GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

Abstract

Spectral graph sparsification aims to find ultrasparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is also introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within two minutes using a single CPU core and about 6GB memory.

Sf-Grass

“…Several test cases have been tested in the experiments. Table 2 shows the spectral graph sparsification results on various graphs when comparing to the state-of-the-art sparsification tool GRASS 1 [8][9][10], where N (M) represents the number of nodes (edges) in the original graph; T дr ass denotes the the sparsifier construction time using GRASS; T r denotes the multilevel graph coarsening time; T spar denotes the multilevel sparsifier construction time by SF-GRASS; |E of f | denotes the number of off-tree edges added for forming the final sparsifier from the initial spanning-tree sparsifier. κ(L G , L P ) denotes the final relative condition number between the Laplacians of the original graph G and the sparsifier P. κ(L G , L S ) denotes the relative condition number between the Laplacians of the original graph G and the initial spanning-tree sparsifier S generated by SF-GRASS.…”

“…Since spectral sparsification aims to approximate the first few eigenvalues and eigenvectors of the original Laplacian with the minimum number of edges, it can be regarded as a low-pass filter on graphs for removing redundant edges. Spectral sparsification usually involves two steps: the first step is to generate a low-stretch spanning tree (LSST) from the original graph using star or petal decompositions [1,7]; the next step is to identify and recover spectrally-critical off-tree edges into the LSST to drastically reduce the condition number, and thereby minimizing the spectral mismatch [10]. However, prior spectral sparsification methods [8,24] usually require solving linear systems of equations with Laplacian solvers, which can still be computationally challenging for large problems.…”

“…However, prior spectral graph sparsification methods strongly rely on fast Laplacian matrix solvers that are usually challenging to implement in practice and inherently-difficult accelerate on parallel processors. For example, effective-resistance sampling-based spectral sparsification method [24] requires multiple Laplacian matrix solutions for computing each edge's leverage score, while the latest spectral-perturbation based algorithm [10] leverages a graphtheoretic algebraic multigrid (AMG) solver for computing dominant generalized eigenvectors key to estimating each edge's spectral importance. As a result, the performance (scalability) of Laplacian matrix solver can become a dominating factor in existing spectral sparsification methods.…”

SF-GRASS: Solver-Free Graph Spectral Sparsification

“…node aggregation set at level l − 1 with respect to the single node i at level l reduce the condition number, and thereby minimizing the spectral mismatch [10]. However, prior spectral sparsification methods [8,24] usually require solving linear systems of equations with Laplacian solvers, which can still be computationally challenging for large problems.…”

High-Performance Spectral Methods for Computer-Aided Design of Integrated Circuits

“…By properly choosing g min , it is possible to filter out the graph signal's "high-frequency" (highly-oscillating) components and only keep "low-frequency" components in x. Since chip temperature distributions mainly contain "slowly-varying" "low-frequency") components due to relative small ambient thermal conductance values, it is thus possible to exploit emerging spectral sparsification techniques [27,28,29] to retain a small number of edges in the sparsified thermal grids while still preserving accurate thermal profiles, since spectrally-sparsified graphs can well-preserve "lowfrequency" graph signals. Based on the above intuition, Algorithm 7 is proposed for scaling up edge weights in the sparsified thermal grid by matching the "low-frequency" responses filtered by the original thermal grids.…”