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Recent research has established a connection between modern Hopfield networks (HNs) and transformer attention heads, with guarantees of exponential storage capacity. However, these models still face challenges scaling storage efficiently.…

Machine Learning · Computer Science 2025-04-11 Saul Santos , António Farinhas , Daniel C. McNamee , André F. T. Martins

Hopfield neural networks are a possible basis for modelling associative memory in living organisms. After summarising previous studies in the field, we take a new look at learning rules, exhibiting them as descent-type algorithms for…

Neural and Evolutionary Computing · Computer Science 2020-10-06 Pavel Tolmachev , Jonathan H. Manton

Dense Associative Memories (DenseAMs) are generalizations of Hopfield networks, which have superior information storage capacity and can store training data points (memories) at local minima of the energy landscape. When the amount of…

Machine Learning · Computer Science 2026-03-18 Bao Pham , Gabriel Raya , Matteo Negri , Mohammed J. Zaki , Luca Ambrogioni , Dmitry Krotov

We introduce a three-dimensional vectorial extension of the Hopfield associative-memory model in which each neuron is a unit vector on $S^2$ and synaptic couplings are $3\times 3$ blocks generated through a vectorial Hebbian rule. The…

Disordered Systems and Neural Networks · Physics 2025-12-10 F. Gallavotti , A. Zaccone

We present a nonparametric interpretation for deep learning compatible modern Hopfield models and utilize this new perspective to debut efficient variants. Our key contribution stems from interpreting the memory storage and retrieval…

Machine Learning · Statistics 2025-06-10 Jerry Yao-Chieh Hu , Bo-Yu Chen , Dennis Wu , Feng Ruan , Han Liu

Deep neural networks (DNN) trained in a supervised way suffer from two known problems. First, the minima of the objective function used in learning correspond to data points (also known as rubbish examples or fooling images) that lack…

Machine Learning · Computer Science 2018-11-27 Dmitry Krotov , John J Hopfield

Associative Memories like the famous Hopfield Networks are elegant models for describing fully recurrent neural networks whose fundamental job is to store and retrieve information. In the past few years they experienced a surge of interest…

Machine Learning · Computer Science 2025-10-06 Dmitry Krotov , Benjamin Hoover , Parikshit Ram , Bao Pham

We examine a previouly introduced attractor neural network model that explains the persistent activities of neurons in the anterior ventral temporal cortex of the brain. In this model, the coexistence of several attractors including…

Disordered Systems and Neural Networks · Physics 2009-11-10 T. Uezu , A. Hirano , M. Okada

Recent studies point to the potential storage of a large number of patterns in the celebrated Hopfield associative memory model, well beyond the limits obtained previously. We investigate the properties of new fixed points to discover that…

Disordered Systems and Neural Networks · Physics 2017-11-22 Jacopo Rocchi , David Saad , Daniele Tantari

We investigate a new method to augment recurrent neural networks with extra memory without increasing the number of network parameters. The system has an associative memory based on complex-valued vectors and is closely related to…

Neural and Evolutionary Computing · Computer Science 2016-05-20 Ivo Danihelka , Greg Wayne , Benigno Uria , Nal Kalchbrenner , Alex Graves

In Hopfield-type associative memory models, memories are stored in the connectivity matrix and can be retrieved subsequently thanks to the collective dynamics of the network. In these models, the retrieval of a particular memory can be…

Neurons and Cognition · Quantitative Biology 2025-10-21 Marco Benedetti , Nicolas Brunel , Enzo Marinari , Ulises Pereira Obilinovic

We investigate how feature correlations influence the capacity of Dense Associative Memory (DAM), a Transformer attention-like model. Practical machine learning scenarios involve feature-correlated data and learn representations in the…

Machine Learning · Computer Science 2025-08-05 Stefan Bielmeier , Gerald Friedland

Associative memory architectures such as the Hopfield network have long been important conceptual and theoretical models for neuroscience and artificial intelligence. However, translating these abstract models into spiking neural networks…

Neurons and Cognition · Quantitative Biology 2025-07-02 William F. Podlaski , Christian K. Machens

Neural population activity in cortical and hippocampal circuits can be flexibly reorganized by context, suggesting that cognition relies on dynamic manifolds rather than static representations. However, how such dynamic organization can be…

Machine Learning · Computer Science 2026-03-04 Chong Li , Taiping Zeng , Xiangyang Xue , Jianfeng Feng

Krotov and Hopfield (2021) proposed a biologically plausible two-layer associative memory network with memory storage capacity exponential in the number of visible neurons. However, the capacity was only linear in the number of hidden…

Neurons and Cognition · Quantitative Biology 2026-03-11 Mohadeseh Shafiei Kafraj , Dmitry Krotov , Peter E. Latham

Associative memory models such as the Hopfield network and its dense generalizations with higher-order interactions exhibit a "blackout catastrophe" -- a discontinuous transition where stable memory states abruptly vanish when the number of…

Disordered Systems and Neural Networks · Physics 2026-03-24 David G. Clark

The brain can reproduce memories from partial data; this ability is critical for memory recall. The process of memory recall has been studied using auto-associative networks such as the Hopfield model. This kind of model reliably converges…

Neurons and Cognition · Quantitative Biology 2016-05-18 James P. Roach , Leonard M Sander , Michal R. Zochowski

We introduce Graph Hopfield Networks, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification. Gradient descent on this joint energy yields an iterative update interleaving Hopfield…

Machine Learning · Computer Science 2026-03-05 Abinav Rao , Alex Wa , Rishi Athavale

Traditional neural networks require enormous amounts of data to build their complex mappings during a slow training procedure that hinders their abilities for relearning and adapting to new data. Memory-augmented neural networks enhance…

Emerging Technologies · Computer Science 2021-06-23 Geethan Karunaratne , Manuel Schmuck , Manuel Le Gallo , Giovanni Cherubini , Luca Benini , Abu Sebastian , Abbas Rahimi

The Hopfield model is a pioneering neural network model with associative memory retrieval. The analytical solution of the model in mean field limit revealed that memories can be retrieved without any error up to a finite storage capacity of…

Disordered Systems and Neural Networks · Physics 2017-10-31 Do-Hyun Kim , Jinha Park , B. Kahng