提出一种新型图信号谱签名，可识别图的几何与对称性。
Power Spectrum Signatures of Graphs
- 基于图傅里叶变换平方构建谱签名，具自同构不变性。
- 在水桶距离下对图扰动保持稳定，适用于点云与图回归任务。
- 适合做图结构特征提取，尤其关注几何与对称性分析。
基于图拉普拉斯算子的点签名在图、点云和流形的机器学习中广泛应用，用于聚类与形状分析。本文提出一种新点签名——功率谱签名，定义为图信号在实数轴上的平方图傅里叶变换。该签名不依赖于拉普拉斯算子的特征向量，具有图自同构不变性。我们证明其在水桶距离下对输入图的扰动具有稳定性。聚焦于指示函数类，该签名可用于生成顶点的描述性特征。通过多个应用展示其在刻画点云几何结构与对称性，以及图回归问题中的实际价值。
原文摘要 · Abstract (English)
Point signatures based on the Laplacian operators on graphs, point clouds, and manifolds have become popular tools in machine learning for graphs, clustering, and shape analysis. In this work, we propose a novel point signature, the power spectrum signature, a measure on $\mathbb{R}$ defined as the squared graph Fourier transform of a graph signal. Unlike eigenvectors of the Laplacian from which it is derived, the power spectrum signature is invariant under graph automorphisms. We show that the power spectrum signature is stable under perturbations of the input graph with respect to the Wasserstein metric. We focus on the signature applied to classes of indicator functions, and its applications to generating descriptive features for vertices of graphs. To demonstrate the practical value of our signature, we showcase several applications in characterizing geometry and symmetries in point cloud data, and graph regression problems.
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