Spectral graph sparsification aims to find ultra-sparse 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 work 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 very large-scale integration computer-aided design, 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 2 min using a single CPU core and about 6-GB memory.

中文翻译：

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GRASS：利用可扩展光谱扰动分析的 GRAph 光谱稀疏化

谱图稀疏化旨在找到超稀疏子图，其拉普拉斯矩阵可以很好地近似原始拉普拉斯特征值和特征向量。近年来，光谱稀疏技术已被广泛研究以加速各种数值和图形相关的应用。先前的近线性时间谱稀疏化方法首先从原始图中提取低拉伸生成树以形成稀疏器的主干，然后将谱关键的树外边的一小部分恢复到生成树以显着提高近似性质量。然而，不清楚应该恢复多少树外边缘才能在稀疏器内实现所需的谱相似度。受最近图形信号处理技术的启发，这项工作提出了一种相似性感知谱图稀疏化框架，该框架利用有效的谱树外边缘嵌入和过滤方案来构建具有保证谱相似性（相对条件数）级别的谱稀疏器。还引入了迭代图密集化方案，以促进对高度病态问题的树外边缘的高效过滤。所提出的方法已使用从公共领域稀疏矩阵集合中获得的各种图形进行了验证，这些图形与超大规模集成计算机辅助设计、有限元分析以及许多机器学习和数据挖掘中经常研究的社会和数据网络相关。应用程序。例如，