### Abstract

Mixtures of Gaussian (or normal) distributions arise in a variety of application areas. Many heuristics have been proposed for the task of finding the component Gaussians given samples from the mixture, such as the EM algorithm, a local-search heuristic from Dempster, Laird and Rubin [J. Roy. Statist. Soc. Ser. B 39 (1977) 1-38]. These do not provably run in polynomial time. We present the first algorithm that provably learns the component Gaussians in time that is polynomial in the dimension. The Gaussians may have arbitrary shape, but they must satisfy a "separation condition" which places a lower bound on the distance between the centers of any two component Gaussians. The mathematical results at the heart of our proof are "distance concentration" results - proved using isoperimetric inequalities - which establish bounds on the probability distribution of the distance between a pair of points generated according to the mixture. We also formalize the more general problem of max-likelihood fit of a Gaussian mixture to unstructured data.

Original language | English (US) |
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Pages (from-to) | 69-92 |

Number of pages | 24 |

Journal | Annals of Applied Probability |

Volume | 15 |

Issue number | 1 A |

DOIs | |

State | Published - Feb 1 2005 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Clustering
- Efficient algorithms
- Estimation
- Gaussian mixtures
- Isoperimetric inequalities
- Learning
- Mixture models

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## Cite this

*Annals of Applied Probability*,

*15*(1 A), 69-92. https://doi.org/10.1214/105051604000000512