Estimation of Smooth Functionals of Covariance and Other Parameters of High-Dimensional Models

在复杂的高维数据的统计推断中，一个关键问题是如何对由高维向量或大矩阵表示的参数进行统计估计。由于“维度诅咒”，这类估计问题的最优错误率往往相当慢，因此识别低维结构和高维参数的特征变得越来越重要，这些特征可以用经典的“低维”统计中常见的错误率进行有效估计。这些特征通常由函数表示，这些函数平滑地依赖于未知参数，目标是利用它们的平滑性来开发有效的估计过程。这种性质的问题经常出现在各种应用中，如信号和图像处理、机器学习和数据分析。本项目的目的是系统地研究这些问题，并开发新的方法来有效地估计光滑泛函。该项目是数学、统计学和计算机科学之间的跨学科领域，它包括一些活动，以促进与这些领域的研究人员的互动，并确保拟议的研究对教育的影响。该项目的主要重点是基于高维或无限维观测的协方差算子的光滑函数估计的一般方法的发展。预计这些方法将适用于其他重要的高维模型，如高斯位移模型（矢量和矩阵情况下）；线性回归模型（包括量子态层析成像中的迹回归和回归模型）；一些非线性模型。要开发的方法包括一种基于迭代应用bootstrap （bootstrap链）的平滑泛函估计问题的减少偏差的新方法，以及建立具有减少偏差的估计量的渐近效率所需的集中和正态近似边界。目标是确定感兴趣的函数的最佳平滑阈值，以确保它们的有效估计，特别是在无维度的高复杂性设置中，问题的复杂性以真实协方差的有效秩为特征。其他研究方向包括参数集正则性假设下光滑泛函的有效估计，以及泛函估计方法在高维参数假设检验中的应用。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。

A crucial problem in statistical inference for complex, high-dimensional data is to develop statistical estimators of parameters represented by high-dimensional vectors or large matrices. Optimal error rates in such estimation problems are often rather slow due to the ``curse of dimensionality", and it becomes increasingly important to identify low-dimensional structures and features of high-dimensional parameters that could be estimated efficiently with error rates common in classical, ``low-dimensional" statistics. Such features are often represented by functionals that depend smoothly of unknown parameters and the goal is to take advantage of their smoothness to develop efficient estimation procedures. The problems of this nature often occur in a variety of applications such as signal and image processing, machine learning and data analytics. The purpose of this project is to study these problems systematically and to develop new approaches to efficient estimation of smooth functionals. The project is in an interdisciplinary area between mathematics, statistics and computer science and it includes a number of activities to facilitate interactions with researchers in these areas and to ensure the impact of proposed research on education. The main focus of the project is on the development of general methods of estimation of smooth functionals of covariance operators based on high-dimensional or infinite-dimensional observations. It is expected that these methods will be applicable to other important high-dimensional models such as Gaussian shift models (both in vector and in matrix case); linear regression models (including trace regression and regression models in quantum state tomography); some non-linear models. The methods to be developed include a new approach to bias reduction in smooth functional estimation problems based on iterative application of bootstrap (bootstrap chains) and concentration and normal approximation bounds needed to establish asymptotic efficiency of estimators with reduced bias. The goal is to determine optimal smoothness thresholds for functionals of interest that ensure their efficient estimation, in particular, in a dimension free high-complexity setting, with complexity of the problem characterized by the effective rank of the true covariance. Other directions include the study of efficient estimation of smooth functionals under regularity assumptions on the parameter set and applications of methods of functional estimation to hypotheses testing for high-dimensional parameters.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.