### 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 ||A - CC^{+}A||_{F} ≤ (1 + ε) ||A - A_{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.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a |

Publisher | Springer-Verlag |

Pages | 316-326 |

Number of pages | 11 |

ISBN (Print) | 3540380442, 9783540380443 |

State | Published - Jan 1 2006 |

Event | 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 - Barcelona, Spain Duration: Aug 28 2006 → Aug 30 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4110 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 |
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Country | Spain |

City | Barcelona |

Period | 8/28/06 → 8/30/06 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a*(pp. 316-326). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4110 LNCS). Springer-Verlag.

**Subspace sampling and relative-error matrix approximation : Column-based methods.** / Drineas, Petros; Mahoney, Michael W.; Muthukrishnan, Shanmugavelayutham.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4110 LNCS, Springer-Verlag, pp. 316-326, 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006, Barcelona, Spain, 8/28/06.

}

TY - GEN

T1 - Subspace sampling and relative-error matrix approximation

T2 - Column-based methods

AU - Drineas, Petros

AU - Mahoney, Michael W.

AU - Muthukrishnan, Shanmugavelayutham

PY - 2006/1/1

Y1 - 2006/1/1

N2 - 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 Ak, which is obtained by projecting A on the top k left singular vectors of A. While Ak 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 ||A - CC+A||F ≤ (1 + ε) ||A - Ak|| 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.

AB - 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 Ak, which is obtained by projecting A on the top k left singular vectors of A. While Ak 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 ||A - CC+A||F ≤ (1 + ε) ||A - Ak|| 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.

UR - http://www.scopus.com/inward/record.url?scp=33750079844&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750079844&partnerID=8YFLogxK

M3 - Conference contribution

SN - 3540380442

SN - 9783540380443

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 316

EP - 326

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a

PB - Springer-Verlag

ER -