Associative recall of memory without errors.

A neural network which is capable of recalling without errors any set of linearly independent patterns is studied. The network is based on a Hamiltonian version of the model of Personnaz et al. The energy of a state of N (+1}neurons is the square of the Euclidean distance —in phase space— between the state and the linear subspace spanned by the patterns. This energy corresponds to nonlocal updatings of the synapses in the learning mode. Results of the mean-field theory (MFT) of the system as well as computer simulations are presented. The stable and metastable states of the network are studied as a function of "temperature" T and a=p/X, where p is the number of ernbedded patterns. The maximum capacity of the network is a=1. For all a (0&a&1}the embedded patterns are not only locally stable but are global minima of the energy. The patterns appear, as metastable states, below a temperature T =T~(a). The temperature T~(a) decreases to zero as a~1. The spurious states of the network are studied in detail in the case of random uncorrelated patterns. At finite p, they are identical to the mixture states of Hopfield s model. At finite a, a spin-glass phase exists as a metastable state. According to the replica symmetric MFT the spinglass state becomes degenerate with the patterns at a=ag:1 2/~ and disappears above it. Possi-