[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86364-en":3,"doc-seo-86364-105":29,"detail-sidebar-cat-0-en-105":82},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},86364,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Redundancy Maximization as a Principle of Associative Memory Learning in Hopfield Networks","Associative memory modeled by Hopfield networks retrieves stored patterns from partial or noisy cues, yet the local computational principles enabling retrieval are not fully understood. This work applies Partial Information Decomposition (PID) to quantify unique, redundant, and synergistic contributions of external pattern input and internal recurrent input at the level of individual neurons. Below capacity, neuron activity is dominated by high redundancy, while synergy and unique information remain near zero until capacity is exceeded and performance drops sharply. Using redundancy maximization as a local information-theoretic learning goal increases memory capacity to 1.59, over tenfold beyond classical Hopfield networks and exceeding recent state-of-the-art methods.","arXiv :2511 .02584v2 [ cs .IT] 13 Jul 2026  \nREDUNDANCY MAXIMIZATION AS A PRINCIPLE OF ASSOCIATIVE MEMORY LEARNING IN HOPFIELD NETWORKS  \nMark Blümel 1,†, Andreas C. Schneider2,1,†, Valentin Neuhaus2,1,†, David A. Ehrlich2,3,1 , Marcel Graetz4 , Michael Wibral3,1 , Abdullah Makkeh3,1 , and Viola Priesemann 1,2  \n1 Complex Systems Theory, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany  \n2Faculty of Physics, Institute for the Dynamics of Complex Systems, University of Göttingen  \n3 Göttingen Campus Institute for Dynamics of Biological Networks, University of Göttingen, Göttingen, Germany  \n4 Champalimaud Centre for the Unknown, Lisbon, Portugal †These authors share first authorship.  \n[viola.priesemann@ds.mpg.de](viola.priesemann@ds.mpg.de)  \nABSTRACT  \nAssociative memory, traditionally modeled by Hopfield networks, enables the retrieval of previously stored patterns from partial or noisy cues. Yet, the local computational principles which are required to enable this function remain incompletely understood. To formally characterize the local information processing in such systems, we employ a recent extension of information theory—Partial Information Decomposition (PID) . PID decomposes the contribution of different inputs to an output into unique information from each input, redundant information across inputs, and synergistic information that emerges from combining different inputs. Applying this framework to individual neurons in classical Hopfield networks we find that below the memory capacity, the information in a neuron’s activity is characterized by high redundancy between the external pattern input and the internal recurrent input, while synergy and unique information are close to zero until the memory capacity is surpassed and performance drops steeply. Inspired by this observation, we use redundancy maximization at each neuron as an information-theoretic learning goal. This dramatically increases the network’s memory capacity to 1.59, a more than tenfold improvement over the 0.14 capacity of classical Hopfield networks, and also outperforming recent state-of-the-art implementations of Hopfield networks.  \nOverall, this work establishes redundancy maximization as a new design principle for associative memories and opens pathways for new associative memory models based on information-theoretic goals.  \nKeywords Information Theory · Associative Memory Learning · Hopfield Networks  \n1 Introduction  \nAssociative memory—the ability to retrieve patterns from noisy or partial inputs—is a fundamental brain function, enabling the retrieval of memories from imperfect sensory stimuli. This type of content-addressable memory can be modeled by recurrent neural networks called “Hopfield networks”, for which their inventor John Hopfield was recognized with the Nobel Prize in physics in 2024 [1, 2] . Recently, continuous-valued extensions of Hopfield networks have also found renewed application in machine learning [3] .  \nDespite decades of development since their first introduction in 1982, the principles underlying associative memory formation remain incompletely understood. Originally, Hopfield networks were trained using the biologically-inspired Hebbian learning rule based on firing coincidences. Since then, new learning rules have been introduced that display improved memory capacity and stability [4, 5] . Nevertheless, a key question remains: Is there an underlying principle that governs the formation of associative memory? And, if so, can it be exploited directly to improve performance?  \nTo answer these questions, we propose to analyze Hopfield networks from an information processing perspective. Hopfield networks store patterns as attractors of their neural dynamics, created by training the network’s weights using the patterns as a teaching signal. How the information of the recurrent dynamics and the teaching signal together predicts the neuron’s firing thus becomes pivotal to the network’s perf","cbCaiepXOUxNWsK1","https://ap.wps.com/l/cbCaiepXOUxNWsK1","pdf",1500887,1,31,"English","en",105,"# Abstract\n# Keywords\n# Introduction\n## Associative memory and Hopfield networks\n## Information-theoretic analysis and PID\n## Local neuron-level contributions\n## Contributions and outline","[{\"question\":\"What learning strategy is proposed, and what performance gain does it achieve?\",\"answer\":\"The authors propose redundancy maximization at each neuron as a learning goal. It increases memory capacity to 1.59, more than tenfold compared with classical Hopfield networks and surpassing recent state-of-the-art implementations.\"}]",1784210965,78,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":77,"head_meta":79,"extra_data":81,"updated_unix":27},"redundancy-maximization-as-a-principle-of-associative-memory-learning-in-hopfield-networks","",{"@graph":35,"@context":76},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/redundancy-maximization-as-a-principle-of-associative-memory-learning-in-hopfield-networks/86364/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70],{"name":71,"@type":72,"acceptedAnswer":73},"What learning strategy is proposed, and what performance gain does it achieve?","Question",{"text":74,"@type":75},"The authors propose redundancy maximization at each neuron as a learning goal. 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