The Stability Of Quaternionic Associative Memory Neural Network And Its Learning Algorithm

In this paper we investigate the stability of quaternionic associative memory neural network and its learning algorithm. The states of each neuron of the network are denoted by bipolar quaternions, of which the four components take the value of either-1or1. About the asynchronous stability of quaternionic Hopfield network, a new stability condition is derived. The obtained result not only permits a little relaxation on the Hermitian conjugate assumption of the weight matrix, but also generalized the existing result. The synchronous stability of quaternionic Hopfield network is also discussed and we indicate that with a Hermitian conjugate symmetric weight matrix the network would converge to a stable state or a two-state limit cycle in finite steps. Further more, if the weight matrix is Hermitian conjugate symmetric and nonnegative definite, the network would converge to a stable state in finite steps. About the stability of quaternionic BAM network, we prove that the network would converge to a stable state in finite steps for any given weight matrix in both synchronous and asynchronous updating mode. The Hebbian learning rule for embedding patterns is introduced and a sufficient condition is derived to ensure that all the stored patterns are fixed points of the network. To improve the storage capacity of the network, an iterating learning algorithm based on relaxation method used to solve linear inequalities is proposed. The algorithm can embed all the patterns into the network and make them to be fixed points of the network.