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

arXiv: https://arxiv.org/abs/2510.16607 · HTML: https://mlsanity.com/paper/2510.16607/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 18 Oct 2025.

## Builds on (most cited first, 10 of 10)

- [robosuite: A Modular Simulation Framework and Benchmark for Robot Learning](https://mlsanity.com/paper/2009.12293/) (2020)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research](https://mlsanity.com/paper/1802.09464/) (2018)
- [Brain-inspired learning in artificial neural networks: a review](https://mlsanity.com/paper/2305.11252/) (2023)
- [Principled Training of Neural Networks with Direct Feedback Alignment](https://mlsanity.com/paper/1906.04554/) (2019)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [Quaternion Gradient and Hessian](https://mlsanity.com/paper/1406.3587/) (2014)
- [A Sparse Quantized Hopfield Network for Online-Continual Memory](https://mlsanity.com/paper/2307.15040/) (2023)
- [Time Series Compression using Quaternion Valued Neural Networks and Quaternion Backpropagation](https://mlsanity.com/paper/2403.11722/) (2024)
- [Quaternion recurrent neural network with real-time recurrent learning and maximum correntropy criterion](https://mlsanity.com/paper/2402.14227/) (2024)

## Built on by (most cited first, 1 of 1)

- [Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning](https://mlsanity.com/paper/2605.26167/) (2026)

## Abstract (arXiv)

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.