Efficient Signed Graph Sampling via Balancing & Gershgorin Disc Perfect Alignment

A basic premise in graph signal processing (GSP) is that a graph encoding pairwise (anti-)correlations of the targeted signal as edge weights is leveraged for graph filtering. Existing fast graph sampling schemes are designed and tested only for positive graphs describing positive correlations. However, there are many real-world datasets exhibiting strong anti-correlations, and thus a suitable model is a signed graph, containing both positive and negative edge weights. In this paper, we propose the first linear-time method for sampling signed graphs, centered on the concept of balanced signed graphs. Specifically, given an empirical covariance data matrix $\bar{{\mathbf {C}}}$, we first learn a sparse inverse matrix ${\mathcal {L}}$, interpreted as a graph Laplacian corresponding to a signed graph ${\mathcal {G}}$. We approximate ${\mathcal {G}}$ with a balanced signed graph ${\mathcal {G}}^{b}$ via fast edge weight augmentation in linear time, where the eigenpairs of Laplacian ${\mathcal {L}}^{b}$ for ${\mathcal {G}}^{b}$ are graph frequencies. Next, we select a node subset for sampling to minimize the error of the signal interpolated from samples in two steps. We first align all Gershgorin disc left-ends of Laplacian ${\mathcal {L}}^{b}$ at the smallest eigenvalue $\lambda _{\min }({\mathcal {L}}^{b})$ via similarity transform ${\mathcal {L}}^{s} = {\mathbf {S}}{\mathcal {L}}^{b} {\mathbf {S}}^{-1}$, leveraging a recent linear algebra theorem called Gershgorin disc perfect alignment (GDPA). We then perform sampling on ${\mathcal {L}}^{s}$ using a previous fast Gershgorin disc alignment sampling (GDAS) scheme. Experiments show that our signed graph sampling method outperformed fast sampling schemes designed for positive graphs on various datasets with anti-correlations.