1 / 88

Hypercubes and Neural Networks

Hypercubes and Neural Networks. bill wolfe 10/23/2005. Modeling. Simple Neural Model. a i Activation e i External input w ij Connection Strength Assume: w ij = w ji (“symmetric” network)  W = (w ij ) is a symmetric matrix . Net Input. Vector Format:. Dynamics. Basic idea:.

mattox
Télécharger la présentation

Hypercubes and Neural Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related