Part 11

Consider the relation between two mothers and their sons as pictured in Figure 4. Observe that if symbols (points) are substituted for the actual physical objects (mothers and sons), the relation is not altered in any way. This is what we mean by simulation and this is how a SOM simulates. It is not even necessary that the objects, used to display the relation, be defined; i.e., these objects may be primitive. (If this were not so, no mathematical or physical theory could model the environment.) The main prerequisite is sufficient resolution to distinguish the objects from each other.

MATHEMATICAL MODEL

The mathematical model must represent both the environment and the SOM and for reasons given in the companion paper each is represented as a metrizable topology. For uniqueness we factor each space into equal parts and represent the environment as the channel

W ⟶ X. (Ref. 10a)

Consider now the SOM to be represented by the cascaded channels

X ⟶ Y ⟶ Z

where X ⟶ Y is a variable which represents the reorganization of the SOM existing input-output relation represented by Y ⟶ Z.

The solution of the three channels-in-cascade problem

W ⟶ X ⟶ Y ⟶ Z,

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.

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.

MECHANIZATION OF THE NPO

Analog computer elements were used to simulate one NPO which was tested in the experimental configuration shown in Figure 6. The NPO operates upon a channel which is artificially generated from the two noise generators i₁ and i₂ and the signal generator i₀ (i₀ may also be a noise generator). The NPO accepts the inputs labelled X₁ and X₂ and provides the three outputs Ξ₁, Ξ₂, and γ. X₁ is the linear combination of the outputs of generators i₁ and i₀, similarly X₂ is obtained from i₂ and i₀.

Obviously, i₀ is an important parameter since it represents the memory relating the spaces X₁ and X₂. Ξ₁ has the property that the magnitude of its projection on i₀ is a maximum while Ξ₂ to the opposite has a zero projection on i₀. γ is the detected version of the eigenvalue of Ch(X₂,X₁).

In the companion paper it was shown how one can provide a Euclidean geometrical representation of the NPO. This representation is shown in Figure 7 which shows the vectors i₀, i₁, i₂, X₁, X₂, Ξ₁, Ξ₂, and the angles Θ₁, Θ₂, and γ. The length of a vector is given by

|X| = κₓ(2πε)⁻¹ᐟ² ∈ H(X)

and the angle between two vectors by

|Θ(X₁,X₂)|-sin⁻¹ ∈ -R(X₁,X₂).

The three vectors i₀, i₁, i₂ provide an orthogonal coordinate system because the corresponding signals are random, i.e.,

κ R(i₀,i₁,i₂) ≡ 0.

As external observers we have a prior knowledge of this coordinate system; however, the NPO is given only the vectors X₁ and X₂ in the i₀ ⨉ i₁ and i₀ ⨉ i₂ planes. The NPO can reconstruct the entire geometry but the actual output Ξ obviously is constrained to lie in the plane of the input vector X. The following formulas are typical of the relations present.

|Ξ₁| tan β = ———— |Ξ₂|

cos Θ = cos 2β csc 2γ