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# Neural Gas

The neural gas algorithm (Martinetz and Schulten, 1991) sorts for each input signal the units of the network according to the distance of their reference vectors to . 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:
1.
Initialize the set to contain N units  with reference vectors chosen randomly according to .

Initialize the time parameter t: 2.
Generate at random an input signal according to .
3.
Order all elements of according to their distance to , i.e., find the sequence of indices such that is the reference vector closest to , is the reference vector second-closest to and is the reference vector such that k vectors exist with . Following Martinetz et al. (1993) we denote with the number k associated with .
4.
Adapt the reference vectors according to with the following time-dependencies:   5.
Increase the time parameter t: 6.
If continue with step 2

For the time-dependent parameters suitable initial values and final values 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: . 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: Competitive Hebbian Learning Up: Soft Competitive Learning without Previous: Soft Competitive Learning without

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