Dense Associative Memory

The absolute capacity of dense associative memory has mainly been analyzed for unbiased patterns. Here we examine the effect of bias in centered binary patterns under the Krotov-Hopfield single-site criterion $P_{\mathrm{error}}=1/N$, where $P_{\mathrm{error}}$ is the probability that a single-site flip lowers the energy of a stored pattern and $N$ is the number of neurons. Each pattern component takes $1-q$ with probability $q$ and $-q$ otherwise, where $0<q\le1/2$. For polynomial interactions of order $n$, a signal-to-noise analysis gives an absolute capacity of order $N^{n-1}/\ln N$ at $q=1/2$. For fixed $q<1/2$, however, the capacity is $O(N^{n/2})$ for even $n\ge4$ and $O(N^{(n+1)/2})$ for odd $n\ge5$. For $n=3$, both the unbiased and fixed-bias capacities remain $O(N^2/\ln N)$. For $n\ge4$, these different asymptotic forms imply a nonuniform large-$N$ limit near $q=1/2$. Asymptotic matching predicts a bias-induced crossover in the region $1-2q=O(\ln N/N^{\lfloor n/2\rfloor-1})$. The crossover originates from a bias-dependent crosstalk mean that reduces the stability of sites carrying the more frequent value $-q$. Computer simulations are compared with the finite-size conditioned-Gaussian predictions. An activity-dependent control potential that cancels the conditional crosstalk mean restores the $N^{n-1}/\ln N$ capacity for fixed $0<q<1/2$ within the conditioned-Gaussian approximation.

Hierarchical correlations are a universal feature of any realistic model of data, and the question of how associative memory models may learn these correlations and generalize beyond them to construct new sensible images is an important step towards understanding more complex modern architectures such as diffusion models. We consider a hierarchical model for memories which are sampled and stored in a dense Hopfield network with polynomial activation. We analytically derive conditions for each level of this hierarchy to be locally stable - that is they are local energy minima. We use prototype reconstruction as a minimal model of generalization and we find that it takes only a quasi-polynomial amount of information to generalize beyond particular memories and even particular groups in the hierarchy. We observe a qualitatively analogous phase diagram in the number of memories, sharpness of the activation function (polynomial degree) for data from Fashion-MNIST.

Linear attention is increasingly used in frontier language models for efficient long-context inference and constant-memory decoding. Its fixed-size recurrent memory, however, requires an online decision at each token: what to write and how strongly to overwrite existing associations before knowing which information future queries will require. Delta-rule models learn this strength from the current token embedding but do not track confidence in the memory estimate, preventing each write from adapting to accumulated evidence. To represent this uncertainty explicitly, we reformulate recurrent associative memory as a linear--Gaussian state-space model, for which the Kalman filter is the optimal recursive estimator, and introduce a new family of models, Kalman Delta Networks (KDNs). Within KDNs, the transition propagates both the memory state and its uncertainty, allowing the Kalman gain to weight each residual write by accumulated evidence and observation reliability. Under this formulation, Delta-style updates emerge as a special case that substitutes a token-wise isotropic surrogate for predictive covariance and omits covariance tracking. Exact tracking, however, entails a dense, state-dependent Riccati recursion that is poorly suited to GPU-parallel linear-attention scans. To address this issue, we introduce two scan-compatible KDN approximations. Diagonal KDN projects each one-step posterior onto the diagonal Gaussian family through online mean-field variational inference, whereas Isotropic KDN uses an isotropic approximation with a single uncertainty scalar per head. Their uncertainty recurrences are Mobius maps, enabling associative scans with logarithmic parallel depth. Across controlled pretraining at 750M and 1.3B parameters, KDN variants consistently improve perplexity and mean downstream accuracy over state-of-the-art linear-attention models.

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.

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.

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).

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.

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.

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 ${\sim}1.6\times$ FlashAttention-2 throughput, and an exact $O(T)$ chunked-state form that reaches ${\sim}11\times$ FlashAttention-2 forward throughput at $131$k tokens. An associative scan of the first-order feature lowers the inter-chunk recurrence depth to $O(\log(T/C))$ for chunk size $C$ and averages ${\sim}2.4\times$ 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 $0.985$ MQAR at a $16\times$ 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.

Extending the context length of large language models (LLMs) is critical for many real-world applications, yet standard transformers remain constrained by quadratic compute and linear memory scaling. In this work, we investigate the Associative Recurrent Memory Transformer (ARMT) as a practical approach for enabling long-context processing in LLMs, constant memory scaling, and better efficiency. We make three main contributions. First, we construct two domain-specific long-context datasets designed to evaluate realistic workloads, focusing on narrow-domain fine-tuning scenarios. Second, we propose a comprehensive training recipe for ARMT-based context extension, combining continued pre-training, synthetic long-context data generation, curriculum learning, and selective integration of associative memory into chosen model layers. Third, we present an extensive experimental study demonstrating that ARMT-augmented models: (i) process inputs well beyond their original context limits without degrading performance relative to in-limit baselines; (ii) generalize more effectively to out-of-distribution context lengths; and (iii) need 30% less FLOPs while preserving baseline performance within the original context window.

Linear-attention and state-space language models compress the prefix into a fixed-size recurrent state, yielding O(1) memory at the cost of a lossy exact memory: when many key--value associations compete, earlier facts are overwritten and needle recall degrades. Inspired by Complementary Learning Systems, we give linear attention a hippocampal complement. HOLA (Hippocampal Linear Attention) keeps the usual delta-rule state as a compressive memory and adds a bounded exact KV cache, forming a semiparametric test-time memory: the state models linearly compressible structure, while the cache stores associations that should not be forced through that state. The cache writes without a learned eviction module, keeping tokens with large beta * ||e||, the prediction residual actually committed to the state; a decoupled RMSNorm-gamma cache read then turns these exact KV pairs into sharp retrieval rather than soft averaging. At 340M parameters trained on 15B SlimPajama tokens, HOLA lowers Wikitext perplexity from 27.32 to 22.92 (-16.1%), below a full-attention Transformer++ (26.88), and improves LAMBADA perplexity from 30.95 to 30.26. It also achieves the best linear in-context retrieval and remains much more robust than GDN or a matched HOLA+recency cache on RULER needle-in-a-haystack recall out to 32k tokens (16x its training length).

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.

Anomaly detection in multivariate time series remains a critical challenge in large-scale distributed systems, where related entities may exhibit transferable precursor behavior prior to anomaly onset. Existing methods typically operate independently on each data stream and therefore remain fundamentally reactive. To address this limitation, we introduce Distributed Hierarchical Temporal Memory (D-HTM), a neuromorphic framework that enables cross-entity preemptive warning through a Shared Associative Memory (SAM). D-HTM combines a Spatial Pooler (SP) that projects observations into a common Sparse Distributed Representation (SDR) space, Temporal Memory (TM) modules that learn entity-specific dynamics online, and a Shared Associative Memory that stores recurring pre-anomaly signatures. By reusing precursor knowledge across related entities, D-HTM can issue warnings prior to local anomaly onset while preserving HTM's online learning capabilities. We evaluate D-HTM on the Server Machine Dataset (SMD), the Soil Moisture Active Passive (SMAP) dataset, the Mars Science Laboratory (MSL) dataset, and a synthetic cascade benchmark designed to isolate precursor transfer. Experimental results demonstrate effective cross-entity warning propagation while maintaining competitive reactive anomaly detection performance. Across the real-world datasets, D-HTM provides an average warning lead time of 8.1 samples prior to anomaly onset. These findings demonstrate that transferable precursor structure can emerge within a shared SDR space and be reused for preemptive warning generation, extending HTM beyond isolated reactive detection toward distributed predictive reasoning.

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.

RGB-Event tracking improves localization robustness by fusing RGB appearance textures and dense temporal motion cues from event sensors. While this multi-modal scheme broadens tracking applicability, real-world scenes suffer diverse structured signal degradations that hinder traditional multi-modal fusion. In harsh environments, either modality can lose reliability drastically, and targets frequently appear incomplete due to occlusion, edge truncation and foreground clutter.To tackle the above challenges, we present a hierarchical perturbation and retrieval framework tailored for RGB-Event tracking with robustness against partial target missing and modal degradation, termed APRTrack. To mimic real-world signal corruption, APRTrack constructs structured degradation via two adversarial perturbation branches at the modality and spatial levels, which separately simulate full-modal failure and localized target region absence. A hierarchical routing mechanism is designed to disentangle the training pipelines of the two perturbation types, effectively eliminating feature collapse induced by superimposed degradation constraints. Furthermore, we devise Footprint-guided Channel-calibrated Hopfield Retrieval (FCHR) for reliable historical information compensation. This module evaluates retrieval confidence based on association footprints between queries and memory banks, and calibrates the retrieval metric space prior to Hopfield matching, realizing controllable historical feature compensation bounded to target regions. Extensive experiments on FE108, COESOT, VisEvent, and FELT datasets demonstrate the effectiveness of our proposed strategies for the RGB-Event visual object tracking. The source code and pre-trained models will be released on https://github.com/Event-AHU/OpenEvTracking

Large Language Models (LLMs) are traditionally viewed as autoregressive generators. However, from the perspective of collective computation, they function as high-dimensional Dense Associative Memories that store complex reasoning patterns as latent attractors. In this work, we investigate the energy landscape of mathematical reasoning. We posit that correct reasoning chains correspond to deep, wide attractor basins ("flat minima") in the model's output distribution, whereas hallucinations manifest as sharp, unstable local minima. To exploit this geometry, we introduce a retrieval mechanism based on a Gibbs measure of the trajectory's spectral entropy. By sampling multiple reasoning paths and weighting them by their inverse energy ($P \propto e^{-βE}$), we approximate the equilibrium distribution of the associative memory, effectively ``relaxing'' the system into a robust solution. Empirically, this physics-inspired mechanism improves Microsoft Phi-3.5 performance on GSM8K by 5.38% (84.7% $\to$ 90.1%), demonstrating that inference is better modeled as a dynamic settling process into an attractor basin rather than greedy next-token prediction.

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 \textbf{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}.

Robotic inspection relies on accurate sensing to acquire high-fidelity geometric measurements for defect detection and metrology. While prior work has focused on robot motion and viewpoint planning, how to configure sensing parameters remains largely underexplored, despite their decisive impact on measurement quality. We propose SenseHD, a robotic sensing system that formulates scanner configuration as an instruction-conditioned sensing decision. Instead of predicting precise parameter values, SenseHD treats sensing parameters as discrete sensing actions and selects stable sensing regimes through hyperdimensional associative memory. Experiments on a real robotic inspection platform demonstrate that SenseHD robustly selects appropriate sensing configurations and significantly improves inspection reliability, while remaining lightweight and efficient compared to baseline methods.

SWave is a complex-valued recurrent language model (169.26M parameters, D=384, L=16, T=2048) trained on FineWeb-Edu using 2xH100 NVL. It was designed around three founding premises: that representing language as complex waves rather than real-valued numbers enables richer information encoding; that a Cayley-parameterised unitary transition provides a mathematical guarantee against state decay or explosion; and that a hidden state which rotates rather than shrinks preserves signal integrity over arbitrarily long contexts. The core of SWave evolved substantially across three development phases. The Resonance Head was found to structurally admit imaginary-channel collapse as a global loss minimum (a failure mode we term cos-domination collapse) and was superseded by an untied head with independent real and imaginary embedding tables from the Phase-Associative Memory (PAM) architecture. This resolved the degenerate minimum and enabled stable 200,000-step training (best-step PPL 22.0 at step 89,861). ComplexNorm and the Wave Propagation Scan proved load-bearing throughout all three phases and were retained to the final architecture. ProtectGatedScan was reframed as a structural prior rather than a learned behaviour. The four multi-scale retention concepts showed no measurable improvement under controlled evaluation and were found non-load-bearing. The ComplexGatedUnit was superseded by a real-valued squared-ReLU channel mixer with fewer parameters. The auxiliary training objectives showed no benefit once structural constraints were resolved. The investigation yields a formal characterisation of cos-domination collapse, a parallel scan with a log-space backward pass for numerical stability, six transferable engineering principles for complex-valued recurrent training, and a plan-to-code traceability methodology for catching structural divergences that conventional test suites miss.

Search, a foundational operation in computer science, maps a query to a matching item in a collection. It is typically implemented as a System-2 like, rule-based pipeline in which a key is computed, an index is probed, and candidates are verified. By contrast, human recognition resembles a System-1 like, associative model of identity recovery, in which even partial cues can trigger a recall without explicitly enumerating, ranking, or even accessing discrete candidates. Here, we show that music sound identification, a difficult search problem, can be performed in a single neural feed-forward pass by a generative transformer. Trained on an audio dataset, the model predicts the corresponding track identifier from a short audio excerpt. This approach surpasses state-of-the-art acoustic fingerprinting, with the largest gains for short audio segments (1 second), demonstrating the method is not only viable but advantageous. Moreover, it reduces external storage to 0.33% of the baseline footprint and improves inference latency by 2.3x (p95). Furthermore, the model can reject queries for unseen tracks, supporting open-set operation while reducing misattribution risk. Using music track identification as an example, this work reframes search, bringing it closer in spirit to human associative recognition and away from algorithmic database lookup.

Memory formation is fundamental to intelligence, yet whether deep neural networks preserve identifiable memory traces analogous to biological memory units remains an open question. This work introduces a geometric framework to identify such "AI engrams" by formalizing the neuroscientific criteria of specificity, reactivation, sufficiency, and necessity into a constrained inverse problem. We derive a closed-form estimator that isolates individual memory traces from globally entangled parameters, and show that this biologically-derived solution corresponds to a natural gradient update on the parameter manifold. AI engrams enable surgical manipulation of learned knowledge: any subset of memories can be composed or erased through linear arithmetic, without iterative optimization. Experiments ranging from simple MLPs to LLMs demonstrate the causal validity and substantial scalability of AI engrams. Together, these results bridge theories of biological memory and artificial representation learning and offer geometric insight into how deep networks simultaneously support functional specificity within distributed storage.

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 ($p < 0.01$). 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.

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.

We characterize the pre-softmax attention matrix $\mathbf{QK^\top}$ in transformers as an associative memory matrix encoding pairwise associations between input features. By decomposing this matrix into its symmetric and skew-symmetric parts, we interpret the symmetric component as governing the structure of the energy landscape, and the skew-symmetric component as driving circulation on that landscape. Leveraging the energy formulation induced by the symmetric component, we derive Hopfield-style stability measures that quantify the stability of retrieved features. We observe meaningful correlations between Hopfield-style stability measures and the fidelity-diversity trade-offs in generation. Finally, we propose a controllable knob to modulate this trade-off by modifying the circulation of the underlying dynamics. Code is available at our GitHub (https://github.com/hyeon-cho/Attention-Symmetric-Decomposition).

What can the success of modern AI tell us about how humans learn? This paper argues that taking AI seriously as a model of human learning supports a modest but genuine associationism. The central finding is that supervised learning -- learning driven by evaluative feedback -- underlies a surprisingly wide range of contemporary AI systems, from large language models to game-playing agents, differing primarily in how much work is required to generate the relevant feedback signal. This vindicates associationist ideals of a uniform, gradual, error-driven learning mechanism operating across domains, and defuses the once-influential argument that associationist mechanisms are too limited to account for human cognitive capacities. At the same time, the successes of deep learning depend on computational architectures that go well beyond anything classical associationists envisaged, and supervised learning operates within these as one component rather than a complete account of learning.

Vector-HaSH and the Tolman-Eichenbaum Machine (TEM) propose the hippocampal-entorhinal circuit factorizes memory via a grid-cell scaffold for compositional replay. Concurrently, human iEEG shows sharp-wave ripples gate recall and multi-hop replay fidelity decays multiplicatively. Yet, these fields lack a shared algebraic foundation. We introduce VaCoAl, an algebro-deterministic hyperdimensional memory architecture built on Galois-field linear-feedback shift registers. Its deterministic Galois-field diffusion offers a substrate-level alternative to Vector-HaSH's random projections, matching quasi-orthogonality while ensuring bit-exact reproducibility. Furthermore, the path-integral Confidence Ratio CR2 provides an algebraically tractable model for the empirically observed multiplicative replay decay. Biologically, VaCoAl's two operating regimes align with the EC-CA3 direct and EC-DG-CA3 trisynaptic pathways, explaining their 520-Myr conservation. Independent cellular evidence supports that the DG-CA3 pathway implements a biophysical homologue of Galois-field arithmetic. We also link this framework to Judea Pearl's Ladder of Causation. Reversible GF(2) binding provides the surgical algebra for the do-operator (Rung 2), and VaCoAl's dual-orthogonalizer architecture supplies the parallel substrate required for counterfactual reasoning (Rung 3). Ultimately, we prove these formal correspondences and derive testable iEEG predictions, uniting computational neuroscience, electrophysiology, and hyperdimensional computing.

Linear attention reduces the quadratic cost of softmax attention to $\mathcal{O}(T)$, but its memory state grows as $\mathcal{O}(T)$ in Frobenius norm, causing progressive interference between stored associations. We introduce \textbf{Variational Linear Attention} (VLA), which reframes the memory update as an online regularised least-squares problem with an adaptive penalty matrix maintained via the Sherman-Morrison rank-1 formula. We prove that normalising the write direction to unit length gives the recurrence Jacobian spectral norm exactly $1$ for all sequence lengths and head dimensions (Proposition 2), and that the state norm is self-limiting under bounded inputs (Proposition 1). Empirically, VLA reduces $\|S_t\|_F$ by $109\times$ relative to standard linear attention at $T{=}1{,}000$, achieves near-perfect exact-match accuracy on multi-query associative recall within the effective per-head memory regime ($n_\text{pairs} < d_h$), maintaining substantially higher retrieval performance than DeltaNet and standard linear attention under increasing memory load, and maintains 62% accuracy at the per-head capacity boundary. A Triton-fused kernel achieves $14\times$ speedup over sequential Python and $\mathcal{O}(T)$ scaling, crossing below softmax attention latency at approximately 43,000 tokens.

Large language models demonstrate remarkable ability in factual recall, yet the fundamental limits of storing and retrieving input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their corresponding~$d$-dimensional targets via a single layer, requiring each mapped input to be well separated from all other targets. Unlike in supervised classification, this strict separation induces~$p$ constraints per association and produces strong correlations between constraints that make a direct characterisation of the storage capacity difficult. Here, we provide a precise characterisation of this capacity in the following way. We first introduce a decoupled model in which each input has its own independent set of competing outputs, and provide numerical and analytical evidence that this decoupled model is equivalent to the original model in terms of storage capacity, spectra of the learnt weights, and storage mechanism. Using tools from statistical physics, we show that the decoupled model can store up to $p_c \log p_c / d^2 = 1 / 2$ associations, and generalise the computation of $p_c$ to linear two-layer architectures. Our analysis also gives mechanistic insight into how the optimal solution improves over a naïve Hebbian learning rule: rather than boosting input-output alignments with broad fluctuations, the optimal solution raises the correct scores just above the extreme-value threshold set by the competing outputs. These findings give a sharp statistical-physics characterisation of factual storage in linear networks and provide a baseline for understanding the memory capacity of more realistic neural architectures.

Adaptive behavior requires the brain to transition between distinct contexts while maintaining representations of prior experience. The ability to reconfigure neural representations without erasing previously acquired knowledge is central to learning in dynamic environments, yet the neural mechanisms that support this balance remain unclear. Understanding these mechanisms is also critical for addressing catastrophic forgetting in artificial systems designed for lifelong learning. Here, we identify joint sparse coding and temporal dynamics in both the mouse medial prefrontal cortex (mPFC) and computational networks as mechanisms that help preserve prior representations during context transitions. Specifically, sparsity in context-dependent representations reduces cross-context interference, whereas temporal dynamics within the network activity further enhance context separability across time. Strikingly, networks endowed with both properties, such as spiking neural networks, exhibit improved retention during lifelong learning without auxiliary heuristics. These findings establish joint sparse coding and temporal dynamics as a core mechanism supporting flexible context reconfiguration in lifelong learning and, through their activity constraining nature, as an energy-efficient architectural principle for stable adaptation. Together, they provide a mechanistic framework for understanding how the brain preserves prior knowledge while flexibly adapting to new contexts.

Dense Associative Memory (DenseAM) is a promising family of AI architectures that is represented by a neural network performing temporal dynamics on an energy landscape. While hyperparameter transfer methods are well-studied for feed-forward networks, these methods have not been developed for settings in which weights are shared across layers and within the layer, which is common in DenseAMs. Additionally, DenseAMs utilize rapidly peaking activation functions that are rarely used in feed-forward architectures. The confluence of these aspects makes DenseAM a challenging framework for using existing methods for hyperparameter transfer. Our work initiates the development of hyperparameter transfer methods for this class of models. We derive explicit prescriptions for how the hyperparameters tuned on small models can be transferred to models trained at scale. We demonstrate excellent agreement between these theoretical findings and empirical results.

Hopfield networks and their generalizations have established deep connections among biological associative memories, statistical physics, and transformers. Yet most models treat retrieval as a fixed query-to-memory mapping, ignoring the role of external context in recall. In this work, we propose a two-stage associative memory architecture, wherein a context-gate subcircuit reshapes the retrieval energy landscape before and during recall. We show theoretically that context gating increases inter-memory separation while inducing sparsity, translating into exponential improvements in retrieval. Crucially, we prove that the system admits a unique self-consistent fixed point, revealing that the resulting retrieval state is driven by both a direct contextual bias and a second-order retrieval-gate feedback loop. We then bridge this theory to transformers; specifically, we evaluate a first-order approximation on Llama-3, confirming that in-context learning acts as context-gated retrieval. Native dynamics mirror our theory: context localizes a memory subspace, enabling the zero-shot query to cleanly discriminate. Ultimately, this framework provides a mechanistic link between associative memory theory and LLM phenomenology.

High-capacity associative memory models, such as Kernel Logistic Regression (KLR) Hopfield networks, have demonstrated strong storage capabilities but typically rely on computationally expensive synchronous updates. This reliance poses a bottleneck for deployment on energy-efficient, event-driven neuromorphic hardware. In this paper, we investigate the asynchronous retrieval dynamics of KLR Hopfield networks. We show empirically that, under appropriately tuned kernel parameters, asynchronous sequential updates exhibit trajectories that are statistically indistinguishable from those of synchronous dynamics, while maintaining high recall accuracy within the tested regime for random patterns. Furthermore, we find that the asynchronous network achieves empirical storage capacities approaching $P/N \approx 30$ in static random pattern regimes, exceeding classical limits. To evaluate computational efficiency, we analyze the total number of state transitions (bit flips) required for error correction. The results show that the network converges using a number of events close to the initial Hamming distance from the target pattern, without observable spurious oscillations. These findings suggest that the large-margin attractors induced by KLR learning create a smooth energy landscape suited for sparse, event-driven computation, providing a basis for scalable and low-power associative memory on neuromorphic architectures.

How many key-value associations can a $d\times d$ linear memory store? We show that the answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. In an isotropic Gaussian model for the stored pairs, we show that top-1 retrieval, where every signal must beat its largest distractor, requires the logarithmic model-size scale $d^2\asymp n\log n$. We prove that the correlation matrix memory construction, which stores associations by superposing key-target outer products, achieves this scale through a sharp phase transition, and that the same scaling is necessary for any linear memory. Thus the logarithm is the intrinsic extreme-value price of winner-take-all decoding. We next consider listwise retrieval, where the correct target need not be the unique top-scoring item but should remain among the strongest candidates. To formalize this regime, we propose the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this listwise retrieval criterion, the capacity follows the quadratic scale $d^2\asymp n$. At load $n/d^2\toα$, we develop an exact asymptotic theory for the TAM empirical-risk minimizer through a two-parameter scalar variational principle. The theory has a rich phenomenology: in the ridgeless limit it yields a closed-form critical load separating satisfiable and unsatisfiable phases, and it predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles. Finally, a small-tail extrapolation further leads to the conjectural sharp top-1 threshold $d^2\sim 2n\log n$.

Adapting pretrained models typically involves a trade-off between the high training costs of backpropagation and the heavy inference overhead of memory-based or in-context learning. We propose FAAST, a forward-only associative adaptation method that analytically compiles labeled examples into fast weights in a single pass. By eliminating memory or context dependence, FAAST achieves constant-time inference and decouples task adaptation from pretrained representation. Across image classification and language modeling benchmarks, FAAST matches or exceeds backprop-based adaptation while reducing adaptation time by over 90% and is competitive to memory/context-based adaptation while saving memory usage by up to 95%. These results demonstrate FAAST as a highly efficient, scalable solution for supervised task adaptation, particularly for resource-constrained models. We release the code and models at https://github.com/baoguangsheng/faast.

Associative memory or content-addressable memory is an important component function in computer science and information processing, and at the same time a key concept in cognitive and computational brain science. Many different neural network architectures and learning rules have been proposed to model the brain's associative memory while investigating key component functions like figure-ground segmentation, perceptual reconstruction and rivalry. A less investigated but equally important capability of associative memory is prototype extraction where the training set comprises distorted prototype instances and the task is to recall the correct generating prototype given a new distorted instance. In this paper we benchmark associative memory function of seven different Hebbian learning rules employed in non-modular and modular recurrent networks with winner-take-all dynamics operating on moderately sparse binary patterns. We measure pattern storage and weight information capacity, prototype extraction capabilities, and sensitivity to correlations in data. The original additive Hebb rule comes out with worst capacity, covariance learning proves to be robust but with moderate capacity, and the Bayesian-Hebbian learning rules show highest capacity in almost all different conditions tested.

Can Neural Assemblies -- groups of neurons that fire together and strengthen through co-activation -- learn the direction of causal influence between variables? While established as a computationally general substrate for classification, parsing, and planning, neural assemblies have not yet been shown to internalize causal directionality. We demonstrate that the inherent operations of neural assemblies -- projection, local plasticity control, and sparse winner selection -- are sufficient for directional learning. We introduce DIRECT (DIRectional Edge Coupling/Training), a mechanism that co-activates source and target assemblies under an adaptive gain schedule to internalize directed relations. Unlike backpropagation-based methods, DIRECT relies solely on local plasticity, making the resulting causal claims auditable at the mechanism level. Our findings are verified through a dual-readout validation strategy: (i) synaptic-strength asymmetry, measuring the emergent weight gap between forward and reverse links, and (ii) functional propagation overlap, quantifying the reliability of directional signal flow. Across multiple domains, the framework achieves perfect structural recovery under a supervised, known-structure setting. These results establish neural assemblies as an auditable bridge between biologically plausible dynamics and formal causal models, offering an "explainable by design" framework where causal claims are traceable to specific neural winners and synaptic asymmetries.

We present a biologically detailed extension of the classical Hopfield/Marr auto-associative memory model for CA3, implementing ten populations (two asymmetric pyramidal subtypes, eight GABAergic interneuron classes), forty-seven compartments, multi-rule plasticity (recurrent Hebb, BCM anti-saturation, mossy-fiber short-term, endocannabinoid iLTD, burst-gated Hebb), and a bimodal cholinergic encoding/consolidation cycle. Evaluated on pattern completion across auto-associative, associative, and temporal regimes, and on a controlled inhibitory-proportion manipulation at $N{=}256$, the full architecture exhibits \emph{three qualitative signatures absent from a minimal Hopfield baseline}: (i)~multi-attractor cross-seed behaviour at $K{=}5$ with biologically realistic inhibitory proportions, where two of five seeds converge to positive attractors with margin ${+}0.10{-}0.22$ (Cohen's $d{=}0.71$, one-sided $p{=}0.08$); (ii)~target-selective associative recall in paired $(A, B)$ memory at $K{\geq}5$, where the full model retrieves $B$ from a partial cue of $A$ while the minimal model echoes $A$ (Pearson margin $Δ{=}{+}0.163$ at $K{=}5$); (iii)~reduced cross-seed variance of the full model below the minimal baseline under clean upstream, with ratios $1.0{-}3.0$. These three signatures are architecture-specific: they appear consistently across independent regimes and are absent from the minimal control.

Standard transformer architectures learn fixed slow-weight representations during training and lack mechanisms for rapid adaptation within an episode. In contrast, biological neural systems address this through fast synaptic updates that form transient associative memories during inference, a property known as Hebbian plasticity. In this paper, we conduct an empirical study of Hebbian Fast-Weight (HFW) modules integrated into multiple transformer backbones, including ViT-Small, DeiT-Small, and Swin-Tiny. We evaluate six model variants: ViT, DeiT, Swin, ViT-Hebbian, DeiT-Hebbian, and Swin-Hebbian on 5-way 1-shot and 5-way 5-shot classification tasks using the Omniglot benchmark under a Prototypical Network meta-learning framework. We propose a single module placement strategy for Swin-Tiny in which one HFW module is applied to the final stage feature map after all hierarchical stages have completed. This design avoids the training instability caused by placing separate Hebbian modules at each stage and achieves the highest test accuracy across all six models (96.2% at 1-shot; 99.2% at 5-shot), outperforming its non-Hebbian baseline by $+0.3$ percentage points at 1-shot. We analyze the interaction between Swin's shifted window inductive bias and episode-level Hebbian binding, discuss why per-block placement fails for ViT and DeiT variants in a low-data regime, and situate the results within the wider literature on fast and slow-weight meta-learning.

Sequential associative memories (SAMs) are difficult to build and maintain in real-world streaming environments, where observations arrive incrementally over time, have imbalanced sampling, and non-stationary temporal dynamics. Vector Symbolic Architectures (VSAs) provide a biologically-inspired framework for building SAMs. Entities and attributes are encoded as quasi-orthogonal hyperdimensional vectors and processed with well defined algebraic operations. Despite this rich framework, most VSA systems rely on simple additive updates, where repeated observations reinforce existing information even when no new information is introduced. In non-stationary environments, this leads to the persistence of stale information after the underlying system changes. In this work, we introduce the Sequential Relevance Memory Unit (SRMU), a domain- and cleanup-agnostic update rule for VSA-based SAMs. The SRMU combines temporal decay with a relevance gating mechanism. Unlike prior approaches that solely rely on cleanup, the SRMU regulates memory formation by filtering redundant, conflicting, and stale information before storage. We evaluate the SRMU on streaming state-tracking tasks that isolate non-uniform sampling and non-stationary temporal dynamics. Our results show that the SRMU increases memory similarity by $12.6\%$ and reduces cumulative memory magnitude by $53.5\%$. This shows that the SRMU produces more stable memory growth and stronger alignment with the ground-truth state.

Physical selectors (lasers choosing a mode, Ising machines settling on a ground state, condensates occupying a spin state) produce high-dimensional signatures at the moment of decision: full field amplitudes, multimode interference patterns, or scattering responses. These signatures are richer than the winner's index, yet they are routinely discarded. We show that when the signatures are repeatable across trials (stereotyped) and linearly independent across routes, a single linear decoder compiled from calibration data maps them to arbitrary payloads, merging selection and memory access into one event and eliminating the fetch that dominates latency and energy in sparse routing architectures. The construction requires one SVD of measured device responses, which certifies capability and bounds worst-case error for any downstream payload before the task is chosen. Runtime error separates into two independently diagnosable channels, decoding fidelity (controlled by dictionary conditioning $σ_{\min}(Φ)$) and routing reliability (controlled by the margin-to-noise ratio $Δ/T_{\mathrm{eff}}$), each with a distinct physical origin and targeted remedy. We derive the full error decomposition, give Ising-machine selector constructions, and validate the predicted scalings on synthetic speckle-signature simulations across three measurement modalities. No hardware demonstration exists; we provide a falsifiable four-step experimental protocol specifying what a first experiment must measure. Whether real device signatures satisfy stereotypy is the central open question.

An associative memory (AM) enables cue-response recall, and it has recently been recognized as a key mechanism underlying modern neural architectures such as Transformers. In this work, we introduce the concept of distributed dynamic associative memory (DDAM), which extends classical AM to settings with multiple agents and time-varying data streams. In DDAM, each agent maintains a local AM that must not only store its own associations but also selectively memorize information from other agents based on a specified interest matrix. To address this problem, we propose a novel tree-based distributed online gradient descent algorithm, termed DDAM-TOGD, which enables each agent to update its memory on the fly via inter-agent communication over designated routing trees. We derive rigorous performance guarantees for DDAM-TOGD, proving sublinear static regret in stationary environments and a path-length dependent dynamic regret bound in non-stationary environments. These theoretical results provide insights into how communication delays and network structure impact performance. Building on the regret analysis, we further introduce a combinatorial tree design strategy that optimizes the routing trees to minimize communication delays, thereby improving regret bounds. Numerical experiments demonstrate that the proposed DDAM-TOGD framework achieves superior accuracy and robustness compared to representative online learning baselines such as consensus-based distributed optimization, confirming the benefits of the proposed approach in dynamic, distributed environments.

Associative 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. Overall, 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.