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230 5 Neural Networks

Figure 5.54. Binary images corresponding to spurious states of a Hopfield network
trained with the prototypes of Figure 5.52.

Notice that the spurious states shown in Figure 5.54 correspond to the
complement of prototype states. Besides the complement, one can obtain spurious
states that are a combination of an odd number of prototype patterns or states that
are not correlated with any of the prototypes.
Another problem that decreases the efficiency of a Hopfield net, besides the
presence of spurious states, is the use of an excessive number of prototypes for the
available number of neurons. As a matter of fact, a prototype pattern with moderate
noise may converge to an incorrect prototype if it shares many feature values with
other prototypes. There is, therefore, an upper limit to the maximum number of
patterns that can be memorized and retrieved with practically no errors, the so-
called storage capacity of the discrete Hopfield net. Formulas and estimates of the
storage capacity are presented in (Looney, 1997) and (Haykin, 1999).

Figure 5.55. Prototype patterns used in CAM experiments with the Hopfield
network Lippmann (1987).