Self-Organization of Attractor Landscapes in High-Capacity Kernel Logistic Regression Hopfield Networks

2025-11-17

AI Summary

While kernel-based Hopfield networks enhance memory capacity, the dynamical mechanisms underlying attractor stability under high loading remain unclear. Method: We systematically investigate the self-organization of attractors in kernelized logistic-regression Hopfield networks via geometric analysis and gradient decomposition of the energy landscape. Contribution/Results: We introduce “peak sharpness” as a novel metric, revealing a strong anti-correlation between direct driving forces and indirect feedback forces. We identify and interpret the “optimization ridge” phenomenon under high load—a structural feature that enhances attractor stability. Combining theoretical modeling with phase-diagram simulations, we characterize how attractor morphology evolves with kernel width and storage load, establishing a physical picture of high-capacity associative memory. Our analysis identifies the parameter regime that maximizes stability, yielding interpretable, physics-grounded design principles for high-performance associative memory systems.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanism behind this enhancement remains poorly understood. We address this gap by conducting a geometric analysis of the network's energy landscape. We introduce a novel metric, ``Pinnacle Sharpness,'' to quantify the local stability of attractors. By systematically varying the kernel width and storage load, we uncover a rich phase diagram of attractor shapes. Our central finding is the emergence of a ``ridge of optimization,'' where the network maximizes attractor stability under challenging high-load and global-kernel conditions. Through a theoretical decomposition of the landscape gradient into a direct ``driving'' force and an indirect ``feedback'' force, we reveal the origin of this phenomenon. The optimization ridge corresponds to a regime of strong anti-correlation between the two forces, where the direct force, amplified by the high storage load, dominates the opposing collective feedback force. This demonstrates a sophisticated self-organization mechanism: the network adaptively harnesses inter-pattern interactions as a cooperative feedback control system to sculpt a robust energy landscape. Our findings provide a new physical picture for the stability of high-capacity associative memories and offer principles for their design.

Problem

• Analyzing energy landscape geometry in high-capacity kernel Hopfield networks
• Quantifying attractor stability through novel Pinnacle Sharpness metric
• Revealing self-organization mechanism behind robust memory formation

Innovation

• Introduces Pinnacle Sharpness metric for attractor stability
• Reveals optimization ridge through force decomposition analysis
• Demonstrates self-organization via cooperative feedback control system