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Energy function: E(S1,…,SN) = - ½ i Σ= j 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 = wij Sj + hext where wij is exchange interaction strength and wij = wji At low temperature Sj = sgn(hi) Effect of temperature: Glauler : +1, with probability g( hi) Si = -1, with probability 1- g( hi) Where g(h) = 1 1 + e –2bh b= 1/ ( K * T) K = Boltzman’s constant T = temperature Stochastic hopfield nets Prob ( Si = + 1) = 1 1 + e +–2bhi b = 1/T 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. 1. Encoding: N x N neurons, (nij ) • 1<= i <= N 1<=j<= N activation of neuron ij: 1, if Ξ link from i to j nij = 0, otherwise. 2. Quantity to minimize: total length L= i<j Σ dij nij 3. Constraints: n = 1, V i 4. Energy function: Quantity to minimize + constraint penalty E ([nij ]) = Σ dij nij + (γ /2) Σ (1- Σj nij)2 i<j i 5. Reduce energy function to summation of quadratic and linear terms. 1. Coefficients of linear terms are thresholds of units. 2. Coefficients of quadratic terms are the weights between neurons. E ( n ) = …= (Nγ)/2 + Σ (d -γ )nij + γ Σ nij nik I<j ij i,j,k So, weightij, kl = - γ, if i,j & k,l have index in common φ, otherwise. Thresholds of node nij : dij-γ 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 Σ dij nia (nj,a+1 + nj,a-1 ) i,j,a • Constraints: Σa nia = 1, V city i Σ nia = 1, V city a i • Energy: e=½Σ d n (n + n ) + (γ /2) [ Σ (1- Σ ij ia j,a+1 j,a-1 a i i,j,a 2 2 nia) + Σ (1- Σ nia) ] i a =…= ½ i,j,a Σ dij nia nj,a+1 +½ i,j,a Σ dij nia nj,a-1 + γ Σ i!=j nia nja + γ Σ nia njb - γ Σ nia + γ N a!=b i,a So, threshold : - γ for each unit. weights : γ , between units on same row or column dij , between units on different columns Reinforcement Learning ( Learn with critic, not teacher) • Associative Reward – penalty algo ( ARP) • Stochastic units: 1 Prob( Si = + 1) = 1 + e +–2bhi hi = Σ wij vj vj = activation of hidden unit or net inputs ξj themselves. • ξiμ = Siμ , if γμ = +1 (reward) -Siμ , if γμ = -1 (penalty)