Finitesize effects and bounds for perceptron models

AblncL In lhis paper we consider two main aspsln of the binary perccptron problem: the maximal capacity when random pal tem are stored (model A), and its generalhation ability (model B). We have extended previous numerical estimates of critical capacilies and studied thermal properties of systems of small sizes to lest recent replica predictions. We have also considered some simpler versions of thest models. The discrete spherical vemions can be salved 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. Iby vemions of models A and B are soivcd in detail and provide information which is useful for interpreting the Bnile-size eEeciects present in the numerical studies of models A and B. 1. Intmduetion Ever since it was realized that tools employed for studying disordered systems could be applied to models of neural networks, the investigation of such models has become a part of theoretical physics (1, 21. An important question is whether a given neural network architecture can perform tasks such as storing information, classifying it, an