分离非球面高斯混合模型的学习
Learning mixtures of separated nonspherical Gaussians
摘要：
高斯（或正态）分布的混合出现在各种应用领域。对于从混合物中找到给定样本的分量高斯的任务，已经提出了许多启发式算法，例如EM算法，来自Dempster，Laird和Rubin的局部搜索启发式算法[J. Roy.中央集权主义者Soc.Ser.B39（1977）1-38]。这些不能证明在多项式时间内运行。我们提出了第一个算法，可证明学习的组件高斯在时间上是多项式的维度。高斯函数可以是任意形状的，但它们必须满足一个"分离条件“，该条件对任意两个高斯函数的中心之间的距离设置了一个下界。我们证明的核心数学结果是"距离集中“结果--使用等周不等式证明--它建立了根据混合物生成的一对点之间距离的概率分布的界限。我们还正式的最大似然适合的高斯混合非结构化数据的更一般的问题。
英文摘要：
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.
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