Stochastic analysis of the Abe formulation of Hopfield networks.

Marie Kratz (1, 2, 3) , Miguel Atencia (4) , Gonzalo Joya (5)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 SAMOS - Statistique Appliquée et MOdélisation Stochastique
3 MATISSE - UMR 8595 - Modélisation Appliquée, Trajectoires Institutionnelles et Stratégies Socio-Économiques
4 Departamento de Matemática Aplicada
5 Departamento de Tecnología Electrónica

This work studies the influence of random noise in the application of Hopfield networks to combinatorial optimization. It has been suggested that the Abe formulation, rather than the original Hopfield formulation, 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 attractive set, assuming the noise intensity is bounded. The relation of the attractor with that of the deterministic model requires further study.

Domaines: Optimisation et contrôle [math.OC]; Réseau de neurones [cs.NE]; Systèmes dynamiques [math.DS]

Marie Kratz, Miguel Atencia, Gonzalo Joya. Stochastic analysis of the Abe formulation of Hopfield networks.. European Symposium on Artificial Neural Networks, Apr 2005, Bruges, Belgium. ⟨hal-00179439⟩