Rigorous results on the Hopfield model of neural networks

We review some recent rigorous results in the theory of neural networks, and in particular on the thermodynamic properties of the Hopfield model. In this context, the model is treated as a Curie-Weiss model with random interactions and large deviation techniques are applied. The tractability of the random interactions depends strongly on how the number, M, of stored patterns scales with the size, N, of the system. We present an exact analysis of the thermodynamic limit under the sole condition that M / N ↓ 0, as N ↑ ∞, i.e. we prove the almost sure convergence of the free energy to a non-random limit and the a.s. convergence of the measures induced on the overlap parameters. We also present results on the structure of local minima of the Hopfield Hamiltonian, originally derived by Newman. All these results are extended to the Hopfield model defined on dilute random graphs