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

We consider the categorization problem in a Hopfield network with an extensive number of concepts p = alpha N and trained with s examples of weight lambda tau, tau = 1,...,s in the presence of synaptic noise represented by a dimensionless "temperature" T. We find that the retrieval capacity of an example with weight lambda 1, and the corresponding categorization error, depend also on the arithmetic mean lambda m of the other weights. The categorization process is similar to that in a network trained with Hebb's rule, but for lambda 1/lambda m > 1 the retrieval phase is enhanced. We present the phase diagram in the T-alpha plane, together with the de Almeida-Thouless line of instability. The phase diagrams in the alpha-s plane are discussed in the absence of synaptic noise and several values of the correlation parameter b.