Quaternionic Hopfield neural networks with twin-multistate activation function

In the present work, a Hopfield neural network with a new quaternionic activation function, referred to as a twin-multistate activation function, is proposed. The multistate activation function has been used in complex-valued Hopfield neural networks (CHNNs). It is useful for representing multilevel information, and the CHNNs with a multistate activation function have been applied to the storage of multilevel data, such as gray-scale images. A twin-multistate activation function consists of two multistate activation functions. Quaternionic Hopfield neural networks (QHNNs) with a twin-multistate activation function can take the place of the CHNNs with a multistate activation function. The QHNNs require half the number of connection parameters of CHNNs. Projection rule is a fast learning algorithm, and is suitable under restricted computational power. However, it requires full-connection, and the sparse connections are not allowed. Projection rule is also available for the QHNNs with a twin-multistate activation function. When the memory resource and computational power are restricted, the QHNNs with a twin-multistate activation function is useful. In the present work, the conventional network topology is only considered. To take advantage of non-commutativity of quaternions, several researchers proposed network topology for QHNNs, such as dual connections. In future, the QHNNs with a twin-multistate activation function will be extended to those with such network topology.