Graph Sparsification by Approximate Matrix Multiplication

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 $\mathit{\text{sparsifying}}$ a $\mathit{\text{weighted}}$ graph, while sharing the same vertices as the original graph but reducing the number of edges, is through $\mathit{\text{spectral}}\mathit{\text{sparsification}}$. 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 $\mathit{\text{additive}}\mathit{\text{spectral}}\mathit{\text{sparsifiers}}$, which has not been considered in the literature.