Title
Sampling and multilevel coarsening algorithms for fast matrix approximations.
Abstract
This paper addresses matrix approximation problems for matrices that are large, sparse, and/or 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. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing partial singular value decomposition, for which a combination of sampling and coarsening yields significantly improved singular value decomposition 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. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.
Year
DOI
Venue
2017
10.1002/nla.2234
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Keywords
Field
DocType
coarsening,multilevel methods,randomization,singular values,subspace iteration,SVD
Discrete mathematics,Graph,Singular value decomposition,Mathematical optimization,Dimensionality reduction,Graph sparsification,Matrix (mathematics),Hypergraph,Algorithm,Low-rank approximation,Sampling (statistics),Mathematics
Journal
Volume
Issue
ISSN
26.0
3.0
1070-5325
Citations 
PageRank 
References 
0
0.34
34
Authors
2
Name
Order
Citations
PageRank
shashanka ubaru1588.97
Yousef Saad21940254.74