where p(W) (11), p(X), p(X|W), p(Y), p(Z), p(Z|Y) are fixed, yields
that middle channel p₀(Y|X), from a set of permissible middle channels
{p(Y|X)}, which maximizes R(Z,W).
Then the resulting middle channel describes that reorganization of the
SOM which yields the optimum simulation of W ⟶ X by the SOM, within the
constraints upon Ch(Z,Y).
The solution (the middle channel) depends of course on the particular
end channels. Obviously the algorithm which is used to find the
solution does not. It follows that if some physical process were
constrained to carrying out the steps specified by the algorithm,
said process would be capable of simulation and would exhibit
self-organization.
Although the formal solution to the three-channels-in-cascade problem
is not complete, the solution is sufficiently well characterized to
permit proceeding with a mechanization of the algorithm. A considerable
portion of the solution is concerned with the decomposition and
metrization of channels and it is upon this feature that we now focus
attention.
As suggested in the companion paper, if the dimensionality of the
spaces is greater than one, the SOM has only one method available (12).
Consider the decomposition of a space without, for the moment, making
the distinction between input and output.
Figure 5 depicts objects represented by a (perhaps multidimensional)
“cloud” of points. In the absence of a preassigned coordinate system,
the SOM computes the center of gravity of the cloud (which can be
done in any coordinate system) and describes the points in terms of
the distance from this center of gravity; or, which is the same, as
concentric spheres with origin at the center of gravity.
[Illustration: Figure 5—Nilpotent decomposition of a three-dimensional
space]
The direction of particular point cannot be specified for there is no
reference radius vector. Since the SOM wants to end up with a cartesian
coordinate system, it must transform the sphere (a two-dimensional
surface) into a plane (a two-dimensional surface). Unfortunately, a
sphere is not homeomorphic to a plane; thus the SOM has to decompose
the sphere into a cartesian product of a hemisphere (12a) and a
denumerable group. The SOM then can transform the hemisphere into a
plane. The points projected onto the plane constitute a space of the
same character as the one with which the SOM started. Thus, it can
repeat all operations on the plane (a space of one less dimension) by
finding the center of gravity and the circle upon which the desired
point is situated. The circle is similarly decomposed into a line times
a denumerable group. By repeating this operation as many times as the
space has dimensions, the SOM eventually arrives at a single point and
has obtained in the process a description of the space. Since this
procedure can be carried on by the repeated use of one operator, this
operator is nilpotent and to reflect this fact as well as the use of a
projection, we have named this a nilpotent projection operator or NPO
for short.