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

Abstract

We consider the categorization problem in a Hopfield network with an extensive number of concepts p =αN and trained with s examples of weight λτ, T=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 λ₁, and the corresponding categorization error, depend also on the arithmetic mean λm of the other weights. The categorization process is similar to that in a network trained with Hebb’s rule, but for λ₁/λm>1 the retrieval phase is enhanced. We present the phase diagram in the T-α plane, together with the de Almeida–Thouless line of instability. The phase diagrams in the α-s plane are discussed in the absence of synaptic noise and several values of the correlation parameter b.