Lassoing Eigenvalues: A Classical and a Robust Approach

分析多变量数据的需求出现在许多学科中，包括计算机科学、工程学、气象学、化学计量学、心理学、社会学、生物学和遗传学等。多元统计分析的主要目标是建立模型并理解不同测量值或变量之间复杂的相互关系。随着目前科学的发展趋势，越来越常见的是对每个单独的样本点或实验单元收集大量信息，即使样本点或实验单元本身的数量可能仍然相对较少。这导致需要考虑的参数或变量之间的相互关系非常多，但没有足够的数据来使用经典统计方法充分建模这些关系。该研究项目旨在研究基于相对较小样本量的高维数据建模的新方法。当在每个样本点上记录许多测量时出现的另一个问题是测量中的大误差或异常值。如果没有检测到异常值，这可能会使基于经典统计方法的结论变得可疑。但是，对于高维数据，检测异常值是有问题的，因此一种替代方法是使用健壮的统计方法，即即使数据包含错误的数据点，也能产生有效结论的方法。将评估研究项目中开发的统计方法的稳健性。这个项目将使用惩罚方法，这在统计学中有很长的历史，为高维协方差矩阵开发模型和估计程序。人们早就认识到，随机矩阵的较大和较小的样本特征值分别严重向上和向下偏置，即使对于中等大的样本量也是如此。这个问题可以通过使用惩罚方法来解决，惩罚方法将特征值一起缩小。然而，这种收缩不能用通常的惩罚来完成，惩罚是精度矩阵的凸函数。这个项目将采用测地线凸惩罚。此外，还引入了一些新的非光滑测地线凸惩罚，它们不仅可以将特征值压缩在一起，而且具有创建相等特征值子集的套索效应。这种非平滑惩罚方法因此产生了一种模型选择方法，或者更具体地说是一种多尖协方差模型选择方法。在经典多元正态条件下，首次提出了测地线凸惩罚方法。然而，如果多元正态模型不成立，在这种情况下开发的方法就会表现不佳。使经典方法更健壮的一种简单且经常使用的方法是插件方法，即简单地用健壮的替代方法替换样本协方差矩阵在方法中的作用。对于相对于数据维度的适度样本量，这样的插件方法在性能上往往与使用样本协方差矩阵的方法差别不大。为了解决这一缺点，需要开发和研究协方差矩阵的非光滑惩罚m估计量。在这里，测地线凸性的概念起着至关重要的作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The need to analyze multivariate data arises in many disciplines, including computer science, engineering, meteorology, chemometrics, psychology, sociology, biology, and genetics, among others. A primary goal of multivariate statistical analysis is to model and understand the complex interrelationships between different measurements or variables. With current trends in the sciences, an increasingly common occurrence is the collection of large amounts of information on each individual sample point or experimental unit, even though the number of sample points or experimental units themselves may remain relatively small. This results in an extremely large number of parameters or interrelationships between variables to consider, but with insufficient data to adequately model these relationships using classical statistical methods. This research project aims to investigate novel ways to model such high-dimensional data based on relatively small sample sizes. Another issue that arises when many measurements are recorded on each sample point is that of large errors or outliers in the measurements. This may make the conclusion based on classical statistical methods suspect if the outliers are not detected. For high-dimensional data, though, detecting outliers is known to be problematic, and so an alternative is to use robust statistical methods, that is, methods producing valid conclusions even if the data contains bad data points. The robustness of the statistical methods developed within the research project will be evaluated.This project will use penalization methods, which have a long history within statistics, for developing models and estimation procedures for high-dimensional covariance matrices. It has long been recognized that the larger and smaller sample eigenvalues of random matrices are heavily biased upwards and downwards respectively, even for moderately large sample sizes. This problem can be addressed by using penalization methods, which shrink eigenvalues together. Such shrinkage, though, cannot be accomplished using the usual penalties which are convex functions of the precision matrix. This project will employ geodesic convex penalties. Furthermore, some novel non-smooth geodesic convex penalties are to be introduced, which not only shrink eigenvalues together but also have a lasso-type effect of creating subsets of equal eigenvalues. This non-smooth penalization approach thus yields a model selection method, or more specifically a multi-spiked covariance model selection method. The geodesic convex penalization approach is to be first developed under the classical multivariate normal setting. Methods developed under this setting, though, are well known to perform poorly if the multivariate normal model does not hold. A simple and often used approach for making classical methods more robust is the plug-in method, that is, to simply replace the role of the sample covariance matrix in a method with a robust alternative. For modest sample sizes relative to the dimension of the data, such plug-in methods tend not to differ greatly in performance from those utilizing the sample covariance matrix. To address this shortcoming, non-smooth penalized M-estimators of the covariance matrix are to be developed and studied. Here, the concept of geodesic convexity plays a crucial role.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.