Estimation of Functionals of High-Dimensional Parameters of Statisical Models

高维参数的低维特征估计是当代复杂高维数据统计分析中的一个重要课题。由于信息理论的限制，由于其高维性，通常不可能对整个未知参数进行可靠的估计，而在经典统计中常见的低维特征的估计可以以更快的错误率有效地完成。这类问题在应用中经常出现，特别是当未知参数是一个大矩阵时，如量子系统的密度矩阵，而感兴趣的特征是这些矩阵的各种光谱特征。尽管这些问题已经研究了许多年，但对于表示感兴趣的特征的泛函的统计估计，几乎没有通用的方法。该项目的主要目标是在一般数学框架下研究泛函估计问题，并开发一般估计方法以及综合理论，显示泛函估计中的错误率如何依赖于目标泛函的潜在属性，如其平滑性。该项目为在高维统计领域培训研究生提供了新的机会，特别是通过开设研究生课程和研讨会。本课题主要研究统计模型中未知高维参数光滑泛函估计的高阶偏置减少方法（自举链偏置减少）。它基于迭代自举法，可以看作是高维参数空间上某些积分方程的近似解方法。在高维高斯模型的情况下，该方法产生具有最佳错误率的光滑函数估计量。本课题将研究各种重要的高维统计模型的估计量的性质，包括对数凹模型、流形模型、稀疏模型和量子统计中的密度矩阵估计模型。目标是确定函数估计中的最小、最大、最优错误率，并研究快速参数率和慢非参数率之间的相变，这取决于函数的平滑程度和问题的复杂参数。这需要解决许多具有挑战性的分析和概率问题，包括通过独立随机过程（随机同伦）的叠加来研究自举马尔可夫链的近似，发展高维正态近似和耦合方法，以及统计估计的集中界限。该项目将导致对高维函数估计问题的更深入理解，并在高维统计推断中开发各种新的概率工具。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。

Estimation of low-dimensional features of high-dimensional parameters is an important subject in contemporary statistical analysis of complex, high-dimensional data. While information-theoretic limitations often make impossible the reliable estimation of the whole unknown parameter due to its high dimensionality, estimation of low-dimensional features could be done efficiently with much faster error rates, common in classical statistics. Such problems often occur in applications, in particular, when the unknown parameter is a large matrix such as the density matrix of a quantum system, and the features of interest are various spectral characteristics of such matrices. Despite the fact that these problems have been studied for many years, there are few general approaches to statistical estimation of functionals representing the features of interest. The main goal of this project is to study functional estimation problem in a general mathematical framework and to develop general estimation methods as well as a comprehensive theory showing how the error rates in functional estimation depend on the underlying properties of the target functional such as its smoothness. The project provides new opportunities for training graduate students in the areas of high-dimensional statistics, in particular, by developing graduate level courses and seminars. The main focus of the project is on the development of a higher order bias reduction method (bootstrap chain bias reduction) in estimation of smooth functionals of unknown high-dimensional parameter of statistical model. It is based on iterative bootstrap and it could be viewed as a method of approximate solution of certain integral equations on high-dimensional parameter spaces. In the case of high-dimensional Gaussian models, this method yields estimators of smooth functionals with optimal error rates. This research project will study the properties of such estimators for a variety of important high-dimensional statistical models, including log-concave models, models on manifolds, sparse models and density matrix estimation models in quantum statistics. The goal is to determine minimax optimal error rates in functional estimation and to study the phase transition between fast parametric and slow nonparametric rates depending on the degree of smoothness of the functional and complexity parameters of the problem. This requires solving a number of challenging analytic and probabilistic problems, including the study of approximation of bootstrap Markov chains by superpositions of independent stochastic processes (random homotopies), the development of high-dimensional normal approximation and coupling methods as well as of concentration bounds for statistical estimators. The project will result in much deeper understanding of functional estimation problems in high dimensions and in the development of a variety of new probabilistic tools in high-dimensional statistical inference.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.