Similarity-Aware Spectral Sparsification by Edge Filtering
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In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies 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 of the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning 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 an efficient off-tree edge filtering scheme to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification framework and a generalized eigenvalue stability checking scheme are introduced to facilitate efficient and effective filtering of off-tree edges even 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.
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