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## Hypercubes and Neural Networks

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**Hypercubes and Neural Networks**bill wolfe 10/23/2005**Simple Neural Model**• aiActivation • ei External input • wij Connection Strength Assume: wij = wji (“symmetric” network) W = (wij) is a symmetric matrix**Net Input**Vector Format:**Dynamics**• Basic idea:**Lower Energy**• da/dt = net = -grad(E) seeks lower energy**Keeps the activation vector inside the hypercube boundaries**Encourages convergence to corners**Summary: The Neural Model**aiActivation eiExternal Input wijConnection Strength W (wij = wji) Symmetric**Example: Inhibitory Networks**• Completely inhibitory • wij = -1 for all i,j • k-winner • Inhibitory Grid • neighborhood inhibition**Traveling Salesman Problem**• Classic combinatorial optimization problem • Find the shortest “tour” through n cities • n!/2n distinct tours**An Effective Heuristic for the Traveling Salesman Problem**S. Lin and B. W. Kernighan Operations Research, 1973 http://www.jstor.org/view/0030364x/ap010105/01a00060/0**Neural Network Approach**neuron**Tours – Permutation Matrices**tour: CDBA permutation matrices correspond to the “feasible” states.**Only one city per time stopOnly one time stop per**cityInhibitory rows and columns inhibitory**Distance Connections:**Inhibit the neighboring cities in proportion to their distances.**Research Questions**• Which architecture is best? • Does the network produce: • feasible solutions? • high quality solutions? • optimal solutions? • How do the initial activations affect network performance? • Is the network similar to “nearest city” or any other traditional heuristic? • How does the particular city configuration affect network performance? • Is there a better way to understand the nonlinear dynamics?**Initial Phase**Fuzzy Tour Neural Activations