Hypercubes and Neural Networks

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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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Saturation
0 lt ai lt 1
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Dynamics
daj/dt Netj (1-aj)(aj)
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3 Neuron Example
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Brain State lta1, a2, a3gt
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Thinking
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Binary Model
aj 0 or 1
Neurons are either on or off
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Binary Stability
aj 1 and Netj gt0 Or aj 0 and Netj lt0
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Hypercubes
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4-Cube
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4-Cube
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5-Cube
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5-Cube
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5-Cube
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Hypercube Computer Game
http//www1.tip.nl/t515027/hypercube.html
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Hypercube Graph
2-Cube
Adjacency Matrix
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Recursive Definition

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Eigenvectors of the Adjacency Matrix
Theorem 1 If v is an eigenvector of Qn-1 with
eigenvalue x then the concatenated vectors
v,v and v,-v are eigenvectors of Qn with
eigenvalues x1 and x-1 respectively.
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Proof

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Generating Eigenvectors and Eigenvalues
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Walsh Functions for n1, 2, 3
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eigenvector
binary number
1 1 1 1 -1 -1 -1 -1
000 001 010 011 100 101 110 111
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n3
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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 2n-k-1 non adjacent subcubes of
dimension k that are labeled 1 on their
vertices, and 2n-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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n5, k 3
n5, 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.