Asymptotically Stable Quaternion-valued Hopfield-structured Neural Network with Periodic Projection-based Supervised Learning Rules

TL;DR

The paper presents a quaternion-valued supervised Hopfield-structured neural network (QSHNN) with continuous-time dynamics that converge asymptotically to externally specified quaternion targets. It provides a rigorous mathematical foundation using quaternion algebra, left multiplication manifolds, and Lyapunov stability, and couples this with Generalized HR calculus to derive quaternion-preserving learning rules. To maintain quaternion structure during training, the authors introduce a periodic Frobenius projection onto the quaternion left-multiplication submanifold, ensuring both convergence and structural consistency with efficient updates. Experiments on a small quaternion network demonstrate fast convergence, bounded trajectory curvature, and robust performance across randomly generated targets, with implications for robotics control and trajectory planning. This work advances hypercomplex neural networks by integrating supervised learning, dynamical stability guarantees, and structural constraints into a coherent quaternionic learning framework.

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.

Figures (7)

• Figure 1: Structure of quaternion neuron and its interfaces. Four real-valued neurons are integrated together as a quaternion with respect to different component coefficients. The internal operation or the self-connection can be fully characterized by a single weight quaternion. Every numerical factors are compatible by quaternions, projection and approximation only exist in the training. Drawn by the authors.
• Figure 2: Robotic arm with each joint posture parameterized by quaternions for each joint. Our model is capable of high fidelity control and path planning. Figure drawn by the authors, where the robotic arm is generated by AI tool just for demonstration aim.
• Figure 3: The integration of quaternion neuron. It has inside structure, self feedback connection and connection interfaces with other neurons. Activation will be updated every unit impulse time. Drawn by the authors.
• Figure 4: Structure of a fully connected 3-neuron network. Every connection from j-th to i-th neuron is parameterized directly by a quaternion $\omega_{ij}$. Drawn by authors.
• Figure 5: (a). Loss and accuracy curves over training iterations of strict gradient descent Eq. (\ref{['gradient descent']}). It implements a sufficient convergence rapidly but will not preserve the quaternionic structure, shown in Fig. \ref{['3']}. (b). Loss and accuracy curved over training iterations with periodic projection Eq. (\ref{['projection formula']}), which perform good error reduction and preserve quaternionic structure simultaneously. Curves fluctuate more sharply since the projection disturb continuous gradient descent, shown in Fig. \ref{['4']}$\thicksim$\ref{['52']}.

Theorems & Definitions (6)

• Definition 2.1: Quaternion rotation rotation
• Definition 2.2: Quaternion (left GHR) derivative GHR
• Theorem 2.1: Lyapunov Theorem Lyapunov
• Theorem 3.1: First weights constraint of QSHNN
• Theorem 3.2: Second weights constraint of QSHNN
• Theorem 3.3: Frobenius orthogonal projection