用于图表示学习的高阶位置编码
Higher-Order Positional Encodings for Graph Representation Learning
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中文总结 AI 辅助
本文提出高阶位置编码，利用霍奇拉普拉斯算子将高阶拓扑信息注入标准图模型，理论证明其可混合图拉普拉斯频率，实验表明能提升图变换器预测性能。
中文摘要 AI 辅助
许多现实世界系统表现出实体组之间的高阶交互，这些交互无法仅通过成对关系来捕获。图变换器（Graph Transformers）和图神经网络（Graph Neural Networks）越来越依赖位置编码来丰富图表示，然而现有的位置编码仅从原始图计算得到，因此无法直接捕获观测到的高阶交互。拓扑深度学习（Topological Deep Learning）通过将图提升为单纯复形来解决这一局限性，但通常需要在高阶神经网络表示上执行消息传递或注意力机制。我们引入了一种表示学习范式，通过位置编码用高阶拓扑丰富图表示，使标准图学习模型能够利用提升的关联结构而无需修改骨干网络。我们对高阶位置编码的表达能力进行了理论刻画，证明由高阶提升诱导的节点级算子能够以标量图谱滤波器无法实现的方式混合图拉普拉斯频率。在此理论的指导下，我们使用由团复形导出的霍奇拉普拉斯算子实例化高阶位置编码。在ZINC和受控合成基准上使用图变换器进行的实验证明了预测性能的提升，而固定1-骨架实验表明，当单元独立于图提供时，该流程能够传递高阶信息。综合来看，我们的结果确立了高阶位置编码作为图位置编码与拓扑深度学习之间原则性桥梁的地位。
英文摘要
Many real-world systems exhibit higher-order interactions among groups of entities that cannot be captured by pairwise relationships alone. Graph Transformers and Graph Neural Networks increasingly rely on positional encodings to enrich graph representations, yet existing positional encodings are computed solely from the original graph and therefore cannot directly capture observed higher-order interactions. Topological Deep Learning addresses this limitation by lifting graphs to simplicial complexes, but typically requires performing message passing or attention on higher-order neural network representations. We introduce a representation learning paradigm that enriches graph representations with higher-order topology through positional encodings, enabling standard graph learning models to exploit lifted incidence structure without modifying the backbone. We derive a theoretical characterization of the expressivity of higher-order positional encodings, proving that node-level operators induced by higher-order lifts can mix graph Laplacian frequencies in ways that scalar graph spectral filters cannot. Guided by this theory, we instantiate higher-order positional encodings using Hodge Laplacians derived from clique complexes. Experiments with Graph Transformers on ZINC and controlled synthetic benchmarks demonstrate improvements in predictive performance, while a fixed-1-skeleton experiment shows that the pipeline can transmit higher-order information when cells are supplied independently of the graph. Together, our results establish higher-order positional encodings as a principled bridge between graph positional encodings and topological deep learning.
发表机构
- University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
机构由 AI 辅助整理，请以论文原文为准。
