On the Dynamics of Hopfield Neural Networks on Unit Quaternions

TL;DR

This paper investigates the dynamics of quaternionic Hopfield neural networks, revealing that multi-valued versions may not always stabilize, while continuous-valued variants reliably reach equilibrium, supported by theoretical examples.

Contribution

It demonstrates that multi-valued quaternionic Hopfield networks can exhibit non-convergent behavior, and introduces a continuous-valued version that always stabilizes, with detailed theoretical analysis.

Findings

MV-QHNN can produce periodic sequences, not just equilibrium states.

CV-QHNN always converges to an equilibrium under usual conditions.

The paper provides illustrative examples for both network types.

Abstract

In this paper, we first address the dynamics of the elegant multi-valued quaternionic Hopfield neural network (MV-QHNN) proposed by Minemoto and collaborators. Contrary to what was expected, we show that the MV-QHNN, as well as one of its variation, does not always come to rest at an equilibrium state under the usual conditions. In fact, we provide simple examples in which the network yields a periodic sequence of quaternionic state vectors. Afterward, we turn our attention to the continuous-valued quaternionic Hopfield neural network (CV-QHNN), which can be derived from the MV-QHNN by means of a limit process. The CV-QHNN can be implemented more easily than the MV-QHNN model. Furthermore, the asynchronous CV-QHNN always settles down into an equilibrium state under the usual conditions. Theoretical issues are all illustrated by examples in this paper.