Mind, Body, World- Foundations of Cognitive Science, 2013a

This geometric observation raises complications, because it implies that a
hidden value unit will not have a single trigger feature. This is because there are
many input patterns that are orthogonal to a vector of connection weights. Any
input vector that lies in the hyperplane that is perpendicular to the vector of connection
weights will serve as a trigger feature for the hidden value unit (Dawson, 2004);
this is illustrated in Figure 4-14.
Another consequence of the geometric account provided above is that there
should be families of other input patterns that share the property of producing the
same hidden unit activity, but one that is lower than the maximum activity produced
by one of the trigger features. These will be patterns that all fall into the
same hyperplane, but this hyperplane is not orthogonal to the vector of connection
weights.
Vector of
Weights
Plane of Trigger
Features
Figure 4-14. Any input pattern (dashed lines) whose vector falls in the plane
orthogonal to the vector of connection weights (solid line) will be a trigger
feature for a hidden value unit.
The upshot of all of this is that if one trains a network of value units and then wiretaps
its hidden units, the resulting hidden unit activities should be highly organized.
Instead of having a rectangular distribution of activation values, there should
be regular groups of activations, where each group is related to a different family
of input patterns (i.e., families related to different hyperplanes of input patterns).
Empirical support for this analysis was provided by the discovery of activity
174 Chapter 4