Energy function:

1

• Energy function
• E(S1,,SN) - ½ S Wij Si Sj C (Wii P/N)
• (Lyapunov function)
• Attractors local minima of energy function.
• Inverse states
• Mixture states Spurious minima
• Spin glass states

i j
2
Magnetic Systems

• Ising Model Si spins
• Field acting on spin i
• hi wij Sj hext where wij is exchange
  interaction strength and wij wji
• At low temperature Sj sgn(hi)

3
Effect of temperature Glauler
1, with probability g( hi)
Si
-1, with probability 1- g( hi)
1
Where g(h)
1 e 2bh
b 1/ ( K T)
K Boltzmans constant T temperature
4
Stochastic hopfield nets
1
Prob ( Si 1)

1 e 2bhi
b 1/T
5
Optimization with HNN

• Weighted Matching Problem
• N points, dij distance between i j
• Link in pairs each point linked to exactly one
  other point and total length MINIMUM.
• Encoding
• N x N neurons, (nij ) 1lt i lt N
• 1ltjlt N
• activation of neuron ij
• 1, if ? link from i to j
• nij 0, otherwise.

6

• Quantity to minimize
• total length L S dij nij
• 3. Constraints n 1, V i
• 4. Energy function
• Quantity to minimize constraint penalty
• E (nij ) S dij nij (? /2) S (1- S nij)2

iltj
j
i
iltj
7

• Reduce energy function to summation of quadratic
  and linear terms.
• Coefficients of linear terms are thresholds of
  units.
• Coefficients of quadratic terms are the weights
  between neurons.
• E ( n ) (N?)/2 S (dij-? )nij ? S nij
  nik
• So, weightij, kl - ?, if i,j k,l have
  index in common
• f, otherwise.
• Thresholds of node nij dij-?

Iltj
i,j,k
8
Traveling salesman problem (TSP)

• NP- Complete
• N x N nodes nij 1 iff city I is visited at j
  th stop in tour.
• Minimize
• L ½ S dij nia (nj,a1 nj,a-1 )
• Constraints
• S nia 1, V city i
• S nia 1, V city a

i,j,a
a
i
9

• Energy
• e ½ S dij nia (nj,a1 nj,a-1 ) (? /2) S
  (1- S nia)2 S (1- S nia)2
• ½ S dij nia nj,a1 ½ S dij nia nj,a-1 ?
  S nia nja ? S nia njb - ? S nia ? N
• So, threshold - ? for each unit.
• weights ? , between units on same row or
  column
• dij , between units on different columns

a
i
i,j,a
i
a
i,j,a
i,j,a
i!j
a!b
i,a
10
Reinforcement Learning ( Learn with critic, not
teacher)

• Associative Reward penalty algo ( ARP)
• Stochastic units
• Prob( Si 1)
• hi S wij vj
• vj activation of hidden unit or net inputs ?j
  themselves.
• ?iµ Siµ , if ?µ 1 (reward)
• -Siµ , if ?µ -1 (penalty)

1

1 e 2bhi