发表机构
University of California, Davis; Fudan University（加州大学戴维斯分校; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出黎曼流形上的内在稠密联想记忆，证明曲率决定记忆保留与创造，推导容量缩放，并展示曲率可作为设计变量。
AI 中文摘要
几何不仅仅约束联想记忆：曲率决定了它记住什么以及它创造哪些状态。我们通过将记忆视为Epanechnikov核密度模式寻找，在黎曼流形上发展了内在稠密联想记忆。我们比较了测地线与体积校正能量，并表明曲率区分了它们的行为。我们证明了测地线记忆总是保留一个孤立的模式，而校正记忆遵循一个尖锐的Ricci曲率阈值：正曲率可以在高维中抹去记忆，而负曲率则强化它们。我们推导出测地线容量缩放为$q_\eta^{-1/2}$（用于保留每个模式）和$q_\eta^{-1}$（用于典型模式），其中$q_\eta$是成对核重叠概率。我们展示了重叠如何创造新记忆：设计的$N$模式配置实现所有$2^N-1$个子集模式，但存储阈值处的随机数据仅产生泊松数。我们使用黎曼均值漂移建立了精确的一步回忆。在模拟中，我们恢复了预测的曲率转变和每个设计的模式。在WordNet的完整名词层次结构上，我们证明了体积校正改善了低容量检索。总之，我们的工作表明曲率是联想记忆的设计变量，而不仅仅是数据的属性。
英文摘要
Geometry does more than constrain an associative memory: curvature determines what it remembers and which states it creates. We develop intrinsic dense associative memories on Riemannian manifolds by casting memory as Epanechnikov kernel-density mode seeking. We compare geodesic and volume-corrected energies and show that curvature separates their behavior. We prove that geodesic memory always retains an isolated pattern, while corrected memory obeys a sharp Ricci-curvature threshold: positive curvature can erase memories in high dimensions, while negative curvature reinforces them. We derive geodesic capacity scalings of $q_β^{-1/2}$ for retaining every pattern and $q_β^{-1}$ for a typical one, where $q_β$ is the pairwise kernel-overlap probability. We show how overlap \emph{creates} novel memories: designed $N$-pattern configurations realize all $2^N-1$ subset modes, but random data at the storage threshold yield only a Poisson number. We establish exact one-step recall using Riemannian mean shift. In simulations, we recover the predicted curvature transition and every designed mode. On WordNet's full noun hierarchy, we demonstrate that volume correction improves low-capacity retrieval. Together, our work shows that curvature is a design variable for associative memory, not merely a property of the data.
