Learning withQ-state clock neurons

Abstract

We study the optimal learning capacity for neural networks withQ-state clock neurons, i.e. the states arecomplex numbers with magnitude 1 and azimuthal anglesn·2π/Q, withn=0, 1, ...,Q−1. Performing a phase space analysis, the learning capacity αc for given stability κ can be expressed by means of a double-integral with a simple geometrical interpretation, which for vanishing κ reduces to αc(Q) = 4Q/(3Q−4), forQ≧3. Then we define a training algorithm, which generalizes the well-known AdaTron algorithm fromQ=2 toQ≧3 and converges very fast to the network with optimal stability, if the numberp of random patterns to be learned is smaller than αc(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.