Correlations between hidden units in multilayer neural networks and replica symmetry breaking

We consider feed-forward neural networks with one hidden layer, tree architecture, and a fixed hidden-to-output Boolean function. Focusing on the saturation limit of the storage problem the influence of replica symmetry breaking on the distribution of local fields at the hidden units is investigated. These field distributions determine the probability of finding a specific activation pattern of the hidden units as well as the corresponding correlation coefficients and therefore quantify the division of labor among the hidden units. We find that although modifying the storage capacity and the distribution of local fields markedly replica symmetry breaking has only a minor effect on the correlation coefficients. Detailed numerical results are provided for the PARITY, COMMITTEE, and AND machines with K=3 hidden units and nonoverlapping receptive fields.