High-Dimensional Covariance Estimation via Convex Optimization

现代技术使研究人员能够测量有关其研究对象的前所未有的大量属性。许多应用程序中的一个重要问题是如何从这些数据中推断出这些属性之间的潜在关系。这样的目标可以通过统计学领域中称为协方差矩阵的基本结构来表达。使用传统的统计技术需要收集大量的数据来可靠地估计这个矩阵。在这项工作中，新的统计方法将被开发出来，使研究人员能够通过更好地利用他们的数据来做出合理的推断，因为他们拥有的研究对象数量有限。所开发的方法将适用于广泛的领域。例如，在生物学中，人们可以根据少量样本推断出大量基因网络的结构。协方差矩阵本身就是一个目的，在许多常见的统计过程中，协方差矩阵是一个关键因素。因此，通过发展从少数受试者中可靠地估计它的能力，这项工作将使许多其他研究人员无法使用的方法得以使用。应用领域包括疾病诊断、基础生物学、传感器网络和社会网络。研究计划将集中在高维协方差估计和利用凸框架的优势来开发新的统计方法。这项工作将包括开发有效的算法，通过理论和模拟的结合彻底研究估计器和算法的性质，并将方法应用于实际数据集。研究重点主要集中在两个方面：(A)在某些应用中，变量有一个已知的顺序。这种结构建议使用以前未应用于协方差估计的凸惩罚。这项工作将仔细研究使用这种惩罚来估计协方差矩阵和逆协方差矩阵。(B)估计协方差矩阵作为一个同时稀疏和正定的矩阵是一个自然的目标，然而标准的惩罚似然方法不是凸的。本研究将开发基于凸优化的估计器，仍然利用似然。对于所有项目，软件将被制作出来，在网上免费提供，并进行维护，以便其他研究人员可以从中受益。

Modern technologies allow researchers to measure an unprecedentedly large number of attributes regarding the subjects of their study. An important question in many applications is how one can infer from such data the underlying relationships between these attributes. Such an objective can be expressed through a fundamental construct in the field of statistics known as the covariance matrix. Using classical statistical techniques would require one to collect data on a prohibitively large number of subjects to reliably estimate this matrix. In this work, novel statistical methods will be developed that allow researchers to make sound inferences by making better use of their data given the limited number of subjects they have available. The methods developed will be applicable in a wide range of fields. For example, in biology, one can infer the structures of massive networks of genes based on a small number of samples. Beyond being an end in itself, the covariance matrix is a key ingredient in many common statistical procedures. Thus, by developing the ability to reliably estimate it from small numbers of subjects, this work will enable the use of many other methods that would otherwise be unavailable to researchers. Application areas include disease diagnosis, basic biology, sensor networks, and social networks.The research program will focus on high-dimensional covariance estimation and capitalize on the strengths of the convex framework to develop novel statistical methodology. This work will involve developing efficient algorithms, thoroughly investigating the properties of estimators and algorithms through a combination of theory and simulation, and applying methods to real datasets. The research focus is in two main areas: (A) In certain applications, the variables have a known ordering. Such structure suggests the use of a convex penalty not previously applied to covariance estimation. This work will carefully study using such a penalty to estimate both the covariance matrix and the inverse covariance matrix. (B) Estimating a covariance matrix as a simultaneously sparse and positive definite matrix is a natural goal, and yet the standard penalized likelihood approach is not convex. This research will develop convex-optimization-based estimators that still make use of the likelihood. For all projects, software will be produced, made freely available online, and maintained so that other researchers can benefit from its use.