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Subspace Sampling and Relative-Error Matrix Approximation: Column-Based Methods

  • Petros Drineas
  • Michael W. Mahoney
  • S. Muthukrishnan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4110)

Abstract

Given an m ×n matrix A and an integer k less than the rank of A, the “best” rank k approximation to A that minimizes the error with respect to the Frobenius norm is A k , which is obtained by projecting A on the top k left singular vectors of A. While A k is routinely used in data analysis, it is difficult to interpret and understand it in terms of the original data, namely the columns and rows of A. For example, these columns and rows often come from some application domain, whereas the singular vectors are linear combinations of (up to all) the columns or rows of A. We address the problem of obtaining low-rank approximations that are directly interpretable in terms of the original columns or rows of A. Our main results are two polynomial time randomized algorithms that take as input a matrix A and return as output a matrix C, consisting of a “small” (i.e., a low-degree polynomial in k, 1/ε, and log(1/δ)) number of actual columns of A such that

||ACC  +  A|| F ≤(1+ε) ||AA k || F

with probability at least 1–δ. Our algorithms are simple, and they take time of the order of the time needed to compute the top k right singular vectors of A. In addition, they sample the columns of A via the method of “subspace sampling,” so-named since the sampling probabilities depend on the lengths of the rows of the top singular vectors and since they ensure that we capture entirely a certain subspace of interest.

Keywords

Singular Value Decomposition Singular Vector Frobenius Norm Matrix Approximation Unitarily Invariant Norm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Petros Drineas
    • 1
  • Michael W. Mahoney
    • 2
  • S. Muthukrishnan
    • 3
  1. 1.Department of Computer ScienceRPI 
  2. 2.Yahoo Research Labs 
  3. 3.Department of Computer ScienceRutgers University 

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