Rotational invariance of quaternionic hopfield neural networks

Abstract

High-dimensional neural networks have been studied by many researchers for their flexible representation. Quaternionic Hopfield neural networks (QHNNs) are one of them. Quaternions have the inherent property of non-commutative multiplication. Connection weights act differently from left and right in QHNNs, and we can have left QHNNs (LQHNNs) and right QHNNs (RQHNNs). Hybrid QHNNs (HQHNNs), which are the compound models of LQHNNs and RQHNNs, have been proposed, and their excellent noise tolerance has been shown through computer simulations. It has been pointed out that the improvement in their noise tolerance is related to the avoidance of rotational invariance. Thus, it is very important to study rotational invariance. In this paper, we investigate the rotational invariance of QHNNs. Rotational invariance in LQHNNs and RQHNNs is different. We define the left and right-rotated vectors, which are fixed in RQHNNs and LQHNNs, respectively. From the standpoint of HQHNNs, the common rotated vectors are important. We also study the common rotated vectors and provide a couple of exceptional examples.

An Introduction to Quaternion-Valued Recurrent Projection Neural Networks

“…To this end, let us define z = C −1 w or, equivalently, w = Cz. From (30) and (31), we conclude that…”

“…The famous Hopfield neural network (HNN) is a recurrent model which can be used to implement associative memories [1]. Quaternion-valued versions of the Hopfield network, which generalize complex-valued models, have been extensively investigated in the past years [28,29,30,36,37,38,39,40,41,42]. A comprehensive review on several types of quaternionic HNN (QHNN) can be found in [26,28].…”

“…In this paper, we consider a quaternion-valued activation function σ given by (6) whose output is obtained by normalizing its argument to length one [29,30]. The resulting network can be implemented and analyzed more easily than the multistate quaternion-valued Hopfield neural network proposed by Isokawa et al [25,26].…”

“…Unfortunately, the multistate quaternion-valued Hopfield neural network may fails to settle at an equilibrium state [27,28]. In contrast, the continuous-valued quaternionic model proposed independently by Valle and Kobayashi always comes to rest at a stable equilibrium point under the usual conditions on the synaptic weights [29,30]. Apart from hypercomplex-valued Hopfield networks, Valle proposed a complex-valued version of the RCNNs [31].…”

Hyperbolic Hopfield neural networks with four‐state neurons

“…Isokawa et al proposed multi-state quaternionic Hopfield neural networks and applied them to store color images [27][28][29][30]. Kobayashi proposed hybrid quaternionic Hopfield neural networks with a split activation function [31,32] [33]. Kobayashi proposed the hyperbolic Hopfield neural networks (HHNNs) and the Hebbian learning rule [34].…”