Categorization in a Hopfield network trained with weighted examples: Extensive number of concepts

We consider the categorization problem in a Hopfield network with an extens ive number of concepts p = alpha N and trained with s examples of weight la mbda(tau) tau=1,...,s in the presence of synaptic noise represented by a di mensionless "temperature" T. We find that the retrieval capacity of an exam ple with weight lambda(l), and the corresponding categorization error, depe nd also on the arithmetic mean lambda(m) of the other weights. The categori zation process is similar to that in a network trained with Hebb's rule, bu t for lambda(l)/lambda(m) > 1 the retrieval phase is enhanced. We present t he phase diagram in the T-alpha plane, together with the de Almeida-Thoules s line of instability. The phase diagrams in the alpha-s plane are discusse d in the absence of synaptic noise and several values of the correlation pa rameter b.