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76 Extensions to the Standard PSOM Algorithm

Both aspects motivate two extensions to the standard PSOM approach: the
“Local-PSOMs” and the “Chebyshev-spaced PSOM”, which are the focus
of the Sec. 6.3 and 6.4.

6.1 The “Multi-Start Technique”

The multi-start technique was developed to overcome the multiple minima
limitations of the simpler best-match start procedure adopted so far (see
Sec. 4.3).

a) X_1 -> X_2 W 4 b) X_1 -> X_2 M
M
W_a W_a
x x W
2 W 2 3
3

W
2 W
2
W W(s) x 1 W x 1
1 1

Figure 6.1: The problem of local and multiple minima can be solved by the
multi-start technique. The solid curve shows the embedded one-dimensional
(m ) PSOM manifold, spanned by the four asterisks marked reference vectors

fw w w w g in IR . The dashed line connects a set of diamont-marked PSOM
mappings x x .
(a) A pathological situation for the standard approach: depending on the starting
location s t , the best-match search can be trapped in a local minimum.
(b) The multi-start technique solves the task correctly and can be employed to
find multiple solutions.

To understand the rationale behind this technique let us consider the
four-node PSOM with the S-shaped manifold introduced before in Fig. 5.10.
On the left Fig. 6.1a the diamonds on the dotted line show a set of PSOM
mappings x x (P=diag(1,0)). Starting at the small x values, the best-
match procedure finds the first node w as start point. When (after the 7th
trial) the third reference vector w gets closer than w , the gradient descent
iteration starts at w and becomes “trapped” in a local minimum, giving
rise to a totally misleading value for x . On the other trials this problem