Abstract
This study proposes the multistate associative memory scheme of high-dimensional Hopfield-type neural networks based on the quaternion algebra, that is a class of hypercomplex numbers. The presented model is an extension of the complex-valued multistate neural network in which the state of a neuron is represented as one of the points on a unit circle, and the state of its neuron is represented as one of the polar coordinates on a three-dimensional unit hyper-sphere, thus expressed by three kinds of phase variables.
The quaternionic signum function, the energy function, and the method for embedding patterns to the network are introduced, and the properties and stability of the network are explored, such as the monotonically decrease of the energy with respect to the change of the neuron state and the basins of the attractors around the local minima in the network. This extended multistate network will be appropriate for processing the color image patterns, because the three multistate phase variables in the quaternionic neuron state can be regarded as the tone levels of three basic colors, i.e., red, blue, and green.
