Hybrid Quaternionic Hopfield Neural Network

In recent years, applications of complex-valued neural networks have become wide spread. Quaternions are an extension of complex numbers, and neural networks with quaternions have been proposed. Because quaternion algebra is non-commutative algebra, we can consider two orders of multiplication to calculate weighted input. However, both orders provide almost the same performance. We propose hybrid quaternionic Hopfield neural networks, which have both orders of multiplication. Using computer simulations, we show that these networks outperformed conventional quaternionic Hopfield neural networks in noise tolerance. We discuss why hybrid quaternionic Hopfield neural networks improve noise tolerance from the standpoint of rotational invariance.