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. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.