FINITE-SIZE EFFECTS AND BOUNDS FOR PERCEPTRON MODELS

In this paper we consider two main aspects of the binary perception problem: the maximal capacity when random patterns are stored (model A), and its generalization ability (model B). We have extended previous numerical estimates of critical capacities and studied thermal properties of systems of small sizes to test recent replica predictions. We have also considered some simpler versions of these models. The discrete spherical versions can be solved exactly using Gardner's replica calculation for the spherical model and are shown to give a rigorous upper bound and lower bound on the capacities of models A and B, respectively. Toy versions of models A and B are solved in detail and provide information which is useful for interpreting the finite-size effects present in the numerical studies of models A and B.