The solution of the Hopfield model with Gaussian patterns

Abstract

We prove that the free energy of the Hopfield model with Gaussian patterns has a thermodynamic limit at every load $α$ and temperature $T$, and we give it as an explicit variational principle. The Hubbard--Stratonovich transformation maps the model onto $N$ Ising spins coupled to $M=αN$ Gaussian hidden units, whose radius, the empirical variance of the hidden layer, is a thermodynamic variable with an entropic cost. At fixed radius the model is an Ising--spherical spin glass with Gaussian couplings, which we solve. The principle (i) reproduces the Amit-Gutfreund-Sompolinsky theory as its replica-symmetric restriction, (ii) identifies $T_g=1+\sqrtα$ as the rigorous boundary of the annealed region, (iii) selects a unique spin-glass solution of the AGS equations, and (iv) gives the radius an equation of state. Moreover, (v) below $T_g$ we prove that replica symmetry is broken throughout: the instability of the AGS solution reduces to the zero-field de Almeida-Thouless condition of the Sherrington-Kirkpatrick model.