The learning algorithm for a three-layered neural structure with novel non-linear quaternionic-valued multiplicative (QVM) neurons is proposed in this paper. The computing capability of non-linear aggregation in the cell body of biological neurons inspired the development of a non-linear neuron model. However, unlike linear neuron models, most non-linear neuron models are built on higher order aggregation, which is more mathematically complex and difficult to train. As a result, building non-linear neuron models with a simple structure is a difficult and time-consuming endeavor in the neurocomputing field. The concept of a QVM neuron model was influenced by the non-linear neuron model, which has a simple structure and the great computational ability. The suggested neuron’s linearity is determined by the weight and bias associated with each quaternionic-valued input. Non-commutative multiplication of all linearly connected quaternionic input-weight terms accommodates the non-linearity. To train three-layered networks with QVM neurons, the standard quaternionic-gradient-based backpropagation (QBP) algorithm is utilized. The computational and generalization capabilities of the QVM neuron are assessed through training and testing in the quaternionic domain utilizing benchmark problems, such as 3D and 4D chaotic time-series predictions, 3D geometrical transformations, and 3D face recognition. The training and testing outcomes are compared to conventional and root-power mean (RPM) neurons in quaternionic domain using training–testing MSEs, network topology (parameters), variance, and AIC as statistical measures. According to these findings, networks with QVM neurons have greater computational and generalization capabilities than networks with conventional and RPM neurons in quaternionic domain.

本文提出了一种具有新型非线性四元值乘法 (QVM) 神经元的三层神经结构的学习算法。生物神经元细胞体中非线性聚集的计算能力激发了非线性神经元模型的发展。然而，与线性神经元模型不同，大多数非线性神经元模型建立在高阶聚合之上，这在数学上更复杂且难以训练。因此，构建具有简单结构的非线性神经元模型在神经计算领域是一项困难且耗时的工作。QVM神经元模型的概念受到非线性神经元模型的影响，该模型结构简单，计算能力强。建议的神经元线性度由与每个四元值输入相关的权重和偏差决定。所有线性连接的四元输入权重项的非交换乘法适应非线性。为了训练具有 QVM 神经元的三层网络，使用了标准的基于四元数梯度的反向传播 (QBP) 算法。QVM 神经元的计算和泛化能力通过使用基准问题在四元数域中进行训练和测试来评估，例如 3D 和 4D 混沌时间序列预测、3D 几何变换和 3D 人脸识别。使用训练-测试 MSE、网络拓扑（参数）、方差、和 AIC 作为统计措施。根据这些发现，具有 QVM 神经元的网络比具有四元数域中的常规神经元和 RPM 神经元的网络具有更强的计算和泛化能力。