  Recent spectral graph sparsification techniques have shown promising
performance in accelerating many numerical and graph algorithms, such as
iterative methods for solving large sparse matrices, spectral partitioning of
undirected graphs, vectorless verification of power/thermal grids,
representation learning of large graphs, etc. However, prior spectral graph
sparsification methods rely on fast Laplacian matrix solvers that are usually
challenging to implement in practice. This work, for the first time, introduces
a solver-free approach (SF-GRASS) for spectral graph sparsification by
leveraging emerging spectral graph coarsening and graph signal processing (GSP)
techniques. We introduce a local spectral embedding scheme for efficiently
identifying spectrally-critical edges that are key to preserving graph spectral
properties, such as the first few Laplacian eigenvalues and eigenvectors. Since
the key kernel functions in SF-GRASS can be efficiently implemented using
sparse-matrix-vector-multiplications (SpMVs), the proposed spectral approach is
simple to implement and inherently parallel friendly. Our extensive
experimental results show that the proposed method can produce a hierarchy of
high-quality spectral sparsifiers in nearly-linear time for a variety of
real-world, large-scale graphs and circuit networks when compared with the
prior state-of-the-art spectral method.
