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

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.

• The paper shows that kernel logistic regression enables a breakthrough capacity (P/N > 4.0) by self-organizing attractor landscapes.
• It introduces Pinnacle Sharpness as a key metric to quantify local attractor stability by decomposing direct and indirect network forces.
• Empirical phase diagrams reveal a ridge regime where antagonistic interactions drive robust memory retrieval even under massive compression.

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

Introduction

This paper provides a thorough geometric analysis of the self-organization mechanisms underlying high-capacity Hopfield networks trained via Kernel Logistic Regression (KLR). Historically, the bottleneck in Hopfield networks has been low storage capacity ($P/N \lesssim 0.14$ for random patterns), which limits their practical use in associative memory tasks due to proliferation of spurious attractors and retrieval failures. By leveraging KLR as the update rule, recent work has shown order-of-magnitude improvements in both capacity ($P/N > 4.0$) and attractor "cleanliness" (Nagasubramaniam et al., 1 Apr 2025). However, the dynamical origin of this phenomenon has remained opaque.

The central objective of this work is to elucidate how KLR learning sculpts the attractor landscape geometry to yield highly stable retrieval in a compressed memory regime. The analysis focuses on the decomposition of landscape dynamics into direct and indirect cooperative forces, and introduces a quantitative measure—"Pinnacle Sharpness"—to characterize local attractor stability.

Model Overview and Analytical Methodology

The network studied consists of $N$ bipolar neurons storing $P$ random bipolar patterns via KLR. Each neuron is updated with a logit obtained from kernel similarities (RBF kernel) between the current state and stored patterns,

$$h_i(\mathbf{s}) = \sum_{\mu=1}^P \alpha_{\mu i} K(\mathbf{s}, \boldsymbol{\xi}^\mu)$$

with $\gamma$ controlling the kernel width and $\alpha_{\mu i}$ learned via regularized logistic loss.

Attractor stability is captured via the landscape gradient of a Lyapunov-like function,

$V(\mathbf{s}) = -\sum_{k=1}^N s_k h_k(\mathbf{s})$

and the key metric, Pinnacle Sharpness, is defined as

$$M(\boldsymbol{\xi}^\mu) = \| \nabla V(\mathbf{s}) \|^2 \big|_{\mathbf{s} = \boldsymbol{\xi}^\mu}$$

This measure quantifies the steepness and restorative force at attractor centers.

Critically, the landscape gradient is decomposed into functionally distinct components:

• Direct force ($F_d$): $-h(\mathbf{s})$, representing immediate alignment pressures for neurons.
• Indirect feedback force ($F_i$): aggregate cooperative interactions stemming from inter-neuron dependencies,

$$\left[F_i(\mathbf{x})\right]_j = -\sum_{k=1}^N x_k \frac{\partial h_k(\mathbf{x})}{\partial x_j}$$

A normalized Force Interference metric, $\rho$, captures the correlation or antagonism between these components.

Phase Diagram and Self-Organization Mechanisms

Empirical simulations across wide regimes of kernel width $\gamma$ and load $P/N$ reveal a rich phase diagram for attractor stability. Three major regimes are identified:

1. Local Non-Cooperative Regime: Large $\gamma$ induces moderate, load-independent Pinnacle Sharpness. Patterns are stabilized with negligible interaction; landscape geometry is essentially additive.
2. Global Inefficient Regime: Small $\gamma$ but low $P/N$ yields poorly defined attractors. The landscape is flat due to insufficient pattern information for global optimization.
3. Ridge of Optimization: At small $\gamma$ and high load, a diagonal ridge emerges where Pinnacle Sharpness is maximized by several orders of magnitude. The network enters a collectively optimized regime, maintaining sharp attractors under severe memory congestion.

Detailed force analysis shows that in the ridge regime, $F_d$ and $F_i$ become strongly anti-correlated ($\rho \approx -1$). Both forces grow exponentially with $P/N$, but the direct force's rate dominates, resulting in net stability governed by $(\|F_d\| - \|F_i\|)^2$. Thus, high attractor sharpness emerges from amplified antagonism—not synergy—between forces.

Implications and Theoretical Considerations

The findings establish that high-capacity memory in KLR Hopfield networks is not merely a product of enhanced classifier complexity but a dynamical act of landscape self-organization. Cooperative antagonism between direct and indirect forces enables the network to resolve dense attractor packing, maintaining stable retrieval far beyond classical limits.

Practically, this unveils a geometric design principle: optimal memory systems should explicitly target regimes where direct forces can overtake but not completely suppress indirect feedback, ensuring both robustness and noise resilience. The work motivates development of learning algorithms that maximize Pinnacle Sharpness, rather than traditional static classification losses.

Theoretically, the transition into the ridge of optimization suggests connections to phase transitions, critical phenomena, and potentially self-organized criticality. A rigorous statistical mechanics treatment could reveal further universality and analytic predictability for such memory systems.

Future Directions

Several research trajectories are opened by this framework:

• Quantitative mapping between geometric phase regimes and retrieval error rates, under various noise models.
• Extension of the phase analysis to non-Gaussian kernels, structured patterns, and sparse network connectivities.
• Development of meta-learning rules that modulate force antagonism or Pinnacle Sharpness as part of regularization and optimization.
• Analysis of biological plausibility and correspondence in cortical associative memory systems, given the cooperative feedback archetype observed here.

Conclusion

This work provides an explicit energy landscape perspective for the operation of high-capacity associative memories configured by kernel logistic regression learning. Pinnacle Sharpness emerges as a critical metric for attractor stability, revealing a phase diagram dominated by a ridge of optimization formed through strong antagonism between direct retrieval forces and cooperative feedback. The principles outlined here lay groundwork for both the practical design and theoretical understanding of next-generation associative memory networks capable of robust storage and retrieval at massive compression ratios.