Stochastic analysis of the Abe formulation of Hopfield networks

This work studies the influence of random noise in the ap- plication of Hopfield networks to combinatorial optimization. It has been suggested that the Abe formulation, rather than the original Hopfield for- mulation, is better suited to optimization, but the eventual presence of noise in the connection weights of this model has not been considered up to now. This consideration leads to a model that is formulated as a stochastic differential equation. In the stochastic setting, the analysis reveals that the model is stable, and the states converge towards an at- tractive set, assuming the noise intensity is bounded. The relation of the attractor with that of the deterministic model requires further study.