Global Hyperbolic Hopfield Neural Networks

In recent years, applications of neural networks with Clifford algebra have become widespread. Hyperbolic numbers are useful Clifford algebra to deal with hyperbolic geometry. It is difficult when Hopfield neural network is extended to hyperbolic versions, though several models have been proposed. Multistate or continuous hyperbolic Hopfield neural networks are promising models. However, the connection weights and domain of activation function are limited to the right quadrant of hyperbolic plane, and the learning algorithms are restricted. In this work, the connection weights and activation function are extended to the entire hyperbolic plane. In addition, the energy is defined and it is proven that the energy does not increase.