Graphs arising in statistical problems, signal processing, large networks, combinatorial optimization, and data analysis are often dense, which causes both computational and storage bottlenecks. One way of \textit{sparsifying} a \textit{weighted} graph, while sharing the same vertices as the original graph but reducing the number of edges, is through \textit{spectral sparsification}. We study this problem through the perspective of RandNLA. Specifically, we utilize randomized matrix multiplication to give a clean and simple analysis of how sampling according to edge weights gives a spectral approximation to graph Laplacians. Through the $CR$-MM algorithm, we attain a simple and computationally efficient sparsifier whose resulting Laplacian estimate is unbiased and of minimum variance. Furthermore, we define a new notion of \textit{additive spectral sparsifiers}, which has not been considered in the literature.

翻译：统计问题、信号处理、大规模网络、组合优化和数据分析中出现的图往往是稠密的，这会导致计算和存储瓶颈。一种对加权图进行稀疏化（即保留原图顶点但减少边数）的方法是通过谱稀疏化。我们从RandNLA的角度研究这一问题。具体而言，我们利用随机矩阵乘法，对按边权重采样如何实现图拉普拉斯矩阵的谱近似给出了简洁清晰的分析。通过CR-MM算法，我们获得了一个简单且计算高效的稀疏化器，其生成的拉普拉斯估计是无偏且方差最小的。此外，我们定义了一个文献中尚未考虑的新概念——加性谱稀疏化。