让图神经网络学会处理高斯分布特征，提升关系数据建模能力
Gaussian Sheaf Neural Networks
- 基于细胞层化理论设计新拉普拉斯算子，保留高斯参数几何结构
- 在合成与真实数据上均优于传统GNN，特征分布建模更精准
- 适合需要概率特征建模的科研与工业场景，如不确定性感知系统
图神经网络（GNN）已成为关系数据学习的标准方法。尽管传统GNN的消息传递机制适用于向量型节点特征，但在某些场景下，节点特征用概率分布表示更合适。例如，当节点特征为高斯分布时（由均值和协方差矩阵定义），若简单地将参数拼接为向量并应用标准消息传递，会丢失均值与协方差所遵循的几何与代数结构。为此，本文提出高斯层化神经网络（GSNN），一个将这些归纳偏置融入图学习的理论框架。基于细胞层化理论，我们推导出一种新的拉普拉斯算子，该算子将层化拉普拉斯推广至高斯分布设置，并保持其关键性质。我们在合成数据与真实世界数据上进行了实验，验证了GSNN的实际有效性。
原文摘要 · Abstract (English)
Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.
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