We investigate how feature correlations influence the capacity of Dense Associative Memory (DAM), a Transformer attention-like model. Practical machine learning scenarios involve feature-correlated data and learn representations in the input space, but current capacity analyses do not account for this. We develop an empirical framework to analyze the effects of data structure on capacity dynamics. Specifically, we systematically construct datasets that vary in feature correlation and pattern separation using Hamming distance from information theory, and compute the model's corresponding storage capacity using a simple binary search algorithm. Our experiments confirm that memory capacity scales exponentially with increasing separation in the input space. Feature correlations do not alter this relationship fundamentally, but reduce capacity slightly at constant separation. This effect is amplified at higher polynomial degrees in the energy function, suggesting that Associative Memory is more limited in depicting higher-order interactions between features than patterns. Our findings bridge theoretical work and practical settings for DAM, and might inspire more data-centric methods.

Networks of phase oscillators can serve as dense associative memories if they incorporate higher-order coupling beyond the classical Kuramoto model's pairwise interactions. Here we introduce a generalized Kuramoto model with combined second-harmonic (pairwise) and fourth-harmonic (quartic) coupling, inspired by dense Hopfield memory theory. Using mean-field theory and its dynamical approximation, we obtain a phase diagram for dense associative memory model that exhibits a tricritical point at which the continuous onset of memory retrieval is supplanted by a discontinuous, hysteretic transition. In the quartic-dominated regime, the system supports bistable phase-locked states corresponding to stored memory patterns, with a sizable energy barrier between memory and incoherent states. We analytically determine this bistable region and show that the escape time from a memory state (due to noise) grows exponentially with network size, indicating robust storage. Extending the theory to finite memory load, we show that higher-order couplings achieve superlinear scaling of memory capacity with system size, far exceeding the limit of pairwise-only oscillators. Large-scale simulations of the oscillator network confirm our theoretical predictions, demonstrating rapid pattern retrieval and robust storage of many phase patterns. These results bridge the Kuramoto synchronization with modern Hopfield memories, pointing toward experimental realization of high-capacity, analog associative memory in oscillator systems.

We propose a novel energy function for Dense Associative Memory (DenseAM) networks, the log-sum-ReLU (LSR), inspired by optimal kernel density estimation. Unlike the common log-sum-exponential (LSE) function, LSR is based on the Epanechnikov kernel and enables exact memory retrieval with exponential capacity without requiring exponential separation functions. Moreover, it introduces abundant additional \emph{emergent} local minima while preserving perfect pattern recovery -- a characteristic previously unseen in DenseAM literature. Empirical results show that LSR energy has significantly more local minima (memories) that have comparable log-likelihood to LSE-based models. Analysis of LSR's emergent memories on image datasets reveals a degree of creativity and novelty, hinting at this method's potential for both large-scale memory storage and generative tasks.

We study the optimal memorization capacity of modern Hopfield models and Kernelized Hopfield Models (KHMs), a transformer-compatible class of Dense Associative Memories.We present a tight analysis by establishing a connection between the memory configuration of KHMs and spherical codes from information theory. Specifically, we treat the stored memory set as a specialized spherical code.This enables us to cast the memorization problem in KHMs into a point arrangement problem on a hypersphere.We show that the optimal capacity of KHMs occurs when the feature space allows memories to form an optimal spherical code.This unique perspective leads to: 1. An analysis of how KHMs achieve optimal memory capacity, and identify corresponding necessary conditions. Importantly, we establish an upper capacity bound that matches the well-known exponential lower bound in the literature. This provides the first tight and optimal asymptotic memory capacity for modern Hopfield models.2.

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.

Associative memory engages in the integration of relevant information for comprehension in the human cognition system. In this work, we seek to improve alignment between language models and human brain while processing speech information by integrating associative memory. After verifying the alignment between language model and brain by mapping language model activations to brain activity, the original text stimuli expanded with simulated associative memory are regarded as input to computational language models. We find the alignment between language model and brain is improved in brain regions closely related to associative memory processing. We also demonstrate large language models after specific supervised fine-tuning better align with brain response, by building the \textit{Association} dataset containing 1000 samples of stories, with instructions encouraging associative memory as input and associated content as output.

Brain-inspired computing aims to mimic cognitive functions like associative memory, the ability to recall complete patterns from partial cues. Memristor technology offers promising hardware for such neuromorphic systems due to its potential for efficient in-memory analog computing. Hopfield Neural Networks (HNNs) are a classic model for associative memory, but implementations on conventional hardware suffer from efficiency bottlenecks, while prior memristor-based HNNs faced challenges with vulnerability to hardware defects due to offline training, limited storage capacity, and difficulty processing analog patterns. Here we introduce and experimentally demonstrate on integrated memristor hardware a new hardware-adaptive learning algorithm for associative memories that significantly improves defect tolerance and capacity, and naturally extends to scalable multilayer architectures capable of handling both binary and continuous patterns. Our approach achieves 3x effective capacity under 50% device faults compared to state-of-the-art methods. Furthermore, its extension to multilayer architectures enables superlinear capacity scaling (\(\propto N^{1.49}\ for binary patterns) and effective recalling of continuous patterns (\propto N^{1.74}\ scaling), as compared to linear capacity scaling for previous HNNs. It also provides flexibility to adjust capacity by tuning hidden neurons for the same-sized patterns. By leveraging the massive parallelism of the hardware enabled by synchronous updates, it reduces energy by 8.8x and latency by 99.7% for 64-dimensional patterns over asynchronous schemes, with greater improvements at scale. This promises the development of more reliable memristor-based associative memory systems and enables new applications research due to the significantly improved capacity, efficiency, and flexibility.

By leveraging tools from the statistical mechanics of complex systems, in these short notes we extend the architecture of a neural network for hetero-associative memory (called three-directional associative memories, TAM) to explore supervised and unsupervised learning protocols. In particular, by providing entropic-heterogeneous datasets to its various layers, we predict and quantify a new emergent phenomenon -- that we term {\em layer's cooperativeness} -- where the interplay of dataset entropies across network's layers enhances their retrieval capabilities Beyond those they would have without reciprocal influence. Naively we would expect layers trained with less informative datasets to develop smaller retrieval regions compared to those pertaining to layers that experienced more information: this does not happen and all the retrieval regions settle to the same amplitude, allowing for optimal retrieval performance globally. This cooperative dynamics marks a significant advancement in understanding emergent computational capabilities within disordered systems.

Memorization is a fundamental ability of Transformer-based Large Language Models, achieved through learning. In this paper, we propose a paradigm shift by designing an architecture to memorize text directly, bearing in mind the principle that memorization precedes learning. We introduce MeMo, a novel architecture for language modeling that explicitly memorizes sequences of tokens in layered associative memories. By design, MeMo offers transparency and the possibility of model editing, including forgetting texts. We experimented with the MeMo architecture, showing the memorization power of the one-layer and the multi-layer configurations.

We introduce in-context denoising, a task that refines the connection between attention-based architectures and dense associative memory (DAM) networks, also known as modern Hopfield networks. Using a Bayesian framework, we show theoretically and empirically that certain restricted denoising problems can be solved optimally even by a single-layer transformer. We demonstrate that a trained attention layer processes each denoising prompt by performing a single gradient descent update on a context-aware DAM energy landscape, where context tokens serve as associative memories and the query token acts as an initial state. This one-step update yields better solutions than exact retrieval of either a context token or a spurious local minimum, providing a concrete example of DAM networks extending beyond the standard retrieval paradigm. Overall, this work solidifies the link between associative memory and attention mechanisms first identified by Ramsauer et al., and demonstrates the relevance of associative memory models in the study of in-context learning.