## Abstract

An ϵ-coreset to the dimensionality reduction problem for a (possibly very large) matrix A ĝ Rn x d is a small scaled subset of its n rows that approximates their sum of squared distances to every affine k-dimensional subspace of Rd, up to a factor of 1±ϵ. Such a coreset is useful for boosting the running time of computing a low-rank approximation (k-SVD/k-PCA) while using small memory. Coresets are also useful for handling streaming, dynamic and distributed data in parallel. With high probability, non-uniform sampling based on the so called leverage score or sensitivity of each row in A yields a coreset. The size of the (sampled) coreset is then near-linear in the total sum of these sensitivity bounds. We provide algorithms that compute provably tight bounds for the sensitivity of each input row. It is based on two ingredients: (i) iterative algorithm that computes the exact sensitivity of each row up to arbitrary small precision for (non-affine) k-subspaces, and (ii) a general reduction for computing a coreset for affine subspaces, given a coreset for (non-affine) subspaces in Rd. Experimental results on real-world datasets, including the English Wikipedia documents-term matrix, show that our bounds provide significantly smaller and data-dependent coresets also in practice. Full open source code is also provided.

Original language | English |
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Title of host publication | KDD 2020 - Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |

Publisher | Association for Computing Machinery |

Pages | 2051-2061 |

Number of pages | 11 |

ISBN (Electronic) | 9781450379984 |

DOIs | |

State | Published - 23 Aug 2020 |

Event | 26th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2020 - Virtual, Online, United States Duration: 23 Aug 2020 → 27 Aug 2020 |

### Publication series

Name | Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |
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### Conference

Conference | 26th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2020 |
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Country/Territory | United States |

City | Virtual, Online |

Period | 23/08/20 → 27/08/20 |

### Bibliographical note

Publisher Copyright:© 2020 ACM.

## Keywords

- PCA
- SVD
- coreset
- dimensionality reduction
- low rank approximation
- sketch

## ASJC Scopus subject areas

- Software
- Information Systems