This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface curvature estimates. The paper presents a comprehensive comparison of the average surface curvature of a tuned QMLP model and a tuned real-valued MLP model on both a regression task and a classification task. The results provide strong support for the improved optimization performance observed in QMLPs across various problem domains.

本文探讨了四元数多层感知器 (QMLP) 神经网络相对于实值多层感知器 (MLP) 神经网络的优越性能，这种现象已被经验观察到，但尚未得到彻底研究。该研究首次利用损失表面可视化和投影技术来检查基于四元数的优化损失表面。这项研究的主要贡献是统计证据表明 QMLP 模型比实值神经网络产生更平滑的损失表面，这是使用基于表面曲率估计的损失表面“优度”的稳健定量测量来测量和比较的。广泛的计算测试验证了这些表面曲率估计的有效性。本文对调优 QMLP 模型和调优实值 MLP 模型在回归任务和分类任务上的平均表面曲率进行了全面比较。结果为在不同问题领域的 QMLP 中观察到的优化性能改进提供了强有力的支持。