🤖 AI Summary
This work investigates the free energy landscape and memory retrieval mechanisms of high-order dense associative memory models. Leveraging large deviation theory and statistical physics, it constructs a free energy functional tailored to polynomial interactions and Log-Sum-Exponential (LSE) activation, enabling a rigorous analysis of temperature-dependent behavior and ground state energy in the finite pattern regime. The study establishes, for the first time, the exact full-retrieval phase transition threshold for LSE-based models, elucidates the critical role of initial conditions in memory recovery within high-order networks, and develops a general analytical framework extensible to complex associative memory architectures. This framework not only reproduces classical results from the Hopfield model but also systematically extends the theoretical foundations of dense associative memory.
📝 Abstract
Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also evaluate the disorder-averaged ground-state energy of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact full-retrieval threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.