AI 中文总结
提出一种基于Dirichlet能量的谱密集联想记忆，用于图数据的存储与检索，具有指数级容量和误差衰减，并在多个数据集上验证了其有效性。
AI 中文摘要
密集联想记忆传统上专注于存储和检索向量值模式。然而，许多现代机器学习问题天然具有图结构，需要针对关系模式、图扩散几何、社区结构以及基于图的归纳偏置的记忆机制。我们提出了一种用于图数据存储与检索的谱密集联想记忆，扩展了经典的向量值记忆。检索通过由Dirichlet能量与谱范数距离诱导的log-sum-exp能量实现，产生存储拉普拉斯矩阵的softmax加权平均，该平均仍保持有效的图拉普拉斯矩阵。我们证明了指数级存储容量和指数衰减的检索误差。在图检索之外，我们为图学习核心的谱量建立了理论保证，包括特征值、特征空间和扩散算子。在合成图数据、真实世界航空网络、蛋白质构象数据和可穿戴传感器数据上的实验表明，我们的方法在保持数据图几何的同时实现了稳健的图检索。我们的框架为图结构数据提供了一种新的联想记忆范式，并将密集联想记忆与现代图学习和生成式AI联系起来。
英文摘要
Dense associative memories have traditionally focused on storing and retrieving vector-valued patterns. Many modern machine learning problems, however, are naturally graph-structured, requiring memory mechanisms for relational patterns, graph diffusion geometries, community structures, and graph-based inductive biases. We propose a spectral dense associative memory for storage and retrieval of graph data, extending the classical vector-valued memories. Retrieval is performed through a log-sum-exp energy induced by Dirichlet energy with spectral norm distances, producing a softmax-weighted average of the stored Laplacians that remains a valid graph Laplacian. We prove exponential storage capacity and exponentially decaying retrieval error. Beyond graph retrieval, we establish theoretical guarantees for spectral quantities central to graph learning, including eigenvalues, eigenspaces, and diffusion operators. Experiments on synthetic graph data, real-world airline network, protein conformation data and wearable sensor data demonstrate robust graph retrieval while preserving the graph geometry of the data. Our framework provides a new associative memory paradigm for graph-structured data and bridges dense associative memory with modern graph learning and generative AI.
