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. A sub-linear time algorithm $\mathtt{U}\text{-}\mathtt{Hop}$+ to reach KHMs' optimal capacity. 3. An analysis of the scaling behavior of the required feature dimension relative to the number of stored memories.These efforts improve both the retrieval capability of KHMs and the representation learning of corresponding transformers.Experimentally, we provide thorough numerical results to back up theoretical findings.

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.

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.

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 extend the existing work on Hopfield network state classification, employing more complex models that remain interpretable, such as densely-connected feed-forward deep neural networks and support vector machines. The states of the Hopfield network can be grouped into several classes, including learned (those presented during training), spurious (stable states that were not learned), and prototype (stable states that were not learned but are representative for a subset of learned states). It is often useful to determine to what class a given state belongs to; for example to ignore spurious states when retrieving from the network. Previous research has approached the state classification task with simple linear methods, most notably the stability ratio. We deepen the research on classifying states from prototype-regime Hopfield networks, investigating how varying the factors strengthening prototypes influences the state classification task. We study the generalizability of different classification models when trained on states derived from different prototype tasks -- for example, can a network trained on a Hopfield network with 10 prototypes classify states from a network with 20 prototypes? We find that simple models often outperform the stability ratio while remaining interpretable. These models require surprisingly little training data and generalize exceptionally well to states generated by a range of Hopfield networks, even those that were trained on exceedingly different datasets.

The theoretical contribution presented in 291--310 is a welcome insight on the computational power of ReLUs. The experimental results for rectified polynomial units reported in figures 2 and 3 are interesting and apparently novel, even in the context of standard feedforward multi-layer networks. Being 291--297 a central point of the paper it should be expanded and better justified. Furthermore, the simple capacity analysis developed in p. 3 for the polynomial energy function is invoked here for the rectified polynomial energy function. This has to be justified. The paper starts from and mostly focuses on the associative memory (Hamiltonian) formulation, but then the findings are restricted to one-step retrieval.

Sequential learning involves learning tasks in a sequence, and proves challenging for most neural networks. Biological neural networks regularly conquer the sequential learning challenge and are even capable of transferring knowledge both forward and backwards between tasks. Artificial neural networks often totally fail to transfer performance between tasks, and regularly suffer from degraded performance or catastrophic forgetting on previous tasks. Models of associative memory have been used to investigate the discrepancy between biological and artificial neural networks due to their biological ties and inspirations, of which the Hopfield network is perhaps the most studied model. The Dense Associative Memory, or modern Hopfield network, generalizes the Hopfield network, allowing for greater capacities and prototype learning behaviors, while still retaining the associative memory structure. We investigate the performance of the Dense Associative Memory in sequential learning problems, and benchmark various sequential learning techniques in the network. We give a substantial review of the sequential learning space with particular respect to the Hopfield network and associative memories, as well as describe the techniques we implement in detail. We also draw parallels between the classical and Dense Associative Memory in the context of sequential learning, and discuss the departures from biological inspiration that may influence the utility of the Dense Associative Memory as a tool for studying biological neural networks. We present our findings, and show that existing sequential learning methods can be applied to the Dense Associative Memory to improve sequential learning performance.

We investigate the behavior of the Nadaraya-Watson kernel smoothing estimator in high dimensions using its relationship to the random energy model and to dense associative memories.

Dense Associative Memories or modern Hopfield networks permit storage and reliable retrieval of an exponentially large (in the dimension of feature space) number of memories. At the same time, their naive implementation is non-biological, since it seemingly requires the existence of many-body synaptic junctions between the neurons. We show that these models are effective descriptions of a more microscopic (written in terms of biological degrees of freedom) theory that has additional (hidden) neurons and only requires two-body interactions between them. For this reason our proposed microscopic theory is a valid model of large associative memory with a degree of biological plausibility. The dynamics of our network and its reduced dimensional equivalent both minimize energy (Lyapunov) functions. When certain dynamical variables (hidden neurons) are integrated out from our microscopic theory, one can recover many of the models that were previously discussed in the literature, e.g. the model presented in ''Hopfield Networks is All You Need'' paper. We also provide an alternative derivation of the energy function and the update rule proposed in the aforementioned paper and clarify the relationships between various models of this class.

A model of associative memory is studied, which stores and reliably retrieves many more patterns than the number of neurons in the network. We propose a simple duality between this dense associative memory and neural networks commonly used in deep learning. On the associative memory side of this duality, a family of models that smoothly interpolates between two limiting cases can be constructed. One limit is referred to as the feature-matching mode of pattern recognition, and the other one as the prototype regime. On the deep learning side of the duality, this family corresponds to feedforward neural networks with one hidden layer and various activation functions, which transmit the activities of the visible neurons to the hidden layer.