混合距离与谱编码可更准定位节点，尤其在语法树中表现更好。
Converse and Collision-Based Achievability for Node Localization with Hybrid Distance-Spectral Graph Positional Encodings
- 融合锚点距离与低频谱坐标，生成混合位置编码。
- 实验表明编码信息量与节点定位成功率正相关，比例接近对数归一化值。
- 适合研究图神经网络结构感知、自然语言处理中的语法几何建模者。
图位置编码广泛应用于图神经网络和图Transformer，但其能否唯一识别节点仍不明确。本文研究一种结合锚点距离轮廓与量化低频拉普拉斯能量坐标的混合编码方法。将编码视为观测映射，推导出单纯形优化的反向结论、精确的碰撞因子分解公式 \\(\kappa_H=\kappa_D\kappa_{S|D}\\) 及碰撞信息量 \\(I_H=-\log\kappa_D-\log\kappa_{S|D}\\)。在随机正则图上，通过有界相关高斯波近似显式表达判据；对真实拉普拉斯能量坐标，给出距离条件下的谱碰撞充分条件，确保条件可实现性。实验显示，\\(I_H/\log n\\) 能有效校准定位成功率；在通用依存树上的仅位置编码结构任务探测中，混合编码比仅距离或仅谱编码基线更优地恢复语法树几何结构。
原文摘要 · Abstract (English)
Graph positional encodings are widely used in graph neural networks and graph Transformers, yet it remains unclear when the code itself can identify nodes. We study a hybrid distance-spectral encoding that combines anchor-distance profiles with quantized low-frequency Laplacian-energy coordinates. Treating the encoding as an observation map yields a simplex-refined converse, an exact collision factorization \(κ_H=κ_Dκ_{S|D}\), and the collision information \(I_H=-\logκ_D-\logκ_{S|D}\). On random regular graphs, the criterion is made explicit through a bounded-correlation Gaussian-wave surrogate; for actual Laplacian-energy coordinates, we give the distance-conditioned spectral collision condition sufficient for conditional actual-coordinate achievability. Experiments show that \(I_H/\log n\) calibrates localization success, and PE-only structural task probes on Universal Dependencies trees show that hybrid encodings better recover syntactic-tree geometry than distance-only or spectral-only baselines.
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