## Abstract

One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or "nearly quadratic" running time; i.e., O(dn ^{2-α(d)}) time where n is the number of data points, d is the dimension of the space and α(d) approaches 0 as d grows. In this paper, we show (for three fairly natural clustering rules) that computing an approximate solution can be done much more efficiently, More specifically, for agglomerative clustering (used, for example, in the Alta Vista™ search engine), for the clustering defined by sparse partitions, and for a clustering based on minimum spanning trees we derive randomized (1 + ε) approximation algorithms with running times Õ(d ^{2}n ^{2-γ}) where γ > 0 depends only on the approximation parameter ε and is independent of the dimension d.

Original language | American English |
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Pages (from-to) | 153-167 |

Number of pages | 15 |

Journal | Machine Learning |

Volume | 56 |

Issue number | 1-3 |

DOIs | |

State | Published - Jul 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗An extended abstract of this paper appeared in STOC 1999. †Work supported by BSF grant 96-00402, by MoS contract number 9480198, and by a grant from the Fund for the Promotion of Research at the Technion.

## Keywords

- Graph-theoretic clustering
- High dimensional spaces
- Sparse partitions