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