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Hypercubes and Neural Networks bill wolfe 9/21/2005
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Modeling
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“activation level” “Net Input”
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0 <= a i <= 1 Saturation
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da j /dt = Net j (1-a j )(a j ) Dynamics
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3 Neuron Example
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Brain State:
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“Thinking”
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Binary Model a j = 0 or 1 Neurons are either “on” or “off”
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Binary Stability a j = 1 and Net j >=0 Or a j = 0 and Net j <=0
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Hypercubes
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4-Cube
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5-Cube
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http://www1.tip.nl/~t515027/hypercube.html Hypercube Computer Game
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2-Cube Adjacency Matrix: Hypercube Graph
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Recursive Definition
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Theorem 1: If v is an eigenvector of Q n-1 with eigenvalue x then the concatenated vectors [v,v] and [v,-v] are eigenvectors of Q n with eigenvalues x+1 and x-1 respectively. Eigenvectors of the Adjacency Matrix
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Proof
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Generating Eigenvectors and Eigenvalues
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Walsh Functions for n=1, 2, 3
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1 000 001 010 011 100 101 110 111 eigenvectorbinary number
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n=3
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Theorem 3: Let k be the number of +1 choices in the recursive construction of the eigenvectors of the n-cube. Then for k not equal to n each Walsh state has 2 n-k-1 non adjacent subcubes of dimension k that are labeled +1 on their vertices, and 2 n-k-1 non adjacent subcubes of dimension k that are labeled -1 on their vertices. If k = n then all the vertices are labeled +1. (Note: Here, "non adjacent" means the subcubes do not share any edges or vertices and there are no edges between the subcubes).
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n=5, k= 3n=5, k= 2
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Inhibitory Hypercube
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Theorem 5: Each Walsh state with positive, zero, or negative eigenvalue is an unstable, weakly stable, or strongly stable state of the inhibitory hypercube network, respectively.
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