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Asymptotically Stable Quaternion-valued Hopfield-structured Neural Network with Periodic Projection-based Supervised Learning Rules
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Proposes a quaternion-valued supervised Hopfield-structured network (QSHNN) with proven fixed-point existence/uniqueness and asymptotic stability under weight constraints, trained by gradient descent with periodic Frobenius projection onto quaternion left-multiplication blocks.
Problem
Hopfield-type networks are limited to unsupervised associative memory with Hebbian/outer-product weights; quaternion-valued versions lack supervised target tracking, guaranteed stability, and structure-preserving learning for applications like robotic joint posture control.
Approach
Embed a continuous-time HNN in the quaternion domain with 4x4 left-multiplication weight blocks, derive stability constraints via Lyapunov analysis, and train with gradient descent periodically projected (least-squares Frobenius) onto the quaternion block structure.
Key results
- QSHNN with four quaternion neurons trained with periodic projection (period 10) reaches accuracy 1.0 with equilibrium error below 1e-6 on random targets (stated training criterion; no aggregate table provided)
- Trained weight blocks visually become quaternion-structured under periodic projection, while strict gradient descent (SHNN) does not preserve structure
- Curvature bound kappa <= 4 derived from gamma=mu=1 and tanh activation; exponential error decay bound stated analytically
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add periodic projection every 10 iterations (ablation) | SHNN · Random target quaternion states | quaternion weight structure | Helped |
| add periodic projection every 10 iterations (ablation) | SHNN · Random target quaternion states | training-curve fluctuation | Hurt |
Surprising findings
- Periodic projection breaks monotone loss decrease, producing sharply fluctuating loss curves, yet still converges
Limitations
- Robotics application (PyBullet, benchmarks, baselines) only preliminary or planned; no quantitative comparison with IK/CHOMP/STOMP/RRT
- Experiments limited to a small four-neuron network on random targets
- Stability constraints restrict weight norms and the periodic projection lacks convergence guarantees to the constrained minimizer
Experimental setup
Datasets: randomly generated quaternion target sets
Benchmarks: random quaternion target sets
Metrics: accuracy maximum iterations to convergence equilibrium error (MSE / Euclidean distance)
Baselines: strict gradient descent SHNN (no projection) damped least-squares IK CHOMP STOMP RRT-Connect + B-spline smoothing
Builds on: continuous-time Hopfield neural network GHR calculus Lyapunov theory
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