Optimal nonlinear training in the multi-class proximity problem

Using a signal-to-noise analysis, the effects of nonlinear modulation of the Hebbian learning rule in the multi-class proximity problem are investigated. Both random classification and classification provided by a Gaussian and a binary teacher are treated. Analytic expressions are derived for the learning and generalization rates around an old and a new prototype. For the proximity problem with binary inputs but Q'-state outputs, it is shown that the optimal modulation is a combination of a hyperbolic tangent and a linear function. As an illustration, numerical results are presented for the two-class and the Q' = 3 multi-class problem.