Abstract
Motivated by the geometric advantages of quaternions in representing rotations and postures, we propose a quaternion-valued supervised learning Hopfield-structured neural network (QSHNN) with a fully connected structure inspired by the classic Hopfield neural network (HNN). Starting from a continuous-time dynamical model of HNNs, we extend the formulation to the quaternionic domain and establish the existence and uniqueness of fixed points with asymptotic stability. For the learning rules, we introduce a periodic projection strategy that modifies standard gradient descent by periodically projecting each 4*4 block of the weight matrix onto the closest quaternionic structure in the least-squares sense. This approach preserves both convergence and quaternionic consistency throughout training. Benefiting from this rigorous mathematical foundation, the experimental model implementation achieves high accuracy, fast convergence, and strong reliability across randomly generated target sets. Moreover, the evolution trajectories of the QSHNN exhibit well-bounded curvature, i.e., sufficient smoothness, which is crucial for applications such as control systems or path planning modules in robotic arms, where joint postures are parameterized by quaternion neurons. Beyond these application scenarios, the proposed model offers a practical implementation framework and a general mathematical methodology for designing neural networks under hypercomplex or non-commutative algebraic structures.
Abstract (translated)
受四元数在表示旋转和姿态方面的几何优势启发，我们提出了一种基于经典霍普菲尔德神经网络（HNN）的全连接结构，并结合监督学习方法设计的一种四元数值霍普菲尔德结构神经网络（QSHNN）。从连续时间动力学模型出发，我们将该模型的形式扩展到四元数域，并建立了固定点的存在性、唯一性和渐近稳定性。在学习规则方面，我们引入了一种周期性投影策略，通过周期性地将权重矩阵中的每个4*4块投影到最小二乘意义上的最近的四元数值结构上来修改标准梯度下降法。这种策略在整个训练过程中保持了收敛性和四元数的一致性。 得益于这一严格的数学基础，实验模型实现了高精度、快速收敛和强鲁棒性的目标，在随机生成的目标集合上表现良好。此外，QSHNN的演化轨迹表现出良好的曲率边界，即足够的平滑度，这对于像控制系统或机器人手臂中的路径规划模块这样的应用场景至关重要，其中关节姿态由四元数神经元参数化。 除了这些应用场景之外，所提出的模型还提供了一个实用的实施框架和一种用于在超复数或非交换代数结构下设计神经网络的一般数学方法。
URL
https://arxiv.org/abs/2510.16607
