### Abstract

We show how to approximate a data matrix A with a much smaller sketch A that can be used to solve a general class of constrained k-rank approximation problems to within (1 + ∈) error. Importantly, this class includes k-means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just O(k) dimensions, we generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For k-means dimensionality reduction, we provide (1 + ∈) relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only 'cover' a good subspace for A, but can be used directly to compute this subspace. Finally, for k-means clustering, we show how to achieve a (9 + ∈) approximation by Johnson-Lindenstrauss projecting data to just O(logk/∈^{2} ) dimensions. This is the first result that leverages the specific structure of k-means to achieve dimension independent of input size and sublinear in k.

Original language | English (US) |
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Title of host publication | STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 163-172 |

Number of pages | 10 |

ISBN (Electronic) | 9781450335362 |

DOIs | |

State | Published - Jun 14 2015 |

Event | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States Duration: Jun 14 2015 → Jun 17 2015 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 14-17-June-2015 |

ISSN (Print) | 0737-8017 |

### Other

Other | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 |
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Country | United States |

City | Portland |

Period | 6/14/15 → 6/17/15 |

### All Science Journal Classification (ASJC) codes

- Software

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## Cite this

*STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing*(pp. 163-172). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. 14-17-June-2015). Association for Computing Machinery. https://doi.org/10.1145/2746539.2746569