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Hopfield network
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==Energy== [[File:Energy landscape.png|thumb|right|500px|Energy Landscape of a Hopfield Network, highlighting the current state of the network (up the hill), an attractor state to which it will eventually converge, a minimum energy level and a basin of attraction shaded in green. Note how the update of the Hopfield Network is always going down in Energy.]] Hopfield nets have a scalar value associated with each state of the network, referred to as the "energy", ''E'', of the network, where: :<math>E = -\frac12\sum_{i,j} w_{ij} s_i s_j -\sum_i \theta_i s_i</math> This quantity is called "energy" because it either decreases or stays the same upon network units being updated. Furthermore, under repeated updating the network will eventually converge to a state which is a [[local minimum]] in the energy function (which is considered to be a [[Lyapunov function]]).<ref name="Hopfield1982" /> Thus, if a state is a local minimum in the energy function it is a stable state for the network. Note that this energy function belongs to a general class of models in [[physics]] under the name of [[Ising model]]s; these in turn are a special case of [[Markov networks]], since the associated [[probability measure]], the [[Gibbs measure]], has the [[Markov property]].
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