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Hypercubes and Neural Networks

Hypercubes and Neural Networks

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Hypercubes and Neural Networks

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  1. Hypercubes and Neural Networks bill wolfe 10/23/2005

  2. Modeling

  3. Simple Neural Model • aiActivation • ei External input • wij Connection Strength Assume: wij = wji (“symmetric” network) W = (wij) is a symmetric matrix

  4. Net Input Vector Format:

  5. Dynamics • Basic idea:

  6. Energy

  7. Lower Energy • da/dt = net = -grad(E)  seeks lower energy

  8. Problem: Divergence

  9. A Fix: Saturation

  10. Keeps the activation vector inside the hypercube boundaries Encourages convergence to corners

  11. Summary: The Neural Model aiActivation eiExternal Input wijConnection Strength W (wij = wji) Symmetric

  12. Example: Inhibitory Networks • Completely inhibitory • wij = -1 for all i,j • k-winner • Inhibitory Grid • neighborhood inhibition

  13. Traveling Salesman Problem • Classic combinatorial optimization problem • Find the shortest “tour” through n cities • n!/2n distinct tours

  14. TSP solution for 15,000 cities in Germany

  15. TSP50 City Example

  16. Random

  17. Nearest-City

  18. 2-OPT

  19. 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

  20. Centroid

  21. Monotonic

  22. Neural Network Approach neuron

  23. Tours – Permutation Matrices tour: CDBA permutation matrices correspond to the “feasible” states.

  24. Not Allowed

  25. Only one city per time stopOnly one time stop per cityInhibitory rows and columns inhibitory

  26. Distance Connections: Inhibit the neighboring cities in proportion to their distances.

  27. putting it all together:

  28. 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?

  29. typical state of the network before convergence

  30. “Fuzzy Readout”

  31. Initial Phase Fuzzy Tour Neural Activations