用大偏差理论推导出密集联想记忆的自由能公式，揭示记忆检索与初始状态的关系。
Free energy landscape of Dense Associative Memory
- 基于大偏差理论建立自由能泛函的一般表达式
- 给出多项式互作与LSE激活下温度依赖的自由能解
- 精确求出LSE模型全回忆阈值，适用于复杂网络分析
通过大偏差理论，我们求解并获得了一类广义联想记忆（包括密集联想记忆）的自由能泛函通式。通过重现霍普菲尔德模型的经典结果来验证方法。针对有限模式数，推导出具有多项式相互作用和对数求和指数（LSE）激活函数的密集联想记忆在温度依赖下的自由能泛函，并在扩展极限下评估了系统的无序平均基态能量。该解析框架揭示了高阶密集网络中记忆检索对初始状态的依赖性，给出了LSE模型的精确全回忆阈值。该方法为分析多样复杂的联想记忆架构提供了系统化途径。
原文摘要 · Abstract (English)
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.
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