Estimation of Functionals of High-Dimensional Parameters of Statisical Models

统计模型高维参数泛函的估计

基本信息

批准号：
2113121
负责人：
Vladimir Koltchinskii
金额：
$ 22万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2113121&HistoricalAwards=false
关键词：
Estimation Functionals Dimensional Parameters Statisical

项目摘要

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.

高维参数的低维特征估计是当代复杂高维数据统计分析的一个重要课题。虽然由于信息论的限制，由于其高维性，通常无法可靠地估计整个未知参数，但低维特征的估计可以以更快的错误率有效地完成，这在经典统计学中很常见。此类问题在应用中经常发生，特别是当未知参数是一个大矩阵（例如量子系统的密度矩阵）时，并且感兴趣的特征是此类矩阵的各种光谱特征。尽管这些问题已经研究了很多年，但对代表感兴趣特征的泛函进行统计估计的通用方法却很少。该项目的主要目标是在通用数学框架中研究函数估计问题，并开发通用估计方法以及显示函数估计中的错误率如何取决于目标函数的基本属性（例如其平滑度）的综合理论。该项目为高维统计领域的研究生培训提供了新的机会，特别是通过开发研究生水平的课程和研讨会。该项目的主要重点是开发一种高阶偏差减少方法（引导链偏差减少）来估计统计模型的未知高维参数的平滑泛函。它基于迭代引导，可以看作是在高维参数空间上近似求解某些积分方程的方法。在高维高斯模型的情况下，该方法产生具有最佳错误率的平滑泛函的估计量。该研究项目将研究各种重要高维统计模型的此类估计量的性质，包括量子统计中的对数凹模型、流形模型、稀疏模型和密度矩阵估计模型。目标是确定函数估计中的极小极大最优错误率，并根据问题函数的平滑程度和复杂性参数来研究快速参数率和慢速非参数率之间的相变。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逼近、高维正态逼近和耦合方法以及浓度界限的开发统计估计器。该项目将导致对高维函数估计问题的更深入理解，并开发高维统计推断中的各种新概率工具。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。