Binary associative memory networks: A review of mathematical framework and capacity analysis

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In recent years, heightened interest has been ignited in associative memory networks, largely attributed to their perceived equivalence with the attention mechanism, a fundamental component of the Transformer architecture. The opaque nature of deep neural networks, often characterized as “black boxes”, has intensified the pursuit of explainability, positioning associative memory networks as promising candidates for illuminating the inherent complexities of deep learning models. Despite their increasing prominence, the mathematical analysis of their capacity remains a significant research gap, which constitutes the central focus of this paper. To address this gap, we commence with a review of the mathematical framework underpinning associative memory networks, with particular emphasis on their binary configurations, drawing insights from the derivation of the dense associative memory model. Additionally, we review a systematic methodology for analyzing the capacity of binary associative memory networks, building upon established studies of dense associative memory networks. Utilizing this analytical framework, we derive the capacity of several prominent associative memory networks, including binary modern Hopfield networks and binary spherical Hopfield networks. Through comprehensive discussions and rigorous deductions, we aim to elucidate the characteristics of binary associative memory networks, thereby providing valuable insights and practical guidance for their effective application in real-world scenarios.