Quaternion-Valued Recurrent Projection Neural Networks on Unit Quaternions

Hypercomplex-valued neural networks, including quaternion-valued neural networks, can treat multidimensional data as a single entity. In this paper, we present the quaternion-valued recurrent projection neural networks (QRPNNs). Briefly, QRPNNs are obtained by combining the non-local projection learning with the quaternion-valued recurrent correlation neural network (QRCNNs). We show that QRPNNs overcome the crosstalk problem of QRCNNs. Thus, they are appropriate to implement associative memories. Furthermore, computational experiments reveal that QRPNNs exhibit greater storage capacity and noise tolerance than their corresponding QRCNNs. Introduction The Hopfield neural network, developed in the early 1980s, is an important and widely-known recurrent neural network which can be used to implement associative memories [1, 2]. Successful applications of the Hopfield network include control [3, 4], computer vision and image processing [5, 6], classification [7, 8], and optimization [2, 9, 10]. Despite its many successful applications, the Hopfield network may suffer from a very low storage capacity when used to implement associative memories. Precisely, due to crosstalk between the stored items, the Hebbian learning adopted by Hopfield in his original work allows for the storage of approximately n/(2 ln n) items, where n denotes the length of the stored vectors [11]. For example, Personnaz et al. [12] as well as Kanter and Sompolinsky [13] proposed the projection rule to determine the synaptic weights of the Hopfield networks. The projection rule increases the storage capacity of the Hopfield network to n 1 items. Another simple but effective improvement on the storage capacity of the original Hopfield networks was achieved by Chiueh and Goodman's recurrent correlation neural networks (RCNNs) [14, 15]. Briefly, an RCNN is obtained by decomposing the Hopfield network with Hebbian learning into a two layer recurrent neural network.