LEARNING WITH Q-STATE CLOCK NEURONS - OPTIMAL STORAGE CAPACITY AND ADATRON-ALGORITHM

We study the optimal learning capacity for neural networks with Q-state clock neurons, i.e. the states are complex numbers with magnitude 1 and azimuthal angles n . 2-pi/Q, with n = 0, 1, ..., Q - 1. Performing a phase space analysis, the learning capacity alpha(c) for given stability kappa can be expressed by means of a double-integral with a simple geometrical interpretation, which for vanishing kappa reduces to alpha(c)(Q) =4Q/(3Q - 4), for Q greater-than-or-equal-to 3. Then we define a training algorithm, which generalizes the well-known Ada Tron algorithm from Q = 2 to Q greater-than-or-equal-to 3 and converges very fast to the network with optimal stability, if the number p of random patterns to be learned is smaller than alpha(c)(Q). Finally, in the conclusions, we also give hints on applications for image recognition and in a "note added in proof" we generalize some results to Potts model networks.