23 research outputs found
Similarity-Aware Spectral Sparsification by Edge Filtering
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
Iterative Row Sampling
There has been significant interest and progress recently in algorithms that
solve regression problems involving tall and thin matrices in input sparsity
time. These algorithms find shorter equivalent of a n*d matrix where n >> d,
which allows one to solve a poly(d) sized problem instead. In practice, the
best performances are often obtained by invoking these routines in an iterative
fashion. We show these iterative methods can be adapted to give theoretical
guarantees comparable and better than the current state of the art.
Our approaches are based on computing the importances of the rows, known as
leverage scores, in an iterative manner. We show that alternating between
computing a short matrix estimate and finding more accurate approximate
leverage scores leads to a series of geometrically smaller instances. This
gives an algorithm that runs in
time for any , where the term is comparable
to the cost of solving a regression problem on the small approximation. Our
results are built upon the close connection between randomized matrix
algorithms, iterative methods, and graph sparsification.Comment: 26 pages, 2 figure