Algebraic Hopfield Network

A Hopfield network is a mathematical model of a spin glass system, a theory of stochastic physics. This binary model has been extended to many high-dimensional models. A persistent challenge in these models has been determining the activation function and stability conditions. In this paper, we propose an algebraic Hopfield network (AHN), which encompasses most extensions of classic Hopfield networks. In AHNs, weights act as operators on neuron outputs in weighted sum inputs. Here we provide the activation function and stability conditions for AHNs, offering a foundation for developing novel Hopfield network models.