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Self-organizing Feature Map


This model stems from Kohonen (1982) and builds upon earlier work of Willshaw and von der Malsburg (1976). The model is similar to the (much later developed) neural gas model (see 5.1) since a decaying neighborhood range and adaptation strength are used. An important difference, however, is the topology which is constrained to be a two-dimensional grid (tex2html_wrap_inline4975) and does not change during self-organization.

The distance on this grid is used to determine how strongly a unit tex2html_wrap_inline4977 is adapted when the unit tex2html_wrap_inline4979 is the winner. The distance measure is the tex2html_wrap_inline4981-norm (a.k.a. ``Manhattan distance''):

Ritter et al. (1991) propose to use the following function to define the relative strength of adaptation for an arbitrary unit r in the network (given that s is the winner):

Thereby, the standard deviation tex2html_wrap_inline4991 of the Gaussian is varied according to
for a suitable initial value tex2html_wrap_inline4993 and a final value tex2html_wrap_inline4995. The complete self-organizing feature map algorithm is the following:

  1. Initialize the set tex2html_wrap_inline4997 to contain tex2html_wrap_inline4999 units tex2html_wrap_inline5001
    with reference vectors tex2html_wrap_inline5003 chosen randomly according to tex2html_wrap_inline5005.

    Initialize the connection set tex2html_wrap_inline5007 to form a rectangular tex2html_wrap_inline5009 grid.

    Initialize the time parameter t:

  2. Generate at random an input signal tex2html_wrap_inline5013 according to tex2html_wrap_inline5015.
  3. Determine the winner tex2html_wrap_inline5017:


  4. Adapt each unit r according to
  5. Increase the time parameter t:
  6. If tex2html_wrap_inline5029 continue with step 2.

Figure 6.1 shows some stages of a simulation for a simple ring-shaped data distribution. Figure 6.2 displays the final results after 40000 adaptation steps for three other distribution. The parameters were tex2html_wrap_inline5031 and tex2html_wrap_inline5033.

Figure 6.1:   Self-organizing feature map 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. Large adaptation rates in the beginning as well as a large neighborhood range cause strong initial adaptations which decrease towards the end.

Figure:   Self-organizing feature map simulation results after 40000 input signals for three different probability distributions (described in the caption of figure 4.4).

next up previous contents
Next: Growing Cell Structures Up: Soft Competitive Learning with Previous: Soft Competitive Learning with

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