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 exploited for graph filtering. However, existing fast graph sampling schemes are designed and tested only for positive graphs describing positive correlations. In this paper, we show that for datasets with strong inherent anti-correlations, a suitable graph contains both positive and negative edge weights. In response, we propose a linear-time signed graph sampling method centered on the concept of balanced signed graphs. Specifically, given an empirical covariance data matrix barbfC, we first learn a sparse inverse matrix (graph Laplacian) L corresponding to a signed graph G. We define the eigenvectors of Laplacian LB for a balanced signed graph GB -- approximating G via edge weight augmentation -- as graph frequency components. Next, we choose samples to minimize the low-pass filter reconstruction error in two steps. We first align all Gershgorin disc left-ends of Laplacian LB at smallest eigenvalue λmin(LB) via similarity transform Lp = S LB S-1, leveraging a recent linear algebra theorem called Gershgorin disc perfect alignment (GDPA). We then perform sampling on Lp using a previous fast Gershgorin disc alignment sampling (GDAS) scheme. Experimental results show that our signed graph sampling method outperformed existing fast sampling schemes noticeably on various datasets.