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 λτ, τ=1,…,s. We find that the retrieval capacity of an example with weight λ1, and the corresponding categorization error, depends also on the arithmetic mean λm=(1/(s−1))∑τ=2sλτ of the other weights. For λ1/λm<1, the categorization process is similar to that in a network trained with Hebb's rule, but for λ1/λm>1 we find that the line of first-order transitions between the retrieval and categorization phases ends at a critical point in the s, T plane. When two solutions are present, the global minimum of the free energy corresponds to the solution with the highest weight.