## Abstract

Clustering of data points is a fundamental tool in data analysis. We consider points X in a relaxed metric space, where the triangle inequality holds within a constant factor. A clustering of X is a partition of X defined by a set of points Q (centroids), according to the closest centroid. The cost of clustering X by Q is V (Q) = _{x} ∈ _{X} dxQ. This formulation generalizes classic k-means clustering, which uses squared distances. Two basic tasks, parametrized by k ≥ 1, are cost estimation, which returns (approximate) V (Q) for queries Q such that |Q| = k and clustering, which returns an (approximate) minimizer of V (Q) of size |Q| = k. When the data set X is very large, we seek efficient constructions of small samples that can act as surrogates for performing these tasks. Existing constructions that provide quality guarantees, however, are either worst-case, and unable to benefit from structure of real data sets, or make explicit strong assumptions on the structure. We show here how to avoid both these pitfalls using adaptive designs. The core of our design are the novel one2all probabilities, computed for a set M of centroids and α ≥ 1: The clustering cost of each Q with cost V (Q) ≥ V (M)/α can be estimated well from a sample of size O(α|M| ^{−} ^{2} ). For cost estimation, we apply one2all with a bicriteria approximate M, while adaptively balancing |M| and α to optimize sample size per quality. For clustering, we present a wrapper that adaptively applies a base clustering algorithm to a sample S, using the smallest sample that provides the desired statistical guarantees on quality. We demonstrate experimentally the huge gains of using our adaptive instead of worst-case methods.

Original language | English |
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Title of host publication | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 |

Publisher | AAAI press |

Pages | 2884-2891 |

Number of pages | 8 |

ISBN (Electronic) | 9781577358008 |

State | Published - 2018 |

Event | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 - New Orleans, United States Duration: 2 Feb 2018 → 7 Feb 2018 |

### Publication series

Name | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 |
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### Conference

Conference | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 2/02/18 → 7/02/18 |