IDENTIFIABILITY OF NONPARAMETRIC MIXTURE MODELS AND BAYES OPTIMAL CLUSTERING
IDENTIFIABILITY OF NONPARAMETRIC MIXTURE MODELS AND BAYES OPTIMAL CLUSTERING
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DOI：
10.1214/19-aos1887
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发表时间：
2020-08-01
影响因子：
4.5
通讯作者：
Ravikumar, Pradeep
中科院分区：
文献类型：
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作者：
Aragam, Bryon;Dan, Chen;Ravikumar, Pradeep
受数据聚类问题的启发，我们通过引入涉及聚类过度拟合参数（即错误指定）混合模型的新颖框架，建立了可识别非参数混合模型族的一般条件。这些可识别性条件概括了文献中的现有条件，并且足够灵活以包括例如无限高斯混合物的混合物。与最近的文献相反，我们允许一般的非参数混合分量，而是对基础混合度量施加正则性假设。作为我们的主要应用，我们将这些结果应用于基于分区的聚类，将贝叶斯最优分区的概念从基于经典参数模型的聚类推广到非参数设置。此外，该框架是建设性的，因此它产生了一种用于学习已识别混合物的实用算法，并通过几个真实数据的示例进行了说明。分析中的关键概念设备是度量空间上概率测度的凸度量几何及其与混合测度的 Wasserstein 收敛性的联系。结果是一个具有形式一致性保证的灵活的非参数聚类框架。
Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable by introducing a novel framework involving clustering overfitted parametric (i.e., misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature and are flexible enough to include, for example, mixtures of infinite Gaussian mixtures. In contrast to the recent literature, we allow for general nonparametric mixture components and instead impose regularity assumptions on the underlying mixing measure. As our primary application we apply these results to partition-based clustering, generalizing the notion of a Bayes optimal partition from classical parametric model-based clustering to nonparametric settings. Furthermore, this framework is constructive, so that it yields a practical algorithm for learning identified mixtures, which is illustrated through several examples on real data. The key conceptual device in the analysis is the convex, metric geometry of probability measures on metric spaces and its connection to the Wasserstein convergence of mixing measures. The result is a flexible framework for nonparametric clustering with formal consistency guarantees.