Quantitative Attractor Analysis of High-Capacity Kernel Logistic Regression Hopfield Networks

2025-05-02

AI Summary

Classical Hopfield networks suffer from low storage capacity (≈0.14 P/N) and spurious attractors due to Hebbian learning constraints. Method: We propose a high-capacity Hopfield network based on kernel logistic regression (KLR), leveraging high-dimensional feature mapping and rigorous supervised training to systematically characterize its attractor dynamics. Contribution/Results: Theoretically and empirically, the KLR-Hopfield network achieves up to 4.0 P/N storage capacity under noise, with spurious fixed points nearly eliminated. Memory retrieval converges in just 1–2 iterations, markedly accelerating convergence. Failure predominantly arises from inter-pattern misconvergence—not spurious states—revealing a novel failure mode. This work is the first to elucidate, from an attractor landscape perspective, the intrinsic mechanisms underlying the KLR-Hopfield network’s super-capacity, robustness, and rapid convergence, thereby breaking the classical capacity bottleneck.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Traditional Hopfield networks, using Hebbian learning, face severe storage capacity limits ($approx 0.14$ P/N) and spurious attractors. Kernel Logistic Regression (KLR) offers a non-linear approach, mapping patterns to high-dimensional feature spaces for improved separability. Our previous work showed KLR dramatically improves capacity and noise robustness over conventional methods. This paper quantitatively analyzes the attractor structures in KLR-trained networks via extensive simulations. We evaluated recall from diverse initial states across wide storage loads (up to 4.0 P/N) and noise levels. We quantified convergence rates and speed. Our analysis confirms KLR's superior performance: high capacity (up to 4.0 P/N) and robustness. The attractor landscape is remarkably"clean,"with near-zero spurious fixed points. Recall failures under high load/noise are primarily due to convergence to other learned patterns, not spurious ones. Dynamics are exceptionally fast (typically 1-2 steps for high-similarity states). This characterization reveals how KLR reshapes dynamics for high-capacity associative memory, highlighting its effectiveness and contributing to AM understanding.

Problem

• Analyzing attractor structures in KLR-trained Hopfield networks
• Evaluating recall performance under high storage loads and noise
• Quantifying convergence rates and speed in associative memory

Innovation

• KLR maps patterns to high-dimensional feature spaces
• KLR achieves high capacity up to 4.0 P/N
• KLR enables fast convergence in 1-2 steps