THRESHOLD-INDUCED PHASE-TRANSITIONS IN PERCEPTRONS

Error rates of a Boolean perceptron with threshold and either spherica l or Ising constraint on the weight vector are calculated for storing patterns from biased input and output distributions derived within a o ne-step replica symmetry breaking (RSB) treatment. For unbiased output distribution and non-zero stability of the patterns, we find a critic al load, alpha(p), above which two solutions to the saddlepoint equati ons appear; one with higher free energy and zero threshold and a domin ant solution with non-zero threshold. We examine this second-order pha se transition and the dependence of alpha(p) on the required pattern s tability, kappa, for both one-step RSB and replica symmetry (RS) in th e spherical case and for one-step RSB in the Ising case.