Spectral density of correlated random matrices and nonmonotonic stability in hetero-associative memory networks

Random matrix theory, which characterizes spectral distributions of infinitely large matrices, plays a central role across diverse fields, including high-dimensional data analysis, ecology, neuroscience, and machine learning. Among its key results, the Marchenko-Pastur law and the elliptic law have provided theoretical foundations for numerous applications. However, despite their importance, the relationship between these two laws has not yet been fully understood. Here, we develop a novel derivation of the spectral density for a correlated random matrix ensemble that unifies the Marchenko-Pastur and elliptic laws as special cases. Interestingly, matrices from this ensemble can be naturally interpreted as connectivity matrices of hetero-associative memory networks, which, from a modern neural network perspective, are essentially equivalent to the linear attention architecture, a variant of the attention layer in the Transformer. Using this result, we find that the stability of the network depends non-monotonically on the number of memorized patterns. By uncovering a nontrivial property of high-dimensional correlated systems, this work deepens our understanding of asymmetric interactions across various scientific fields.