Statistical Foundations of Model-Based Variable Clustering

变量聚类问题是许多领域的基石，例如遗传学，神经科学，社会学，宏观经济学，仅举几例。在神经科学中，它有助于发现新的功能连接区域。在遗传学中，它有助于发现功能未被充分探索或未知的基因。在宏观经济学中，它可以帮助创建新的经济指数。尽管它的广泛传播的重要性和潜在的影响，这个问题还没有得到一个系统的方法和理论在文献中的治疗。虽然聚类算法比比皆是，并且有很长的历史，但评估其输入的有效性有些武断。在这个项目中提出了一个概率，基于模型的方法。这将使原则性统计变量聚类的统一框架得以发展。具体而言，本项目将介绍和研究重叠和非重叠变量聚类的潜在变量模型类别。重点是：（1）引入可识别的潜变量聚类模型。这将提供明确界定的估计目标，并将促进对各组的科学解释。(II)开发适合这些模型的多项式时间算法。(III)为聚类算法的理论分析建立一个统一的框架，重点是极大极小最优性和高维推理。(IV)研究基于模型的聚类算法对下游分析的影响，重点是图形模型，回归和分类。

The problem of variable clustering is a corner stone in a multitude of areas such as genetics, neuroscience, sociology, macroeconomics, to name a few. In neuroscience, it aids in finding new functionally connected areas. In genetics, it helps advance the discovery of genes with under-explored or unknown functions. In macro-economics it can assist with the creation of new economic indices. Despite its wide-spread importance and potential impact, this problem has not received a systematic methodological and theoretical treatment in the literature. Although clustering algorithms abound, and have a very long history, assessing the validity of their input is somewhat arbitrary. A probabilistic, model-based approach is put forward in this project. This will enable the development of a unified framework for principled statistical variable clustering.Specifically, this project will introduce and investigate classes of latent variable models for overlapping and non-overlapping variable clustering. The focal points are: (I) The introduction of identifiable latent variable models for clustering. This will provide well defined targets for estimation, and will facilitate the scientific interpretation of the clusters. (II) The development of polynomial time algorithms tailored to these models. (III) The creation of a unifying framework for the theoretical analysis of clustering algorithms, with emphasis on minimax optimality and high dimensional inference. (IV) The study of the impact of model based clustering algorithms on downstream analyses, with emphasis on graphical models, regression and classification.