Associative Memory

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Overview

Associative memory asks how a system can store information so that a cue brings back a related pattern, fact, or sequence, a problem central to neural memory and efficient AI. Classical Hopfield networks, dense and kernel associative memories, and Hebbian learning use attractor dynamics or similarity-based retrieval; newer recurrent and linear-attention models add sparse key-value stores, while agent systems use linked episodic memories. Theory examines capacity, retrieval basins, stability, and interference, while experiments test noisy pattern completion, long-range recall, language-model memory, and hardware implementations. A persistent tension is that more storage does not guarantee reliable, selective recall: finite state and overlapping representations can cause interference, forgetting, and false associations, even as explicit retrieval improves access at added cost.

Latest papers

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120 papers16 papers in the last four weeks, down 16% on the four weeks before. 0.1% of all new papers.

Sep 7, 2026

On the Recall Scaling Laws in Mamba: A Theoretical and Mechanistic Study via Hashing

Associative Recall (AR) is the cognitive ability to learn and retrieve links between items in memory. In NLP, AR is used as a benchmark for evaluating the in-context memory capacity of architectures such as Mamba, and has been found to strongly correlate with language modeling performance. This paper explores AR from the perspective of mechanistic interpretability, aiming to reverse-engineer the exact internal algorithm used by Mamba to perform recall. Our key insight is that Mamba performs recall by implicitly learning linear hash functions, and we identify the low-level circuit that enables this behavior. Building on these findings and inspired by theoretical tools in similarity-preserving hashing, such as the Johnson-Lindenstrauss lemma, we develop a theoretical framework for analyzing AR, which we term Recall Scaling Laws. Given the vocabulary size and the number of facts in context, this framework allows us to (1) predict the embedding and state dimensions required for Mamba to achieve perfect recall, (2) predict recall success probability given the model dimensions, and (3) analyze multi-layer models and multi-head SSM patterns. Empirical results show that our theoretical findings are accurate and predictive, offering insights into how AR capacity scales with vocabulary, state, embedding size, and architecture.

Yuval Koren, Assaf Ben-Kish, Raja Giryes +2

Blavatnik School of Computer Science and AI, Tel Aviv University

Mechanistic InterpretabilityScaling LawsAssociative MemoryState Space Models
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Sep 2, 2026

Memory as an Energy Landscape---Hopfield

This chapter reconstructs the Hopfield network as a physical theory of memory rather than merely an early neural-network algorithm. It begins with the problem as it stood before 1982-threshold logic, Hebbian association, correlation memories, and recurrent binary networks-and isolates what Hopfield's synthesis added: a dynamical definition of content-addressable memory, a symmetric recurrent architecture with a Lyapunov function, a Hebbian embedding of patterns in its couplings, and a physical account of basins, robustness, and graceful degradation. The binary and graded-response energy functions are derived in full, together with the signal-crosstalk decomposition governing pattern stability, the mean-field theory of retrieval at extensive load, and the zero-temperature retrieval spinodal at (alpha 0.138) established by Amit, Gutfreund, and Sompolinsky. The energy-based program is then followed through analog optimization networks, polynomial dense associative memories, exponential interactions, and modern continuous Hopfield updates, including the precise conditions under which the update becomes scaled dot-product attention. Throughout, capacity claims are tied to their disorder ensemble, scaling limit, and success criterion, showing why numerically different storage limits need not conflict. A closing assessment distinguishes established results from surviving principles, assumption-bound limitations, and open problems, treating the Hopfield network as an effective theory whose symmetry, locality, and point-neuron assumptions delimit its biological reach. Fixed-seed numerical experiments expose the mechanisms discussed but do not substitute for analytical results.

Nima Dehghani

McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA · The NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI), Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Hopfield NetworksAssociative MemoryEnergy-Based ModelsStatistical Physics of Learning
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Sep 1, 2026

Basin Geometry and Reliable Recall of Dynamical Memories in Reservoir Computing

Reliable attractor recall conventionally requires broad basins of attraction. However, in reservoir-computing based associative memory, temporal cues reliably recover dynamical memories despite basins dominated by unpredictable, riddled-like regions. We reveal that memory basins exhibit an octopus-like'' structure: a robust head'' near the attractor and thin, intertwined ``tentacles'' spanning state space. Initial states in tentacular regions yield near-zero uncertainty exponents, making the recalled memory effectively unpredictable at finite precision. Yet, cue-driven generalized synchronization bypasses this unpredictability, driving the system into the robust basin head. This mechanism yields a quantitative relation linking minimum cue duration, synchronization rate, and basin-head radius. Trained recurrent neural networks exhibit similar geometry, suggesting this phenomenon extends beyond reservoir computing.

Ling-Wei Kong, Ying-Cheng Lai

Department of Computational Biology, Cornell University, Ithaca, NY 14850, USA · School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA · Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

Dynamical SystemsReservoir ComputingChaotic Dynamical SystemsAssociative Memory
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Sep 1, 2026

SSAKG 2.0: An Open-Source Package for Structural Associative Sequence Memory and Context-Based Retrieval

This article presents SSAKG 2.0, an open-source software package for constructing and operating Structural Sequential Associative Knowledge Graphs (SSAKGs). An SSAKG represents objects as graph vertices and ordered sequences as structural patterns of graph connections. The resulting sparse graph is used as an associative memory in which complete sequences can be reconstructed from a partial, unordered context. Version 2.0 introduces new algorithms that exploit individual bits of computer memory to efficiently search graph connections. The package is implemented in Python, while performance-critical graph operations are implemented in C and exposed through a Python interface. This hybrid implementation provides a flexible high-level programming environment while reducing the memory and computational overhead associated with large sparse graphs. The algorithms were evaluated using randomly generated numerical sequences, sequences derived from sentences in the NLTK corpus, and mRNA sequences. The experiments demonstrate the ability of the package to store and reconstruct sequences from partial contexts and provide a basis for evaluating the effects of graph density, sequence length, and memory size on retrieval performance. SSAKG 2.0 is distributed under the Apache 2.0 open-source license. The package includes documentation and reproducible examples and is publicly available through GitHub and the Python Package Index (PyPI).

Przemysław Stokłosa, Janusz A. Starzyk, Paweł Raif

Institute of Management and Information Technology Bielsko-Biała, Poland · University of Information Technology and Management Rzeszów, Poland · Silesian University of Technology Gliwice, Poland

Graph-Based RetrievalKnowledge GraphsAssociative MemorySequence Modeling
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Aug 31, 2026

Kathleen Remembers: Length-Invariant One-Shot Recall Without Attention

Recurrent, attention-free sequence models share a structural weakness: a fading state cannot perform exact recall of something seen once, far in the past. We add to the Kathleen trunk a second memory layer -- a "notebook": a fixed-key holographic (HRR) associative store with a learned local write gate, a self-gating raw read, and write-triggered forgetting -- 25K parameters that attach to the logits of any trunk. (1) Mechanism: on a controlled needle-in-haystack task the notebook reaches 80-82% one-shot recall at 4x the training length, where the bare trunk scores ~4% and a parameter-matched attention head scores 100% inside its training length and 0% beyond it. Addressing is length-invariant by construction; the untrained memory alone recalls at 90% accuracy identically at 512, 2048 and 4096 bytes. Because the store is a linear superposition, two capabilities follow from arithmetic alone: selective unlearning (one subtraction erases one fact to chance, retained facts unharmed) and per-token attribution (counterfactual erasure names the source fact of every correct byte, 100% provenance). (2) Real text: on WikiText-2 bytes the notebook improves prediction of repeated rare words by +0.15-0.27 bits/byte, the gain growing with the distance between mentions and holding zero-shot at 4x training length; write-triggered forgetting eliminates memory pollution at 8x length (first-mention cost +0.33 -> -0.004). (3) Scope and scale: a parameter-matched attention head does generalize on natural-text repetition, so the notebook's claim is exact recall at O(L); on a WikiText-103 ladder (8 to 512 MB) the zero-shot repeat gain rises monotonically. All experiments are pre-registered, seeds reported, and reproducible on a single free-tier GPU.

George Fountzoulas

Department of Computer Engineering & Informatics Frederick University, Nicosia, Cyprus

Memory-Augmented Neural NetworksPersistent Memory for Language ModelsLong-Context ModelingAssociative Memory
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Aug 29, 2026

Rate-Coding Bundle Memory: A Unified Model of Memory and Control for Symbolic Computation in the Brain

We propose a neurobiologically plausible model of cognition that combines the advantages of connectionist and symbolic systems, and that can explain a wide range of cognitive phenomena. This model, called Rate-Coding Bundle Memory (RCBM), is based on the Symbolic Subsystem Hypothesis, which posits that the brain implements a symbolic subsystem within its fundamentally connectionist nature. RCBM is a hybrid model that uses rate coding to represent symbols in a continuous space, and it uses a bundle memory system to store and retrieve these symbols. The model is capable of solving a wide range of cognitive phenomena, including one-shot learning, pattern separation, and the binding problem. We argue that RCBM provides a promising framework for understanding the nature of cognition, and that it can be used to develop more sophisticated models of cognition in the future.

Teun van Gils, Rowan P. Sommers, Markus Ostarek +1

Neurobiology of Language Department, Max Planck Institute for Psycholinguistics · Donders Institute for Brain, Cognition and Behaviour, Radboud University

Computational Cognitive ModelingNeurosymbolic LearningAssociative Memory
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Aug 13, 2026

RippleMem: From Isolated Retrieval to Associative Recollection for Long-Term Agent Memory

LLM-based agents increasingly rely on external memory to support long-horizon reasoning and interaction. However, the main bottleneck is not simply storing past experience, but recovering the right set of evidence when relevant information is distributed across many interactions. Existing approaches struggle with this access problem. Full-context methods require noisy long-context search, flat retrieval often returns isolated and incomplete records, and graph-based memory systems can be expensive to construct while compressing rich event context. We introduce RippleMem, a long-term memory system that replaces one-shot retrieval with adaptive associative recollection. Inspired by cue-dependent episodic retrieval and associative completion, RippleMem stores interaction history as cue-rich episodic memory units and organizes them in an event-centric memory graph. Given a query, it first recalls relevant memory anchors through hybrid cues, then expands from these anchors along semantic and structural associations to recover missing supporting evidence. In this way, initially recalled memories serve not only as answer context, but also as cues for completing the evidence needed to answer. Experiments on LoCoMo and LongMemEval-S show that RippleMem achieves the best overall performance across evaluated settings, improving LLM-as-a-Judge accuracy by 3.95% on LoCoMo and up to 11.87% on LongMemEval-S, while reducing graph construction cost by about 30x.

Jingbo Ji, Lingyi Li, Xilong Cheng +4

Communication University of China, Beijing, China. · Zhilian Yinghe Technology Co., Ltd., Beijing, China · State Key Laboratory of Media Convergence and Communication, Beijing, China

Agent MemoryPersistent Memory for Language ModelsGraph-Based Agent MemoryAssociative Memory
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Aug 12, 2026

Consolidator: Learning Persistent Routed Memory Across Context Boundaries

Copying short-term memory (STM) into a slower store can preserve state across a context boundary, but persistence alone does not ensure that the retained state influences subsequent memory access. We test this distinction in a Phasor Memory Network (PMNet) using Consolidator, a shared slot-local operator that transforms routed STM before accumulating it into long-term memory (LTM), without replaying the source tokens. After each consolidation, the KV cache and STM are cleared. The retained LTM can still be read and is also fed into the hierarchical router, thereby conditioning which explicit-memory slots subsequent inputs access. We evaluate this mechanism on a two-segment modulo-10 mapping task in which the second segment updates the mapping at the same memory address. Following a second consolidation and reset, a held-out query must recover the updated mapping from LTM. The backbone and memory interface are frozen, leaving only 12.35K Consolidator parameters trainable (0.041% of a 29.95M model). Across five paired runs from the same STM-pretraining checkpoint, direct LTM routing raises updated-mapping recall from 44.38±1.94%44.38\pm1.94\% to 87.02±1.76%87.02\pm1.76\% (+42.64±1.10+42.64\pm1.10 percentage points), while immediate STM recall remains 89.90% in both conditions; both train separate Consolidators and retain the same LTM read paths. Learned consolidation outperforms forced identity accumulation by 21.40±1.9121.40\pm1.91 percentage points without routing and 68.70±1.7668.70\pm1.76 with routing. Thus, on this task, consolidated LTM serves as both retrievable content and an access state that shapes subsequent slot selection.

Sungwoo Goo, Hwi-yeol Yun, Sangkeun Jung

College of Pharmacy, Chungnam National University · Department of Computer Science & Engineering, Chungnam National University

Memory-Augmented Neural NetworksPersistent Memory for Language ModelsHierarchical MemoryAssociative MemoryMemory Consolidation
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Aug 8, 2026

High-Capacity Generalized Hopfield Networks

Generalized Hopfield networks are introduced where memories and neurons are continuous variables that lie on a Riemannian manifold. We explicitly focus on symmetric spaces associated with the special unitary groups SU(d), and use both numerical and analytical (replica) techniques to demonstrate an almost order of magnitude enhancement in critical capacity over the vector networks starting with d=3 and further rapidly growing with d. To circumvent the non-linear geometric constraints, we use a Lie algebraic method [following V. Galitski, Phys. Rev. A 84, 012118 (2011)] to exactly describe the classical neural network in terms of linear algebra in an auxiliary Hilbert space. It is shown that in contrast to the traditional Hopfield networks, memory recall in SU(d) Hopfields corresponds to neuron alignment along a top eigenvector of a spiked matrix, which is less susceptible to random matrix crosstalk than other models with continuous neuron variables. Physical platforms to realize SU(d) Hopfields are briefly discussed and physical (in addition to algorithmic) recall mechanism is demonstrated, where memory recovery occurs naturally through generalized Landau-Lifshitz-Gilbert dynamics. To illustrate SU(3) memory recall, we introduce a color (RGB) image encoding/decoding protocol and explicitly run image recovery on corrupted cues. Finally, we quantize the generalized Hopfields which are shown to reduce to Sachdev-Ye glassy type of models. Their many-body spectra generally feature two types of dark and memory bands, where the latter exhibits chaotic Wigner-Dyson level statistics that hides Hebbian data.

Victor Galitski

Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA

Hopfield NetworksAssociative Memory
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Aug 3, 2026

DART: Decoded Attention over Recurrent States for Efficient Long-Context Sequence Modeling

Modern language models are built primarily from Transformers, recurrent models, and their hybrid architectures. Transformers rely on token-level attention memories, while recurrent models such as state space models (SSMs) and linear attention maintain compact recurrent states. These architectures are typically instantiated separately or interleaved at the layer level, leaving open whether a shared memory representation can support both recurrent compression and attention-style retrieval. We study this question through the state space duality (SSD) view of Mamba-2, where the SSM state can be interpreted as a compressed associative key--value (KV) cache. We observe that Mamba-2 decodes token-conditioned values from this state but does not decode token-conditioned keys. Based on this observation, we propose DART (Decoded Attention over Recurrent sTates), which retains the chunk state contributions produced by the Mamba-2 chunked scan as chunk state memories, decodes token-conditioned keys and values from these memories, and performs state-memory attention (SMA) over the resulting KV pairs. The retrieved output is then combined with the native Mamba-2 output through a gated residual connection. DART supports practical training by reusing the Mamba-2 chunked scan and implementing SMA as a FlashAttention-style computation. Our analysis and experiments show that DART substantially reduces the length-dependent inference cache compared with a matched attention baseline (e.g., 75%75\% savings when the chunk size is S=256S=256 and the state size is N=128N=128). Compared with Mamba-2, DART substantially improves associative recall and retrieval while preserving general language-modeling quality.

Yixiao Qian, Song Chen, Pengkai Wang +3

College of Control Science and Engineering, Zhejiang University · Department of Mathematics, National University of Singapore · Hong Kong Polytechnic University +1

Memory-Augmented Language ModelsKV CachingLong-Context Language ModelingAttention MechanismsAssociative MemoryState Space Models
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Jul 22, 2026

Memoir: Should a Model Write to Its Memory While It Thinks?

Memoir combines per-sample fast memory, shared slow parameters, variable-depth latent recurrence, and a future-latent energy objective. We test its riskiest coupling: each pondering iteration may rewrite the fast tier that the same iteration reads. On procedural associative recall with key interference, we compare a coupled arm against an otherwise identical read-only pondering arm. Both arms contain 81,738 parameters, including 76,362 trainable parameters, and use matched declared forward multiply-accumulate counts, data, optimizer, schedule, and seeds. After 240 training steps across 12 seeds, coupled recall is 0.5203 with a 95 percent interval of [0.4522, 0.5883], while read-only recall is 0.6557 with [0.5953, 0.7160]. The arms are paired per seed, and the read-only lead of 0.1354 gives a paired t of 3.23 on 11 degrees of freedom with a 95 percent interval of [0.0431, 0.2277] on the difference, winning on 10 of 12 seeds. After 960 steps across 8 seeds, both arms reach 1.0000, so the measured effect is a learning-speed penalty at a fixed budget, not a demonstrated capability penalty. That longer control is ceiling limited, leaving convergence on a non-saturating task unmeasured. A predicted failure in which memory rewriting corrupts the energy signal did not occur: the energy margin grew and held. Kernel restructuring also reduced delta-rule forward time from 0.907 ms to 0.351 ms on the stated device. Code and evidence are available at https://github.com/RightNow-AI/Memoir

Jaber Jaber, Osama Jaber

RightNow AI

Memory-Augmented Neural NetworksRecurrent Neural NetworksAssociative Memory
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Jul 21, 2026

Associative Emotional Learning in Convolutional Neural Networks

Associative emotional learning enables organisms to adaptively link pleasant or unpleasant outcomes to the presence of predictive stimuli. Whereas computational models such as the Rescorla-Wagner model have shed light on this important function, the limitations of these models are also known, especially when they are applied to neural data. The advent of deep neural networks has opened another avenue for modeling associative emotional learning. In this work we proposed a deep neural network model of visual valence processing, consisting of a visual module that encodes complex natural scenes and a module that recognizes their emotional significance in terms of valence, a key dimension of emotion, and tested a novel Pavlovian learning paradigm on the model. The results showed that with learning, the model reproduced several observations from human associative learning studies, including association formation and generalization, and that the neural representations of the conditioned and the unconditioned stimuli became increasingly aligned both at the single unit and at the neural population level. Comparison between the model and human experimental data provided further validation of our approach. This study thus suggests that deep neural network models, when combined with appropriate learning algorithms, can be used to model behavioral and neural signatures of associative emotion/valence learning.

Seowung Leem, Andreas Keil, Mingzhou Ding +1

J. Crayton Pruitt Family Dept. of Biomedical Engineering, University of Florida, Gainesville, FL 32611, USA · Center for the Study of Emotion and Attention, University of Florida, Gainesville, Florida, USA · Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA +3

Convolutional Neural NetworksAssociative MemoryAffective Computing
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Jul 21, 2026

Free energy landscape of Dense Associative Memory

Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also evaluate the disorder-averaged ground-state energy of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact full-retrieval threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.

Sumedha, Abhishek Singh

School of Physical Sciences, National Institute of Science Education and Research, Bhubaneswar, P.O. Jatni, 752050, India · 2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar 400094, India

Free Energy CalculationNeural Network MemorizationHopfield NetworksAssociative Memory
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Jul 19, 2026

Kernelized Linear Attention: Breaking the Capacity Wall with Symmetric Cones

Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first principles by certifying nonnegative attention weights through a self-dual homogeneous cone. Building on this observation, we show that rank-one positive semi-definite (PSD) features offer a favorable capacity--interference tradeoff. KATA recovers a parameter-free convex output gate and characterizes associative capacity through the Welch interference floor. For tolerances above this floor, KATA enlarges the state without adding parameters and admits spherical codes with exponentially many keys in the projection dimension. We implement KATA as fused Triton kernels at two operating points: a flash-attention-style forward up to ∼1.6×{\sim}1.6\times FlashAttention-2 throughput, and an exact O(T)O(T) chunked-state form that reaches ∼11×{\sim}11\times FlashAttention-2 forward throughput at 131131k tokens. An associative scan of the first-order feature lowers the inter-chunk recurrence depth to O(log⁡(T/C))O(\log(T/C)) for chunk size CC and averages ∼2.4×{\sim}2.4\times the throughput of a matched sequential linear-attention baseline. On long-range MQAR and repeated-key overwrite, several KATA variants outperform Gated DeltaNet, with parameter counts and state sizes reported alongside accuracy. Induction preserves near-perfect recall, while kernel benchmarks show that the maps can be implemented efficiently. KATA retains 0.9850.985 MQAR at a 16×16\times out-of-distribution length, approaching the softmax with roughly one quarter of the KV-cache entries. Experiments on 340M-parameter LLMs reveal a feature-dependent fluency trade-off and clarify how positional embeddings, delta rules, and decay gates interact with feature geometry.

Ayoub Ghriss, Sourav Chakraborty

Department of Computer Science, University of Colorado Boulder

Efficient AttentionLinear AttentionAssociative Memory
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Jul 10, 2026

MLPs are Hebbians: Constructing Efficient Fact-Storing MLPs for Transformers

Large language models (LLMs) store factual knowledge in their parameters. While recent work has shown that this knowledge resides in MLP layers, existing constructive and mechanistic interpretability models of fact-storage in LLMs fail to explain the surprising empirical phenomenon that they store facts at an information-theoretically optimal rate. In this work, we develop a theoretical account of this phenomenon. We develop the first Transformer-compatible fact-storing MLP closed-form construction that satisfies the following three properties empirically observed in LLMs: it (i) attains optimal fact storage scaling, (ii) handles arbitrary input/output geometries, and (iii) works inside Transformers. Key to our work is to analyze the decoding margin of MLPs, whereas prior work only studies MLP fact storage. Under isotropic embeddings, our construction achieves information-theoretically optimal storage capacity scaling and requires 1010-104×104\times fewer parameters at matched fact count than prior constructions. For arbitrary key and value embeddings, we show that our construction attains the same storage capacity scaling, up to penalization factors depending on the embedding geometries. Moreover, we demonstrate that our constructed MLPs can be used within Transformer blocks for factual recall tasks at optimal capacity scaling, requiring 1515-63×63\times fewer parameters at matched fact count than prior constructions. Finally, as a proof-of-concept, we show that fact-storing MLPs enable modular fact editing by swapping a Transformer's MLP with a new one.

Roberto Garcia, Jerry Liu, Ronny Junkins +3

Institute for Computational & Mathematical Engineering, Stanford University · Department of Computer Science, Stanford University · Department of Computer Science and Engineering, University at Buffalo

TransformerMemorization in Language ModelsFactual Knowledge in Language ModelsAssociative Memory
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Jul 10, 2026

Remembering Distinct Items, Not Tokens: A Learnable Dirichlet-Process Cache Between State-Space Models and Attention

Fixed-state sequence models compress an unbounded past into a bounded state, which caps their associative recall at roughly the state dimension; attention escapes the cap by keeping a key-value entry for every token, at quadratic compute and a cache that grows with the sequence. We study the middle ground: a sparse cache that allocates a slot only when an input is novel, so its size tracks the number of distinct items rather than the number of tokens. The allocation rule is the DP-means clustering rule, the small-variance limit of a Dirichlet-process mixture, used not as latent-variable inference but as the key-value memory operator for a deep recurrent backbone. We develop it in two forms, a static cache with a fixed concentration and a surprise-adaptive variant whose concentration follows the recent novelty rate. On a controlled associative-recall benchmark with redundancy we show that the cache matches full-attention recall while storing only the distinct items, that it dominates a fixed-budget eviction cache on the recall-versus-size frontier, and that on a state-space backbone it answers both a recall query and a long-range aggregate at the lowest memory of any model tested. The allocation is learnable end to end: a two-parameter novelty-threshold gate trained on the task loss alone recovers the rule exactly, whereas an over-parameterized gate fails, so the operative ingredient is the inductive bias rather than capacity. The evidence is a family of controlled mechanism studies at modest scale, with the distinct-items property confirmed on four real streams (recommendation, systems logs, clinical events, and insurance claims); a real-backbone, real-corpus language validation is pursued in a companion study.

Siddharth Pal, Viktoria Rojkova

Memory-Augmented Neural NetworksKV CachingAssociative Memory
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Jul 2, 2026

The Orthogonalized Read Is a Removable Training Scaffold for Recurrent Memory

Orthogonalizing the mLSTM memory matrix at read time with five differentiable Newton-Schulz iterations improves noisy associative recall. We replicate this effect and investigate its mechanism. Training on MAD noisy recall exhibits a long chance-level plateau followed by a sharp increase in accuracy. The orthogonalized read improves conditioning during this plateau and can be removed after escape. Ablations support three findings. First, the benefit requires a self-consistent read and gradient: an exact recursive least-squares read (the Mesa layer) yields a similar benefit, while straight-through variants, delta-rule writes, frozen random keys, and Frobenius normalization show no improvement over baseline. Second, across a learning-rate x task-difficulty grid, orthogonalization multiplies escape hazard roughly six-fold, with no detectable dependence on difficulty, and widens the range of learning rates that produce successful runs. Third, adding orthogonalization at inference leaves chance-level failures unresolved, while removing it gradually after escape yields standard mLSTMs at near-perfect accuracy. Schedule changes alone recover much of the reported gain. A batch-size x learning-rate analysis separates the effects of per-step learning rate and gradient noise on escape hazard (elasticities +3.0 and -1.65, respectively), linking the original vocab-96 result to its large-batch training regime. Direct decoding of the memory state recovers roughly half of the associations in behaviorally failed models, indicating a readout-learning limitation despite substantial stored information. These results show that fixed-budget recall benchmarks are sensitive to trainability and provide a tractable setting for investigating abrupt behavioral transitions through measurements of internal representations.

Keston Aquino-Michaels

No Way Labs

Long Short-Term Memory NetworksAssociative Memory
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Jul 1, 2026

Self-Organized Learning in Oscillatory Neural Networks with Memristive Signed Couplings

Oscillatory neural networks (ONNs) have emerged as a promising neuromorphic architecture, leveraging coupled dynamical systems to perform computation and represent information through phase relationships. Their interactions can be designed to support intrinsic energy-minimizing dynamics, enabling tasks such as associative memory and optimization, and positioning them as a candidate architecture for continuous learning and inference. We present a neuromorphic primitive implemented using memristive edges with inhibitory couplings as a potential design for autonomous learning, and provide circuit simulation validation that the system is capable of denoising noisy inputs on an auto-associative task. While numerical Hopfield/Ising models routinely assume signed weights, neuromorphic implementations of ONNs often fail to realize negative weights due to device and circuit constraints. A practically implementable route to inhibitory (negative) weights is particularly valuable: it expands the class of attractor structures accessible to oscillator networks beyond purely synchronous couplings, and supports phase-coded memories where anti-phase constraints are not merely transiently enforced during training but can persist autonomously after release. We provide circuit simulations and theoretical analyses demonstrating that signed effective weights are necessary for anti-phase attractors to persist autonomously.

Riley Acker, Aman Desai, Garrett Kenyon +1

1Computing and Artificial Intelligence (CAI) Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA · Center for Nonlinear Studies (CNLS), Los Alamos National Laboratory, Los Alamos, NM 87545, USA · 3Theoretical Division , Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Oscillatory Neural NetworksNeuromorphic ComputingAssociative Memory
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Jun 30, 2026

Creating Intelligence: A Computational Foundation for AGI

This work introduces a new computational theory of mind grounded in set theory and hyperdimensional computing. Whereas traditional neural networks rely on continuous weights and matrix multiplication, this framework works with sparse binary data. It represents information as discrete sets, directly modeling biological neural population codes. I demonstrate that associative memory emerges naturally from network topologies featuring a combinatorially expanded hidden layer. Learning is driven by topological plasticity rather than scalar weight adjustments. This architecture unifies auto-associative and hetero-associative learning under a single core algorithm: information retrieval via subset pattern matching and exact nearest-neighbor search. Operating with constant-time complexity, these mechanisms bridge perceptual data (sparse distributed representations) and symbols (sparse holographic representations) without continuous bottlenecks. Mapping this framework to neuroanatomy, I propose that both the cerebellum and the neocortex implement variants of this algorithm, making subset pattern matching the fundamental engine of cognition. Because it relies on discrete logic rather than matrix arithmetic, this algorithm translates directly into in-memory hardware. This opens a new route toward synthetic intelligence with human-level energy efficiency.

Peter Overmann

Computational Cognitive ModelingHyperdimensional ComputingVector Symbolic ArchitecturesAssociative Memory
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Jun 30, 2026

Explaining Machine Learning and Memorization with Statistical Mechanics

Artificial neural networks (NNs) and machine learning (ML) algorithms are poorly understood from a theoretical perspective, which makes it difficult to fully realize their potential and overcome their weaknesses. For instance, ML algorithms train NN weights by moving them along a low-dimensional subspace of their allowed values, but this implicitly low-dimensional learning structure is not properly exploited to improve training because its nature is not well understood. Moreover, trained NNs are easily confused by pervasive adversarial attacks whose theoretical underpinnings are still unclear. This thesis aims to improve our theoretical understanding of NNs and ML, with a particular focus on adversarial attacks and implicitly low-dimensional learning. For this purpose, we use mathematical tools from statistical mechanics to study different types of NNs and ways in which they can fit the data. In particular, we study two classes of models that fit the data with various degrees of learning and memorization: dense associative memory (DAM) and restricted Boltzmann machines (RBM). In the process, we investigate connections between different versions of these models that are useful to make analytical investigations more efficient.

Robin Theriault

Faculty of Science

Statistical Physics of LearningNeural Network MemorizationAdversarial RobustnessMemorization in Generative ModelsStatistical Learning TheoryAssociative Memory+1
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Jun 28, 2026

Supervised Hebbian learning in Deep Counterstream Associative Networks

Modern machine learning applications employ deep neural networks training with the error backpropagation algorithm. Although this algorithm is very effective, it lacks biological realism. For example, backpropagation requires symmetric connectivity, and a separate neural processing channel for error signals. Prior works have therefore proposed a number of more realistic alternatives for error backpropagation. However, most of them still suffer from demanding preassumptions that may be not fulfilled in the real brain, for example, they often still require either symmetric connectivity or two separate processing channels, and often require also special mathematical operations like subtractions or function inversions. Here I propose supervised counterstream learning in deep associative networks as a simpler approach that requires only recognition of errors during training, and then backpropagates correcting target activity through the same activity channel as used for forward propagation. For this, two activity waves are initiated at the same time in input and output layers and then traveling in opposite directions to meet in one of the hidden layers. By employing simple local Hebbian-type learning rules, the corresponding activity pattern sequences get linked bidirectionally, thereby decreasing error rates over time. Despite its simplicity and an incomplete hyperparameter optimzation, a high high test accuracy is achieved on the (binarized) MNIST data set that is comparable to more demanding architectures.

Andreas Knoblauch

Albstadt-Sigmaringen University, KEIM Institute, Poststrasse 6, 72458 Albstadt-Ebingen, Germany

Hebbian LearningAssociative Memory
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Jun 23, 2026

Parallel Manifold Steering: Efficient Adaptation of Large Associative Memories via Residual Energy Shaping

Large Transformer models function as Dense Associative Memories (DAMs), retrieving knowledge via high-dimensional attractor dynamics driven by the self-attention mechanism \citep{ramsauer2020hopfield, wu2024attention}. However, adapting these frozen memory systems to new tasks presents a fundamental ``Plasticity-Stability'' dilemma. Current methods either risk catastrophic interference by modifying synaptic weights directly (e.g., LoRA) \citep{hu2021lora} or degrade associative capacity by clogging the retrieval buffer with static prompt tokens (e.g., VPT) \citep{jia2022vpt}. In this work, we propose H-Res (Hierarchical Residual Steering), a mechanism that modulates the effective energy landscape of the Transformer without altering its global equilibrium or expanding its sequence length. By formulating adaptation as a control problem on the activation manifold \citep{chen2018neuralode}, H-Res learns a state-dependent vector field that steers token trajectories into task-specific basins of attraction. We formally prove that H-Res preserves the attention entropy of the foundation model and facilitates Neural Collapse \citep{papyan2020prevalence}. Empirically, Manifold Steering outperforms global weight modification by 26% on associative retrieval tasks and eliminates the computational overhead of prompt-based methods, scaling effectively to structured domains \citep{zha2023vtab}.

Kanishk Awadhiya

Independent Researcher

Memory-Augmented Neural NetworksTransformer AttentionActivation SteeringAssociative MemoryParameter-Efficient Fine-Tuning
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Jun 23, 2026

Towards Version-aware Operations and Transaction Memories for Multi-layer MeMo

MeMo proposes language models with explicit multi-layer correlation matrix memories (CMMs), where memorization, retrieval, and forgetting are architectural operations. This paper asks how such memories can reduce the need for retraining when knowledge changes. For changes expressible as MeMo memory associations, the model's accessible knowledge can be updated by editing explicit memories rather than retraining the whole model. We propose a version-aware operation layer in which high-level operations such as replace, obsolete, keep-history, rollback, and trace are compiled into MeMo-native primitive calls over sequences and tokens. The key observation is that a version-aware operation is rarely a single MeMo association. It is an ordered transaction of primitive edits, for example forgetting one sequence-token chain, memorizing another, preserving a historical chain, and recording an inverse program. The framework introduces two auxiliary CMMs: a Version CMM (V-CMM) for mapping version transitions to transaction handles, and a Transaction CMM (T-CMM) for storing reusable change contents and inverse programs. It supports both direct sequence-level edits and structured diff-level inputs, and outlines an evaluation route for update success, rollback, traceability, locality, and transaction reuse.

Peiran Li

Freie Universität Berlin, Kaiserswerther Str. 16-18, 14195 Berlin, Germany

Knowledge EditingContinual Learning for LLMsMemory-Augmented Language ModelsAssociative Memory
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Jun 13, 2026

T-Mem: Memory That Anticipates, Not Archives

Long-term memory is essential for conversational agents to remain coherent across extended dialogues, follow through on commitments made many sessions earlier, and adapt their behaviour to each user. Current LLM-backed long-term conversational memory, however, is reachability-bounded by the similarity between a query and stored content, both lexical and dense-vector. The approach is effective when query and memory share surface features such as wording or named entities (we call this descriptive). But it misses another, equally valuable class of cases, where query and memory do not share surface features and are tied only by a latent semantic arc (associative). On this regime prevailing long-term memory systems collectively fail. Covering this other half is what allows an assistant, for the first time, to actively draw on past dialogue as a semantic asset. On the memory side, this is the engineering counterpart of what cognitive science calls episodic future thinking: rehearsing past experience for the future contexts under which it will need to be found. We call these write-time rehearsals triggers. We propose T-Mem, the first long-term conversational memory architecture that covers both descriptive and associative recall. At each of two evidence granularities, single facts and full exchanges, T-Mem instantiates one descriptive trigger family and one associative trigger family, so that every memory remains reachable from both surface-similar and relevance-bound queries. As empirical validation, T-Mem reaches state-of-the-art on both LoCoMo and LoCoMo-Plus.

Weidong Guo, Dakai Wang, Zixuan Wang +2

1Tencent · 2The Hong Kong University of Science and Technology (Guangzhou)

Persistent Memory for Language ModelsLLM MemoryLong-Term Conversational MemoryAssociative MemoryAgent Memory Management
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Jun 13, 2026

Controlled Dynamics Attractor Transformer

Transformer architectures have dramatically advanced representation learning and inference in deep models through self-attention mechanisms. In parallel,associative memory (AM) frameworks map representations onto energy landscapes, offering interpretable retrieval mechanisms. However, their continuous-time inference dynamics lack the biological plausibility of classical Continuous Attractor Neural Networks (CANNs). To bridge this gap, we propose Controlled Dynamics Attractor Transformer (CDAT), which couples a mixture von Mises-Fisher (Mo-vMF) attention energy with a Hopfield refinement energy, while augmenting energy descent with a CANN-inspired excitation-inhibition modulation. CDAT instantiates a topology-constrained dynamical system whose couplings encode relational structure among tokens, thereby linking attractor-style dynamics to modern energy-based attention. We further provide a constructive dissipation analysis to formally establish their controlled inference dynamics. Benefiting from these robust and structured dynamics, CDAT achieves state-of-the-art performance across multiple benchmarks in graph anomaly detection and graph classification.

Cheng Zhang, Minnan Luo, Zesheng Yang +3

School of Computer Science and Technology, MOEK-LINNS Lab, Xi’an Jiaotong University, Xi’an, Shaanxi, China · Department of Computer Science and Technology, Tsinghua University, Beijing, China

Energy-Based ModelsContinuous Attractor Neural NetworksGraph ClassificationAssociative MemoryGraph TransformerAnomaly Detection
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Jun 11, 2026

Robust Auto-associative Memory via Convolutional Restricted Hopfield Networks

Associative memory models play a fundamental role in pattern retrieval, but their performance often degrades under adversarial perturbations and severe input corruptions. Existing approaches, including Modern Hopfield Networks (MHNs), and Predictive Coding Networks (PCNs), exhibit limitations in balancing storage capacity, computational efficiency, and robustness. In this paper, we propose a Convolutional Restricted Hopfield Networks (CRHNs), which integrates convolutional feature extraction with attractor-based memory retrieval in a structured latent space. The proposed model leverages subspace representations and fixed-point dynamics, trained via a gradient-free Subspace Rotation Algorithm (SRA), to enhance both robustness and memory capacity. Extensive experiments on Self-Taught Learning (STL) dataset demonstrate that CRHNs consistently achieve significantly lower reconstruction error compared to MHNs and PCNs across a wide range of adversarial attacks and input degradations. In many cases, CRHNs reduce reconstruction error by an order of magnitude and maintains stable retrieval performance under increasing perturbation strength. Statistical analysis further confirms that these improvements are significant (p<0.01p < 0.01). These results highlight the effectiveness of attractor-based memory mechanisms and suggest that CRHNs provide a promising framework for building robust and scalable associative memory systems.

Ci Lin, Tet Yeap, Iluju Kiringa

Electrical Engineering and Computer Science University of Ottawa

Adversarial RobustnessHopfield NetworksAssociative Memory
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Jun 9, 2026

Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memories

Tensor Product Representations provide the structural fidelity required for symbolic reasoning in models but suffer from exponential dimensionality growth when encoding deep recursive structures. Conversely, Vector Symbolic Architectures maintain constant dimensionality but sacrifice capacity and fidelity due to noisy compression via superposition. In this work, we propose Orthogonal Subspace Carving (OSC), a memory architecture that binds fillers to roles by projecting onto the null space of the role basis before aggregating into a fixed order-p tensor. OSC uses projections to enforce geometric orthogonality between bound structures within a static memory trace. We show that this mechanism decouples the tensor order from the structural depth, enabling deep recursive binding within a constant memory footprint. By performing retrieval via recognition, this construction allows for component vectors that are orders of magnitude smaller than the memory tensor, giving superior memory efficiency in settings involving high superposition. We also show that TPR is a special case of binding in Clifford algebra, and give a Clifford formulation of OSC.

Travis Pence, Daisuke Yamada, Vikas Singh

Department of Computer Science, University of Wisconsin-Madison, United States · Department of Biostatistics and Med. Info., University of Wisconsin-Madison, United States

Memory-Efficient InferenceVector Symbolic ArchitecturesAssociative Memory
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Jun 9, 2026

Parallel Causal Associative Fields: Gated Sparse Memory for Long-Context Language Modeling

Transformers achieve strong language modeling performance by providing direct token-to-token communication paths, but causal self-attention scales quadratically with context length. Recurrent and state-space models reduce this cost, yet compress history into sequentially updated fixed-size states. This paper studies a third primitive: a parallel content-addressed memory over causal successor records. The proposed Parallel Causal Associative Field (PCAF) writes local records from a context window into hash buckets, retrieves a bounded candidate set for the current query, forms a sparse cache distribution over successor tokens, and mixes that cache with a parametric local language model through a learned gate. The resulting model maintains sparse long-context access while avoiding a single fixed recurrent state bottleneck. We evaluate PCAF under full autoregressive pretraining on WikiText-103 and PG-19 using a distributed Google Cloud TPU v4-32 pod. At 303M parameters and context length T = 2048, PCAF-semantic reaches 36.31 perplexity on WikiText-103 and 52.45 perplexity on PG-19, compared with 47.49 and 53.84 for a matched dense Transformer. PCAF-semantic simultaneously processes 0.61-0.62M tokens/s across the TPU pod, versus 0.43M tokens/s for dense and local attention baselines. Supporting 41M-parameter multi-seed sweeps and single-GPU component ablations show that the associative cache, retrieval capacity, and learned gate materially affect the speed-quality trade-off.

Muhammad Ahmed

Independent Researcher

Memory-Augmented Language ModelsLong-Context Language Model InferenceLong-Context Language ModelingAutoregressive Language ModelingAssociative Memory
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Jun 8, 2026

Hyperbolic Neural Population Geometry Benefits Computation

Neural population geometry shapes downstream computation. Recent empirical findings in neurobiology suggest that a hyperbolic structure underlies population activity in the hippocampus. Here we provide a theoretical framework for this phenomenon. First, we propose a plausible construction of hippocampal tuning curves that statistically induces hyperbolic geometry. Next, we establish a connection between neural decoding and associative memory by demonstrating that the Modern Hopfield Network update rule computes the minimum mean-squared-error (MMSE) estimator. Finally, we introduce a novel associative memory model defined in hyperbolic space that yields significantly larger capacity than leading models. Our results suggest that animals encode spatial information as a latent hyperbolic cognitive map, improving both memory capacity and decoding accuracy.

Dennis Wu, Yi-Chun Hung, Braden Yuille +2

Department of Computer Science, Northwestern University · Center for Foundation Models and Generative AI, Northwestern University · Integrated Science Program, Northwestern University +3

Neural Representation GeometryNeural ProcessesNeural DecodingHopfield NetworksNeural EncodingAssociative Memory
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Jun 2, 2026

An RRAM-based Hardware Implementation of a Radial Basis Function Neuron for Edge Classifiers

The deployment of modern machine learning (ML) solutions on resource-constrained edge devices highlights implementation challenges. This is especially true for extreme edge applications that include safety-critical components, such as autonomous navigation tasks. This paper demonstrates an artificial neural network (ANN) design leveraging Metal-Oxide Resistive RAM (RRAM) -based Analogue Content Addressable Memory (ACAM) as an efficient hardware substrate for performing metric-based classification and online adaptation on the edge. The proposed design is based on a custom Template piXeL (TXL) cell used for building the ACAM module, where each TXL cell acts as a configurable receptive field neuron. These cells employ a Radial Basis activation function to calculate the distance of an input from the programmed receptive field. The TXL can be organised into dense arrays for calculating the distance of a high-dimensional input against all stored prototypes, effectively performing fast and energy efficient similarity search. This hardware engine enables on-the-fly learning, where the receptive field parameters can be tuned to track domain shift. Through simulation of the proposed TXL-RBF classifier we can achieve 89.1% accuracy on the MNIST dataset while consuming 185fJ per cell per operation when operating at 10MHz.

Georgios Papandroulidakis, Shady Agwa, Themis Prodromakis

Centre for Electronics Frontiers, Institute of Micro and Nano Systems, School of Engineering, The University of Edinburgh, EH9 3BF, UK.

Compute-in-MemoryEdge MLAssociative MemoryOnline LearningImage Classification
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