$O(2)$ -Valued Hopfield Neural Networks

In complex-valued Hopfield neural networks (CHNNs), the neuron states are complex numbers whose amplitudes are: 1) they can also be described in special orthogonal matrices of order and 2) here, we propose a new Hopfield model, the O(2)-valued Hopfield neural network [O(2)-HNN], whose neuron states are extended to orthogonal matrices. Its neuron states are embedded in 4-D space, while those of CHNNs are embedded in 2-D space. Computer simulations were conducted to compare the noise tolerance (NT) and storage capacity (SC) of CHNNs, O(2)-HNNs, and rotor Hopfield neural networks. In terms of SC, O(2)-HNNs outperformed the others, while in NT, they outdid CHNNs.