Learning mixtures of separated nonspherical gaussians

• 成果类型: Article
• 署名作者: Arora, S; Kannan, R
• 署名单位: Princeton University; Yale University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000512
• 发表日期: 2005
• 页码: 69-92
• 关键词: maximum-likelihood; volume

摘要

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.