Power Spectrum Signatures of Graphs

TL;DR

The paper introduces the power spectrum signature, a vertex-centered spectral feature derived from the squared graph Fourier transform with respect to the normalized Laplacian. The authors prove automorphism invariance, establish a Wasserstein-based stability bound under Laplacian perturbations, and show that power spectra can be succinctly represented via quantiles while preserving discriminative power. They connect the signature to established spectral descriptors (e.g., heat kernel signature, diffusion distances, GPS, wavelets) and demonstrate its utility in unsupervised point-cloud clustering and supervised graph regression, where incorporating power spectrum features improves performance across multiple GNN architectures. The work advances robust, basis-invariant, and scale-aware vertex features that capture global spectral energy distributions, enabling improved geometry understanding and learning on graphs and point clouds.

Abstract

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.

Figures (10)

• Figure 1: The power spectrum of the vertex function supported on the blue vertex.
• Figure 2: Torus. We use PCA to obtain a low-dimensional projection of the quantile vectors of power spectra. We vary the matrix $S_{\alpha, \epsilon}$ from $\epsilon = 0.5, 1.0, 1.5$ while fixing $\alpha = \frac{1}{2}$. We use the DBSCAN algorithm to cluster the quantile vectors, and color the points by their cluster affiliation. Note DBSCAN groups the whole torus into one cluster.
• Figure 3: Cyclo-octane. We use PCA to obtain a low-dimensional projection of the quantile vectors of power spectra. We vary the matrix $S_{\alpha, \epsilon}$ from $\epsilon = 0.5, 0.75, 1.0$ while fixing $\alpha = 1/2$. We use the DBSCAN algorithm to cluster the quantile vectors, and color the points by their cluster affiliation. Note that different clusters may appear to overlap in the PCA visualization as the clustering is
• Figure 4: Torus. We plot the principal components of the quantile vectors derived from power spectra. Here the quantile vectors are obtained from the vertex power spectra of $S_{\alpha, \epsilon}$ from $\epsilon = 0.5, 1.0, 1.5$ while fixing $\alpha = \frac{1}{2}$.
• Figure 5: Cyclo-octane. We plot the principal components of the quantile vectors derived from power spectra. Here the quantile vectors are obtained from the vertex power spectra of $S_{\alpha, \epsilon}$ from $\epsilon = 0.5, 0.75, 1.0$ while fixing $\alpha = \frac{1}{2}$.

Theorems & Definitions (27)

• Definition 2.1: Power Spectrum Signature
• Example 2.2: A power spectrum computation
• Remark 2.3
• Proposition 2.4: Injectivity
• proof
• Proposition 2.5: Automorphism invariance
• proof
• Theorem 3.1
• Remark 3.2
• Remark 3.3