  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 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 paper 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 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 VLSI CAD, finite element analysis,
as well as social and data networks frequently studied in many machine learning
and data mining applications.
