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Sampling and multilevel coarsening algorithms for fast matrix approximations
Author(s) -
Ubaru Shashanka,
Saad Yousef
Publication year - 2019
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2234
Subject(s) - singular value decomposition , hypergraph , mathematics , sampling (signal processing) , matrix (chemical analysis) , matching (statistics) , algorithm , dimension (graph theory) , singular value , low rank approximation , column (typography) , graph , sparse matrix , mathematical optimization , computer science , discrete mathematics , combinatorics , mathematical analysis , statistics , materials science , eigenvalues and eigenvectors , physics , geometry , filter (signal processing) , quantum mechanics , hankel matrix , connection (principal bundle) , composite material , computer vision , gaussian
Summary 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.