Twin-multistate commutative quaternion Hopfield neural networks

The complex-valued Hopfield neural network (CHNN) can deal with multi-level information, and has often been applied to the storage of image data. The quaternion Hopfield neural network (QHNN) is a multistate model of a Hopfield neural network, and requires half the connection weight parameters of CHNN. In this study, we propose a commutative quaternion Hopfield neural network (CQHNN) as the analogy of QHNN. The multiplication of commutative quaternions is commutative and convenient, unlike that of quaternions. We compared the noise tolerance of CQHNNs and QHNNs by computer simulation, and discuss the simulation results from the perspective of rotational invariance and self loops.