Categorization in a Hopfield network trained with weighted examples. (I). Finite number of concepts

We consider the categorization problem in a Hopfield network with a finite number of concepts and trained with s examples of weight lambda(tau), tau = 1,...,s. We find that the retrieval capacity of an example with weight lam bda(1), and the corresponding categorization error, depends also on the ari thmetic mean lambda(m) = (1/(s - 1)) Sigma(tau=2)(s) lambda(tau) of the oth er weights. For lambda(1)/lambda(m) < 1, the categorization process is simi lar to that in a network trained with Hebb's rule, but for lambda(1)/lambda (m) >1 we find that the line of first-order transitions between the retriev al and categorization phases ends at a critical point in the s, T plane. Wh en two solutions are present, the global minimum of the free energy corresp onds to the solution with the highest weight. (C) 1999 Elsevier Science B.V . All rights reserved.