Energy function: E(S 1 ,…,S N ) = - ½ S W ij S i S j + C (W ii = P/N) (Lyapunov function)

i = j 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

Magnetic Systems • Ising Model: Si spins • Field acting on spin i: hi = Sj wij Sj + hexternal where wij is exchange interaction strength and wij = wji At low temperature: Sj = sgn(hj)

Effect of temperature: Glauber dynamics: +1, with probability g( hi) Si = -1, with probability 1- g( hi) 1 Where g(h) = 1 + e –2bh b= 1/ ( K * T) K = Boltzman’s constant T = temperature

Stochastic Hopfield nets 1 Prob ( Si = + 1) = - 1 + e + 2bhi Where: b = 1/T

Optimization with HNN --Weighted Matching Problem N points, dij distance between i & j Link points in pairs : each point is linked to exactly one other point and total length is MINIMUM. • Encoding: N x N neurons, (nij ) 1<= i <= N, 1<=j<= N • activation of neuron ij: 1, if there is a link from i to j nij = 0, otherwise.

i<j 2. Quantity to minimize: Total length: L= S dij nij 3. Constraints: Sj nij = 1, for each i 4. Energy function: Quantity to minimize + constraint penalty E ([nij ]) = S dij nij + (g /2) S (1- S nij )2 i j i<j

5. 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 ) = …= (Ng)/2 + S (dij-g )nij + gSnij nik So, weightij, kl = - g, if (i,j) and (k,l) have index in common 0, otherwise. Threshold of node nij : dij-g i<j i,j,k

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 Sdij nia (nj,a+1 + nj,a-1 ) • Constraints: Snia = 1, for each city i Snia = 1, for each visit a i,j,a a i

i,j,a • Energy: E = ½ Sdij nia (nj,a+1 + nj,a-1 ) + (g /2) [S (1- S nia)2 + S (1- S nia)2] =…= ½Sdij nia nj,a+1 +½Sdij nia nj,a-1 + gSnia nja + gSnia nib - gSnia + gN So, threshold : -g for each unit. weights : -g , between units on same row or column -dij , between units on different columns a a i i i,j,a i,j,a i!=j a!=b i,a

Reinforcement Learning ( Learn with critic, not teacher) • Associative Reward –Penalty algo ( ARP) • Stochastic units: Prob( Si = + 1) = hi = Sjwij vj vj = activation of hidden unit or net inputs zj themselves. • ziμ = Siμ , if rμ = +1 (reward) -Siμ , if rμ = -1 (penalty) 1 - 1 + e +2bhi

Energy function: E(S 1 ,…,S N ) = - ½ S W ij S i S j + C (W ii = P/N) (Lyapunov function)