arXiv:2505.01218v4 Announce Type: replace 
Abstract: Kernel-based learning methods such as Kernel Logistic Regression (KLR) can substantially increase the storage capacity of Hopfield networks, but the principles governing their performance and stability remain largely uncharacterized. This paper presents a comprehensive quantitative analysis of the attractor landscape in KLR-trained networks to establish a solid foundation for their design and application. Through extensive, statistically validated simulations, we address critical questions of generality, scalability, and robustness. Our comparative analysis shows that KLR and Kernel Ridge Regression (KRR) exhibit similarly high storage capacities and clean attractor landscapes under typical operating conditions, suggesting that this behavior is a general property of kernel regression methods, although KRR is computationally much faster. We identify a non-trivial, scale-dependent law for the kernel width $\gamma$, demonstrating that optimal capacity requires $\gamma$ to be scaled such that $\gamma N$ increases with network size $N$. This finding implies that larger networks require more localized kernels, in which each pattern's influence is more spatially confined, to mitigate inter-pattern interference. Under this optimized scaling, we provide clear evidence that storage capacity scales linearly with network size~($P \propto N$). Furthermore, our sensitivity analysis shows that performance is remarkably robust with respect to the choice of the regularization parameter $\lambda$. Collectively, these findings provide a concise set of empirical principles for designing high-capacity and robust associative memories and clarify the mechanisms that enable kernel methods to overcome the classical limitations of Hopfield-type models.

يكشف تحليل كمي شامل لأساليب الانحدار اللوجستي القائم على النواة (KLR) في شبكات هوبفيلد عن رؤى حول أدائها واستقرارها، مما يبرز زيادة سعة التخزين التي يمكن أن تحققها هذه الشبكات من خلال أساليب التعلم القائمة على النواة. تستخدم الدراسة محاكاة موسعة لاستكشاف مشهد الجاذبية، مع معالجة أسئلة رئيسية حول العمومية والقابلية للتوسع والموثوقية.

Un análisis cuantitativo integral de la regresión logística por núcleo (KLR) en redes de Hopfield revela información sobre su rendimiento y estabilidad, destacando el aumento de la capacidad de almacenamiento que estas redes pueden lograr a través de métodos de aprendizaje basados en núcleos. El estudio utiliza simulaciones extensas para explorar el paisaje de atractores, abordando preguntas clave sobre generalidad, escalabilidad y robustez.

Une analyse quantitative complète de la régression logistique par noyau (KLR) dans les réseaux de Hopfield révèle des informations sur leur performance et leur stabilité, mettant en évidence l'augmentation de la capacité de stockage que ces réseaux peuvent atteindre grâce aux méthodes d'apprentissage par noyau. L'étude utilise des simulations étendues pour explorer le paysage des attracteurs, abordant des questions clés de généralité, d'évolutivité et de robustesse.

A comprehensive quantitative analysis of Kernel Logistic Regression (KLR) in Hopfield networks reveals insights into their performance and stability, highlighting the increased storage capacity these networks can achieve through kernel-based learning methods. The study utilizes extensive simulations to explore the attractor landscape, addressing key questions of generality, scalability, and robustness.

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

arXiv:2511.13053v3 Announce Type: replace 
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 unifying the geometric analysis of the attractor landscape with the spectral theory of kernel machines. Using a novel metric, "Pinnacle Sharpness," we first uncover a rich phase diagram of attractor stability, identifying a "Ridge of Optimization" where the network achieves maximal robustness under high-load conditions. Phenomenologically, this ridge is characterized by a "Force Antagonism," where a strong driving force is balanced by a collective feedback force. Theoretically, we reveal that this phenomenon arises from a specific reorganization of the weight spectrum, which we term \textit{Spectral Concentration}. Unlike a simple rank-1 collapse, our analysis shows that the network on the ridge self-organizes into a critical state: the leading eigenvalue is amplified to maximize global stability (Direct Force), while the trailing eigenvalues are preserved to maintain high memory capacity (Indirect Force). These findings provide a complete physical picture of how high-capacity associative memories are formed, demonstrating that optimal performance is achieved by tuning the system to a spectral "Goldilocks zone" between rank collapse and diffusion.

كشفت دراسة حديثة عن آليات التنظيم الذاتي والطيفية للمناظر الجذابة في شبكات هوبفيلد عالية السعة المعتمدة على النوى، مع تسليط الضوء على مقياس جديد يسمى 'حدة القمة.' تحدد هذه الدراسة 'قمة التحسين' التي تعزز متانة الشبكة في ظل ظروف الحمل العالي، والتي تتميز بتوازن بين القوى الدافعة وقوى التغذية الراجعة. تشير النتائج إلى إعادة تنظيم طيف الأوزان، والذي يُطلق عليه 'التركيز الطيفي'، وهو أمر حاسم لفهم ديناميات هذه الشبكات.

Un estudio reciente ha revelado los mecanismos de autoorganización y espectrales de los paisajes de atractores en redes Hopfield de alta capacidad basadas en núcleos, destacando una nueva métrica llamada 'Pinnacle Sharpness.' Esta investigación identifica una 'Cresta de Optimización' que mejora la robustez de la red bajo condiciones de alta carga, caracterizada por un equilibrio entre fuerzas impulsoras y fuerzas de retroalimentación. Los hallazgos sugieren una reorganización del espectro de pesos, denominada 'Concentración Espectral,' que es crucial para entender la dinámica de estas redes.

Une étude récente a révélé les mécanismes d'auto-organisation et spectraux des paysages d'attracteurs dans des réseaux Hopfield à haute capacité basés sur des noyaux, mettant en lumière une nouvelle métrique appelée 'Pinnacle Sharpness.' Cette recherche identifie une 'Crête d'Optimisation' qui améliore la robustesse du réseau dans des conditions de forte charge, caractérisée par un équilibre entre forces motrices et forces de rétroaction. Les résultats suggèrent une réorganisation du spectre de poids, appelée 'Concentration Spectrale,' essentielle pour comprendre la dynamique de ces réseaux.

A recent study has unveiled the self-organization and spectral mechanisms of attractor landscapes in high-capacity kernel Hopfield networks, highlighting a novel metric called 'Pinnacle Sharpness.' This research identifies a 'Ridge of Optimization' that enhances network robustness under high-load conditions, characterized by a balance of driving and feedback forces. The findings suggest a reorganization of the weight spectrum, termed 'Spectral Concentration,' which is crucial for understanding the dynamics of these networks.

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

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