Annealing by two sets of interactive dynamics

This work derives the mean field approximation to the mean configuration of a stochastic Hopfield neural network under the Boltzmann assumption. The new approximation is realized by two sets of interactive mean field equations, respectively estimating mean activations subject to mean correlations and mean correlations subject to mean activations. The two sets of interactive dynamics are derived based on two dual mathematical frameworks. Each aims to optimize the objective quantified by a combination of the Kullback-Leibler (KL) divergence and the correlation strength between any two distinct fluctuated variables subject to fixed mean correlations or activations. The new method is applied to the graph bisection problem. By numerical simulations, we show that the new method effectively improves in both performance and relaxation efficiency against the naive mean field equation.