the reference vector second-closest to tex2html_wrap_inline4713 and tex2html_wrap_inline4715 is the reference vector such that k vectors tex2html_wrap_inline4719 exist with tex2html_wrap_inline4721. Following Martinetz et al. (1993) we denote with tex2html_wrap_inline4723 the number k associated with tex2html_wrap_inline4727.
Adapt the reference vectors according to
with the following time-dependencies:


Increase the time parameter t:
If tex2html_wrap_inline4735 continue with step 2

For the time-dependent parameters suitable initial values tex2html_wrap_inline4737 and final values tex2html_wrap_inline4739 have to be chosen. Figure 5.1 shows some stages of a simulation for a simple ring-shaped data distribution. Figure 5.2 displays the final results after 40000 adaptation steps for three other distribution. Following Martinetz et al. (1993) we used the following parameters: tex2html_wrap_inline4741.

Figure 5.1:   Neural gas simulation sequence for a ring-shaped uniform probability distribution. a) Initial state. b-f) Intermediate states. g) Final state. h) Voronoi tessellation corresponding to the final state. Initially strong neighborhood interaction leads to a clustering of the reference vectors which then relaxes until at the end a rather even distribution of reference vectors is found.

Figure:   Neural gas simulation results after 40000 input signals for three different probability distributions (described in the caption of figure 4.4).

next up previous contents
Next: Competitive Hebbian Learning Up: Soft Competitive Learning without Previous: Soft Competitive Learning without

Bernd Fritzke
Sat Apr 5 18:17:58 MET DST 1997