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Using Matrices to Store Associations

By taking the outer product of one vector (representing an item) with another we can form a matrix representing the association between to items.

To determine what is associated with a given item we can take the inner product of the item with the assocaition matrix. The resulting vector will be the vector that was associated with the cue vector (presuming orthognal vectors of length one).

This operation can be represented in network form by a two layer network:

Figure 9: The network representation of a matrix.

Suppose we have three people represented by the vectors v, w, and x and two gangs represented by j and s.

Suppose the first two people belong to the j gang and the last belongs to the s gang.

We can form a memory of these associations by finding outer products and adding:

M = vj + wj + xs

To retrieve the gang of a given person we calculate the dot product of the item with the memory matrix.

w.M = w.(vj + wj + xs)
= (w.v)j + (w.w)j + (w.x) s
= j

This is the task in the second set of exercises