Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment

Yuanchao Bai, Fen Wang, Gene Cheung, Yuji Nakatsukasa, Wen Gao

Abstract: Graph sampling set selection, where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal, is a fundamental problem in graph signal processing (GSP). Previous works employ an unbiased least-squares (LS) signal reconstruction scheme and select samples via expensive extreme eigenvector computation. Instead, we assume a biased graph Laplacian regularization (GLR) based scheme that solves a system of linear equations for reconstruction. We then choose samples to minimize the condition number of the coefficient matrix-specifically, maximize the smallest eigenvalue λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> . Circumventing explicit eigenvalue computation, we maximize instead the lower bound of λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , designated by the smallest left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we first convert the optimization to a dual problem, where we minimize the number of samples needed to align all Gershgorin disc left-ends at a chosen lower-bound target T. Algebraically, the dual problem amounts to optimizing two disc operations: i) shifting of disc centers due to sampling, and ii) scaling of disc radii due to a similarity transformation of the matrix. We further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous NP-hard set cover (SC) problem. The reinterpretation enables us to derive a fast approximation scheme from a known SC error-bounded approximation algorithm. We find an appropriate target T efficiently via binary search. Extensive simulation experiments show that our disc-based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen-decomposition-free sampling schemes in reconstruction error.