Rigorous Results for Spin Glass Neural Network Associative Memories

We use the methods of mathematical physics to carry out a rigorous calculation of the average free energy density and average magnetization of a spin glass type neural network associative memory which uses the pseudoinverse learning rule and has an infinite storage capacity (as the size of the network goes to infinity). We perform the calculation in the regime where the number of patterns can grow logarithmically with the size of the network. Rather than using the replica method, which relies on analytic continuation and is not mathematically rigorous, we use an exact sublattice representation. We also define and study a modified, symmetric version of the Gardner learning algorithm, which is capable of storing patterns with optimal stability. We show that in the logarithmic storage regime, this algorithm produces a network with the same average free energy density and magnetization as the pseudoinverse and Hebb rule networks. Finally, we show that previously derived results for the spurious states and speed of recall in the Hopfield model with Hebbian learning also apply to the pseudoinverse and modified symmetric Gardner rule networks.