Neural Networks Chapter 4

1
Neural NetworksChapter 4

• Joost N. Kok
• Universiteit Leiden

2
Hopfield Networks

• Optimization Problems (like Traveling Salesman)
  can be encoded into Hopfield Networks
• Fitness corresponds to energy of network
• Good solutions are stable points of the network

3
Hopfield Networks

• Three Problems
• Weighted Matching
• Traveling Salesman
• Graph Bipartitioning

4
Hopfield Networks

• Weighted matching Problem
• Let be given N points with distances dij
• Connect points together in pairs such that the
  total sum of distances is as small as possible

5
Hopfield Networks

• Variables nij (iltj) with values 0/1
• Constraint Sj nij 1 for all i
• Optimize Siltj dij nij

6
Hopfield Networks

• Penalty Term approach put constraints in
  optimization criterion
• Weights and thresholds of Hopfield Network can be
  derived from

7
Hopfield Networks

• Travelling Salesman Problem (TSP)Given N cities
  with distances dij .What is the shortest tour?

8
(No Transcript)
9
Hopfield Networks

• Construct a Hopfield network with N2 nodes
• Semantics nia 1 iff town i on position a in
  tour

10
Hopfield Networks

• Constraints

11
Hopfield Networks

• 0/1 Nodes
• Nodes within each row connected with weight g
• Nodes within each column connected with weight g
• Each node is connected to nodes in columns left
  and right with weight dij
• (Often) continuous activation

12
Hopfield Networks
postion city 1 2 3 4
A 0 1 0 0
B 1 0 0 0
C 0 0 0 1
D 0 0 1 0
13
Hopfield Networks

• Graph bipartitioning divide nodes in two sets of
  equal size in such a way as to minimize the
  number of edges going between the sets
• 1/-1 Nodes
• 0/1 Connection matrix Cij

14
(No Transcript)
15
Hopfield Networks
16
Hopfield Networks