Editing Hopfield network (section)

==Spurious patterns==
{{Double image
| direction         = horizontal
| image1            = Hopfield_dynamics.gif
| image2            = Hopfield_dynamics_spurious.gif
| caption1          = 10 random bits initially flipped
| caption2          = 15 random bits initially flipped (converging to the spurious pattern)
| footer            = Hopfield dynamics of a network of 5x5=25 neurons trained to store a single pattern "E" in black
}}
Patterns that the network uses for training (called ''retrieval states'') become attractors of the system. Repeated updates would eventually lead to convergence to one of the retrieval states. However, sometimes the network will converge to spurious patterns (different from the training patterns).<ref name="hertz1991neural">{{harvnb|Hertz|1991}}</ref> In fact, the number of spurious patterns can be exponential in the number of stored patterns, even if the stored patterns are orthogonal.<ref>{{Cite journal |last1=Bruck |first1=J. |last2=Roychowdhury |first2=V.P. |date=1990 |title=On the number of spurious memories in the Hopfield model (neural network) |url=https://ieeexplore.ieee.org/document/52486 |journal=IEEE Transactions on Information Theory |volume=36 |issue=2 |pages=393–397 |doi=10.1109/18.52486|url-access=subscription }}</ref> The energy in these spurious patterns is also a local minimum. For each stored pattern x, the negation -x is also a spurious pattern.

A spurious state can also be a [[linear combination]] of an odd number of retrieval states. For example, when using 3 patterns <math> \mu_1, \mu_2, \mu_3</math>, one can get the following spurious state:

<math> \epsilon_{i}^{\rm{mix}} = \pm \sgn(\pm \epsilon_{i}^{\mu_{1}} 
			         \pm \epsilon_{i}^{\mu_{2}}
			         \pm \epsilon_{i}^{\mu_{3}})
</math>

Spurious patterns that have an even number of states cannot exist, since they might sum up to zero<ref name="hertz1991neural" />