Free energy landscape of Dense Associative Memory

Associative memory in the Hopfield network is attractor dynamics in a disordered many-body system, and higher-order and exponential extensions turn its retrieval update into softmax attention. The polynomial and exponential regimes have been analyzed by different methods, with no common architecture in which to ask what fixes the storage scale. In this paper we study the bipartite architecture of Krotov and Hopfield, which we call the class $H$, whose model is fixed by a Lagrangian for each layer, taking the hidden neurons as the order parameter of retrieval. At polynomial load the replica method yields the replica-symmetric phase diagrams and closed-form capacities, and the crosstalk moment is common to Ising and spherical visible neurons, so their differences come from the visible entropy. With a softmax hidden layer the load is exponential, and a copy representation maps the thermodynamics onto random-energy-model counting, with paramagnetic, condensed, and frozen phases. Heating destabilizes retrieval by quantized reassignments of attention, and typical Gaussian patterns remain metastable at every load. The regimes differ in their crosstalk statistics, central-limit at polynomial load and large-deviation at exponential load, and the class $H$ splits retrieval into two roles, the visible Lagrangian fixing stability and the hidden one the storage scale, two axes that may also guide the design of new Lagrangians.

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.

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.