STORAGE CAPACITY AND OPTIMAL LEARNING OF POTTS-MODEL PERCEPTRONS BY A CAVITY METHOD

By means of a general formulation for the optimal learning capacity of perceptrons with multi-state neurons and real-valued couplings with spherical constraints, which we derive by a cavity method, we calculate the optimal learning capacity alpha(c)(Q', kappa) := p(max)/[N(Q - 1)] for perceptrons with a Q- resp. Q'-state Potts-model input resp. output neurons as a function of Q' and the stability parameter kappa. Among other results, the asymptote for Q' --> infinity is found, and it is shown that for kappa = 0 the information gain per coupling, DELTA I = (alpha(c) ln Q')/(Q' - 1), converges slowly to 1/2 in this limit. Moreover, for Q' --> infinity the same asymptotics also apply for the simple case of Hebbian learning.