发表机构
University of Calgary; Hotchkiss Brain Institute, University of Calgary; Huazhong University of Science and Technology（卡尔加里大学; 卡尔加里大学霍奇基斯脑研究所; 华中科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出基于几何的有限特征Hebbian联想记忆容量理论，解析检索干扰来源，预测容量上限与特征预算，并在合成、视觉及医学图像数据上验证，连接表示几何与记忆容量。
AI 中文摘要
我们为压缩有限特征Hebbian联想记忆中的精确键检索开发了一种基于几何的容量理论。对于随机或近似各向同性的值，检索干扰分为有限特征噪声（随特征维度增加而减小）和结构干扰（由存储键之间的平方核重叠决定，并在无限特征极限下持续存在）。这产生了无需拟合的检索质量预测，揭示了依赖于几何的容量上限，并预测了达到目标检索质量所需的特征预算。当存储值相关时，我们表明检索同时依赖于键核和值Gram矩阵，并推导出考虑这种相互作用的有限特征近似。我们在合成、视觉和医学图像表示上验证了该理论。总体而言，该框架将表示几何直接与记忆容量联系起来，并区分了何时可以通过增加特征预算来改进性能以及何时必须改变表示本身。在这些设置中，预测的检索曲线与经验行为紧密匹配，并正确识别了首选记忆设计的变化。
英文摘要
We develop a geometry-based capacity theory for exact-key retrieval in compressed finite-feature Hebbian associative memory. For random or approximately isotropic values, retrieval interference separates into finite-feature noise, which decreases with feature dimension, and structural interference, which is determined by squared kernel overlap among stored keys and persists in the infinite-feature limit. This yields a fit-free prediction of retrieval quality, reveals a geometry-dependent capacity ceiling, and predicts the feature budget required for a target retrieval quality. When stored values are correlated, we show that retrieval depends jointly on the key kernel and value Gram matrix, and derive finite-feature approximations that account for this interaction. We validate the theory on synthetic, visual, and medical-image representations. Overall, the framework links representation geometry directly to memory capacity and distinguishes when performance can be improved by increasing the feature budget and when the representation itself must be changed. Across these settings, the predicted retrieval curves closely match empirical behavior and correctly identify changes in the preferred memory design.
