The phase space of interactions in neural networks with definite symmetry

The authors calculate the typical fraction of the phase space of interactions which solve the problem of storing a given set of p patterns represented as N-spin configurations, as a function of the storage ratio, alpha =p/N, of the stability parameter, kappa , and of the symmetry, eta , of the interaction matrices. The calculation is performed for strongly diluted networks, where the connectivity of each spin, C, is of the order of ln N. For each value of kappa and eta , there is a maximal value of alpha , above which the volume of solutions vanishes. For each value of kappa and alpha , there is a typical value of eta at which this volume is maximal. The analytical studies are supplemented by numerical simulations on fully connected and diluted networks, using specific learning algorithms.