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Neural Networks Chapter 2 Joost N. Kok Universiteit Leiden Hopfield Networks • Network of McCulloch-Pitts neurons • Output is 1 iff wij S j i and is -1 j otherwise Hopfield Networks Hopfield Networks Hopfield Networks Hopfield Networks • Associative Memory Problem: Store a set of patterns in such a way that when presented with a new pattern, the network responds by producing whichever of the stored patterns most closely resembles the new pattern. Hopfield Networks • Resembles = Hamming distance • Configuration space = all possible states of the network • Stored patterns should be attractors • Basins of attractors Hopfield Networks • N neurons • Two states: -1 (silent) and 1 (firing) • Fully connected • Symmetric Weights • Thresholds Hopfield Networks w13 w16 w57 -1 +1 Hopfield Networks • State: • Weights: • Dynamics: S S1 ... S25 w1,1 w1, 25 w w w 25, 25 25,1 Si : sgn 25 wS i 1 ij j Hopfield Networks • Hebb’s learning rule: – Make connection stronger if neurons have the same state – Make connection weaker if the neurons have a different state Hopfield Networks neuron 1 synapse neuron 2 Hopfield Networks • Weight between neuron i and neuron j is given by wij p 1 p 1 ( ) i ( ) j Hopfield Networks • Opposite patterns give the same weights • This implies that they are also stable points of the network • Capacity of Hopfield Networks is limited: 0.14 N Hopfield Networks • Hopfield defines the energy of a network: E = - ½ ij SiSjwij + i Sii • If we pick unit i and the firing rule does not change its Si, it will not change E. • If we pick unit i and the firing rule does change its Si, it will decrease E. Hopfield Networks • Energy function: H 1 wij Si S j 2 ij • Alternative Form: H C wij Si S j (ij ) • Updates: S sgn wij S j j ' i Hopfield Networks H H wij S S j wij Si S j ' ' i j i H H 2Si ' w S j i ij j i j 2Si w S ij j j 2wii Hopfield Networks • Extension: use stochastic fire rule – Si := +1 with probability g(hi) – Si := -1 with probability 1-g(hi) Hopfield Networks 1 • Nonlinear function: g(x) = 1 + e – xb b g(x) b0 x Hopfield Networks • Replace the binary threshold units by binary stochastic units. • Define b = 1/T • Use “temperature” T to make it easier to cross energy barriers. – Start at high temperature where its easy to cross energy barriers. – Reduce slowly to low temperature where good states are much more probable than bad ones. A B C Hopfield Networks • Kick the network our of spurious local minima • Equilibrium: Si becomes time independent 1 Pr Si 1 f b hi 1 exp( 2.b .hi ) 1 b T