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By Clark Jeffries (auth.), Clark Jeffries (eds.)

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Extra info for Code Recognition and Set Selection with Neural Networks

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In summary, for some applications in content addressable memory and robot arm control, neural networks should only have constant trajectories as attractors. 2, one design route to this feature is the organization of neurons in mutually inhibitory or mutually excitatory interaction subsets (hypergraphs with same sign barycenters) together with the balanced cycle condition. 1 ) The goal of this section is to derive an algebraic equivalence between this form and a sum of products, n terms in each product, each multiplicand in each product being either gi or l-gi ; in certain cases the latter form turns out to be more tractable in the analysis of neural models.

Jp} the associated coefficient simplex. 1) with analogous coefficients {a_ } of the same sign comprises an equivalence class; we call such a set a sign equivalence class. We let N denote the number of coefficient simplexes in a sign equivalence class. 2) are {I. 2. 3}. 3. 4}. {1,4}. 5}. 2) is a hypergraph H with the following types of components. The hypergraph vertices are points labelled {vI. • v n }. • ~} of H are additional vertices. •jp}' Graphically each barycenter (open circle) is joined to these p+ 1 vertices by p+ 1 edges (line segments).

PROOF. Suppose A is an answer set. If neuron i is in the answer set, defme the i th component of an n-vector by xi = + 1; otherwise define xi = -1. 9). Suppose Xj = 1. Then gj = 1 and for all j ¢. i in the two subsets Ra and Cb containing i, gj = O. 9) adds up to zero. Suppose Xj = -1. Then gj = O. By the covering property, at least one and possibly two vertices in the union ofRa and Cb must have gk = 1; other g values in such subsets are all 0 by the intersection property. 9) again adds up to zero.

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