Storage capacity and optimal learning of Potts-model perceptrons by a cavity method

Abstract

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', K) :=pmax/(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 K. Among other results, the asymptote for Q' to infinity is found, and it is shown that for K=0 the information gain per coupling, Delta I=( alpha c In Q')/(Q'-1), converges slowly to 1/2 in this limit. Moreover, for Q' to infinity the same asymptotics also apply for the simple case of Hebbian learning.