Reconstruction-cognizant Graph Sampling Using Gershgorin Disc Alignment

Abstract

Graph sampling with noise is a fundamental problem in graph signal processing (GSP). Previous works assume an unbiased least square (LS) signal reconstruction scheme and select samples greedily via expensive extreme eigenvector computation. A popular biased scheme using graph Laplacian regularization (GLR) solves a system of linear equations for its reconstruction. Assuming this GLR-based scheme, we propose a reconstruction-cognizant sampling strategy to maximize the numerical stability of the linear system-i.e., minimize the condition number of the coefficient matrix. Specifically, we maximize the eigenvalue lower bounds of the matrix, represented by left-ends of Gershgorin discs of the coefficient matrix. To accomplish this efficiently, we propose an iterative algorithm to traverse the graph nodes via Breadth First Search (BFS) and align the leftends of all corresponding Gershgorin discs at lower-bound threshold T using two basic operations: disc shifting and scaling. We then perform binary search to maximize T given a sample budget K. Experiments on real graph data show that the proposed algorithm can effectively promote large eigenvalue lower bounds, and the reconstruction MSE is the same or smaller than existing sampling methods for different budget K at much lower complexity.

Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment

“…Nonetheless, this sampling idea utilizes local message heuristically and has no global performance guarantee. Orthogonally, our early work [22] proposed a fast graph sampling algorithm via Gershgorin disc alignment without any eigen-pair computation, but its performance was sub-optimal due to the node sampling strategy simply based on breadth first search (BFS). It is known that random sampling [23], [24] can lead to very low computational complexity, but it typically requires more samples for the same signal reconstruction quality compared to its deterministic counterparts.…”

Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

“…Recent schemes pro-actively chose samples to preserve geometric features like corners and edges [7]- [9], but do not ensure overall quality reconstruction of the original PC. In this paper, we orchestrate a graph sampling method [10], [11] to choose 3D points in a PC to minimize a worst-case reconstruction error. To our knowledge, this is the first PC sub-sampling work that systematically minimizes a global reconstruction error metric.…”

“…majority of existing graph sampling methods [14]- [17] require computing extreme eigenvectors (corresponding to the smallest or largest eigenvalues) of a large adjacency or graph Laplacian (sub-)matrix per iteration, which is expensive. In contrast, [10], [11] proposed an eigen-decomposition-free method called Gershgorin Disc Alignment Sampling (GDAS) 1 based on the well-known Gershgorin Circle Theorem (GCT) [18]. In summary, the goal is to maximize the lower bound of the smallest eigenvalue λ min (B) of a coefficient matrix B = H H + µL in a linear system, where L is a combinatorial graph Laplacian matrix.…”

“…and v is the first eigenvector of L B , computed using a known fast algorithm 2 Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) [22]. Finally, we adopt GDAS [10], [11] to choose 3D points for PC sub-sampling to maximize λ − min (H H + µL p ) via selection of H. Extensive experimental results show that our PC sub-sampling method outperforms competing methods [4], [5], [7], [9], [23]- [26] in superresolved quality both numerically evaluated with two common PC metrics [27], and visually for a wide range of PC datasets.…”

Bayesian Design of Sampling Set for Bandlimited Graph Signals

“…Inspired by [17], we analyze which vertices are preferred to be sampled for a known topology and priors and explain how stochastic prior makes the Bayesian DoS different from non-Bayesian DoS using Gershgorin circle theorem. However, the work of [17] aims to design a sampling set that improve the condition number of the metric matrix of signal estimator, while we try to maximize the upper bound of eigenvalues of our metric matrix in the object function. Finally, we propose a heuristic algorithm which does not need to solve an optimization problem for Bayesian DoS.…”