## Abstract

We develop and analyze a method to reduce the size of a very large set of data points in a high-dimensional Euclidean space R^{d} to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first k principal components of the reduced set will return approximately the first k principal components of the original set or computing the centers of a k-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new feature of our construction is that the cardinality of the reduced set is independent of the dimension d of the input space and that the sets are mergeable [P. K. Agarwal et al., Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principals of Database Systems, 2012, pp. 23-34]. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets. It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as k-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on data-dependently projecting the points on a low-dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including k-means clustering, principal component analysis, and subspace clustering. The main conceptual contribution is a new coreset definition that allows charging costs that appear for every solution to an additive constant.

Original language | English |
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Pages (from-to) | 601-657 |

Number of pages | 57 |

Journal | SIAM Journal on Computing |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Funding Information:The third author acknowledges the support of Collaborative Research Center 876, Project A2, funded by the German Science Foundation.

Publisher Copyright:

© 2020 Society for Industrial and Applied Mathematics.

## Keywords

- Big data
- Coresets
- K-means
- PCA
- Projective clustering
- Streaming

## ASJC Scopus subject areas

- Computer Science (all)
- Mathematics (all)