arXiv:2511.13053v4 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 الأبحاث الحديثة عن آليات التنظيم الذاتي والطيفية للمناظر الجذابة في الشبكات هوبفيلد ذات السعة العالية المعتمدة على النوى، مع التركيز على مقياس جديد يسمى 'حدة القمة.' تحدد هذه الدراسة 'قمة التحسين' التي تعزز من قوة الشبكة تحت ظروف الحمل العالي، والتي تتميز بتوازن بين القوى الدافعة وقوى التغذية الراجعة. تشير النتائج إلى إعادة تنظيم طيف الأوزان، والتي تُسمى 'التركيز الطيفي'، مما يساهم في استقرار هذه الشبكات.

Investigaciones recientes han 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.' Este estudio 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 contribuye a la estabilidad de estas redes.

Des recherches récentes ont révélé les mécanismes d'auto-organisation et spectraux des paysages d'attracteurs dans des réseaux Hopfield à haute capacité basés sur des noyaux, en mettant l'accent sur une nouvelle métrique appelée 'Pinnacle Sharpness.' Cette étude 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,' qui contribue à la stabilité de ces réseaux.

Recent research has unveiled the self-organization and spectral mechanisms of attractor landscapes in high-capacity kernel Hopfield networks, emphasizing a novel metric called 'Pinnacle Sharpness.' This study 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 contributes to the stability of these networks.

Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networks

arXiv:2511.20698v1 Announce Type: new 
Abstract: Associative memory models based on Hopfield networks and self-attention based on key-value mechanisms have been popular approaches in the study of memory mechanisms in deep learning. It has been pointed out that the state update rule of the modern Hopfield network (MHN) in the adiabatic approximation is in agreement with the self-attention layer of Transformer. In this paper, we go beyond this approximation and investigate the relationship between MHN and self-attention. Our results show that the correspondence between Hopfield networks and Transformers can be established in a more generalized form by adding a new variable, the hidden state derived from the MHN, to self-attention. This new attention mechanism, modern Hopfield attention (MHA), allows the inheritance of attention scores from the input layer of the Transformer to the output layer, which greatly improves the nature of attention weights. In particular, we show both theoretically and empirically that MHA hidden states significantly improve serious problem of deep Transformers known as rank collapse and token uniformity. We also confirm that MHA can systematically improve accuracy without adding training parameters to the Vision Transformer or GPT. Our results provide a new case in which Hopfield networks can be a useful perspective for improving the Transformer architecture.

أظهرت دراسة حديثة وجود صلة بين الشبكات الحديثة من نوع هوبفيلد (MHN) وهياكل المحولات، خاصة في كيفية تحسين الحالات المخفية لآليات الانتباه الذاتي. تشير الأبحاث إلى أنه من خلال دمج متغير جديد، وهو الحالة المخفية المستمدة من MHN، في طبقة الانتباه الذاتي، يمكن تطوير آلية انتباه جديدة تُسمى انتباه هوبفيلد الحديث (MHA). يُحسن هذا التقدم من نقل درجات الانتباه من طبقات الإدخال إلى طبقات الإخراج في المحولات.

Un estudio reciente ha establecido una conexión entre las redes de Hopfield modernas (MHN) y las arquitecturas Transformer, especialmente en cómo los estados ocultos pueden mejorar los mecanismos de autoatención. La investigación indica que al incorporar una nueva variable, el estado oculto derivado de MHN, en la capa de autoatención, se puede desarrollar un nuevo mecanismo de atención llamado atención de Hopfield moderna (MHA). Este avance mejora la transferencia de puntajes de atención de las capas de entrada a las de salida en los Transformers.

Une étude récente a établi un lien entre les réseaux de Hopfield modernes (MHN) et les architectures Transformer, en particulier sur la façon dont les états cachés peuvent améliorer les mécanismes d'auto-attention. La recherche indique qu'en incorporant une nouvelle variable, l'état caché provenant du MHN, dans la couche d'auto-attention, un nouveau mécanisme d'attention appelé attention de Hopfield moderne (MHA) peut être développé. Cette avancée améliore le transfert des scores d'attention des couches d'entrée vers les couches de sortie dans les Transformers.

A recent study has established a connection between modern Hopfield networks (MHN) and Transformer architectures, particularly in how hidden states can enhance self-attention mechanisms. The research indicates that by incorporating a new variable, the hidden state from MHN, into the self-attention layer, a novel attention mechanism called modern Hopfield attention (MHA) can be developed. This advancement improves the transfer of attention scores from input to output layers in Transformers.

Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networks

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