  This paper addresses matrix approximation problems for matrices that are
large, sparse and/or that are representations of large graphs. To tackle these
problems, we consider algorithms that are based primarily on coarsening
techniques, possibly combined with random sampling. A multilevel coarsening
technique is proposed which utilizes a hypergraph associated with the data
matrix and a graph coarsening strategy based on column matching. Theoretical
results are established that characterize the quality of the dimension
reduction achieved by a coarsening step, when a proper column matching strategy
is employed. We consider a number of standard applications of this technique as
well as a few new ones. Among the standard applications we first consider the
problem of computing the partial SVD for which a combination of sampling and
coarsening yields significantly improved SVD results relative to sampling
alone. We also consider the Column subset selection problem, a popular low rank
approximation method used in data related applications, and show how multilevel
coarsening can be adapted for this problem. Similarly, we consider the problem
of graph sparsification and show how coarsening techniques can be employed to
solve it. Numerical experiments illustrate the performances of the methods in
various applications.
