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 $d^2\sim 2n\log n$.

We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.

We study the optimal memorization capacity of modern Hopfield models and Kernelized Hopfield Models (KHMs), a transformer-compatible class of Dense Associative Memories.

Thetwoindependent modelsleadtothenon-end-to-end pipeline, or called two-stage pipeline. Moreover, the human detector involves extra memory as well as computational cost. The bottom-up strategy [16, 17, 18, 19] uses the keypoint heatmap to locate all person keypoints at first and then assigns them to individuals via heuristic grouping process,asshowninFigure1(a).

Dense Associative Memory (DAM) models generalize the classical Hopfield model by incorporating n-body or exponential interactions that greatly enhance storage capacity. While the criticality of DAM models has been largely investigated, mainly within a statistical equilibrium picture, little attention has been devoted to the temporal self-organizing behavior induced by learning. In this work, we investigate the behavior of a stochastic exponential DAM (SEDAM) model through the lens of Temporal Complexity (TC), a framework that characterizes complex systems by intermittent transition events between order and disorder and by scale-free temporal statistics. Transition events associated with birth-death of neural avalanche structures are exploited for the TC analyses and compared with analogous transition events based on coincidence structures. We systematically explore how TC indicators depend on control parameters, i.e., noise intensity and memory load. Our results reveal that the SEDAM model exhibits regimes of complex intermittency characterized by nontrivial temporal correlations and scale-free behavior, indicating the spontaneous emergence of self-organizing dynamics. These regimes emerge in small intervals of noise intensity values, which, in agreement with the extended criticality concept, never shrink to a single critical point. Further, the noise intensity range needed to reach the critical region, where self-organizing behavior emerges, slightly decreases as the memory load increases. This study highlights the relevance of TC as a complementary framework for understanding learning and information processing in artificial and biological neural systems, revealing the link between the memory load and the self-organizing capacity of the network.

Dense Associative Memories are high storage capacity variants of the Hopfield networks that are capable of storing a large number of memory patterns in the weights of the network of a given size. Their common formulations typically require storing each pattern in a separate set of synaptic weights, which leads to the increase of the number of synaptic weights when new patterns are introduced. In this work we propose an alternative formulation of this class of models using random features, commonly used in kernel methods. In this formulation the number of network's parameters remains fixed. At the same time, new memories can be added to the network by modifying existing weights. We show that this novel network closely approximates the energy function and dynamics of conventional Dense Associative Memories and shares their desirable computational properties.

Transformers face scalability challenges due to the quadratic cost of attention, which involves dense similarity computations between queries and keys. We propose CAMformer, a novel accelerator that reinterprets attention as an associative memory operation and computes attention scores using a voltage-domain Binary Attention Content Addressable Memory (BA-CAM). This enables constant-time similarity search through analog charge sharing, replacing digital arithmetic with physical similarity sensing. CAMformer integrates hierarchical two-stage top-k filtering, pipelined execution, and high-precision contextualization to achieve both algorithmic accuracy and architectural efficiency. Evaluated on BERT and Vision Transformer workloads, CAMformer achieves over 10x energy efficiency, up to 4x higher throughput, and 6-8x lower area compared to state-of-the-art accelerators--while maintaining near-lossless accuracy.

We study the optimal memorization capacity of modern Hopfield models and Kernelized Hopfield Models (KHMs), a transformer-compatible class of Dense Associative Memories.

The Muon optimizer is consistently faster than Adam in training Large Language Models (LLMs), yet the mechanism underlying its success remains unclear. This paper demystifies this mechanism through the lens of associative memory. By ablating the transformer components optimized by Muon, we reveal that the associative memory parameters of LLMs, namely the Value and Output (VO) attention weights and Feed-Forward Networks (FFNs), are the primary contributors to Muon's superiority. Motivated by this associative memory view, we then explain Muon's superiority on real-world corpora, which are intrinsically heavy-tailed: a few classes (tail classes) appear far less frequently than others. The superiority is explained through two key properties: (i) its update rule consistently yields a more isotropic singular spectrum than Adam; and as a result, (ii) on heavy-tailed data, it optimizes tail classes more effectively than Adam. Beyond empirical evidence, we theoretically confirm these findings by analyzing a one-layer associative memory model under class-imbalanced data. We prove that Muon consistently achieves balanced learning across classes regardless of feature embeddings, whereas Adam can induce large disparities in learning errors depending on embedding properties. In summary, our empirical observations and theoretical analyses reveal Muon's core advantage: its update rule aligns with the outer-product structure of linear associative memories, enabling more balanced and effective learning of tail classes in heavy-tailed distributions than Adam.

Associative Memories like the famous Hopfield Networks are elegant models for describing fully recurrent neural networks whose fundamental job is to store and retrieve information. In the past few years they experienced a surge of interest due to novel theoretical results pertaining to their information storage capabilities, and their relationship with SOTA AI architectures, such as Transformers and Diffusion Models. These connections open up possibilities for interpreting the computation of traditional AI networks through the theoretical lens of Associative Memories. Additionally, novel Lagrangian formulations of these networks make it possible to design powerful distributed models that learn useful representations and inform the design of novel architectures. This tutorial provides an approachable introduction to Associative Memories, emphasizing the modern language and methods used in this area of research, with practical hands-on mathematical derivations and coding notebooks.