用锚点距离重构扩散几何，统一分子图位置编码方法。
Bridging Distance and Spectral Positional Encodings via Anchor-Based Diffusion Geometry Approximation
- 以锚点距离为输入，通过三边定位映射重建扩散坐标。
- 在DrugBank数据上，距离编码性能接近谱编码，显著优于无编码基线。
- 理论保证随机正则图上的逼近误差，适合图神经网络位置编码研究者。
分子图学习依赖于捕捉局部邻域和全局拓扑的位置信号。目前广泛使用的两类编码分别是基于拉普拉斯或扩散算子的谱编码，以及基于最短路径信息的锚点距离编码，但两者之间的精确关系尚不清晰。本文将距离编码解释为扩散几何的低秩近似，并推导出一个显式的三边定位映射，可从变换后的锚点距离与锚点谱位置重建截断的扩散坐标，在随机正则图上具备逐点与Frobenius范数误差保证。在使用共享GNP基干的DrugBank分子图上，基于距离的Nyström方案能有效恢复扩散几何，且拉普拉斯编码与距离编码均显著优于无编码基线。
原文摘要 · Abstract (English)
Molecular graph learning benefits from positional signals that capture both local neighborhoods and global topology. Two widely used families are spectral encodings derived from Laplacian or diffusion operators and anchor-based distance encodings built from shortest-path information, yet their precise relationship is poorly understood. We interpret distance encodings as a low-rank surrogate of diffusion geometry and derive an explicit trilateration map that reconstructs truncated diffusion coordinates from transformed anchor distances and anchor spectral positions, with pointwise and Frobenius-gap guarantees on random regular graphs. On DrugBank molecular graphs using a shared GNP-based DDI prediction backbone, a distance-driven Nyström scheme closely recovers diffusion geometry, and both Laplacian and distance encodings substantially outperform a no-encoding baseline.
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