Asymptotics and concentration in spectral estimation for large matrices

大型矩阵及其谱特征的估计在处理大型高维数据集的各种科学和工程问题中至关重要。谱方法在核机器学习、流形学习、函数数据分析、大型网络中的社区检测、量子统计和量子信息以及许多其他应用中至关重要。该项目的目的是开发分析高维和无限维矩阵估计问题所需的新数学工具，这些工具可能会导致更强大的复杂高维数据统计推断方法。该项目还包括一些对研究生教育和统计学、计算机科学和其他领域之间的研究合作产生影响的活动。该项目的重点是光谱特性的浓度性质和渐近性（特征值，特征向量，光谱投影仪）几类重要的随机矩阵和算子在高维和无穷维统计推断中起着至关重要的作用，机器学习它们包括样本协方差算子，机器学习中的核矩阵，流形数据的经验热核和拉普拉斯算子，图上谱聚类问题中涉及的矩阵，迹回归问题中的矩阵估计（如矩阵完成和量子态层析成像）。主要目的是研究目标矩阵存在算子范数相合估计的问题（算子），但它收敛速度慢，并制定了广泛的浓度界和渐近结果的具体泛函的基础随机矩阵（算子），如其特征值，其谱投影算子的双线性形式，经验谱投影算子与其真实对应算子的偏差的范数。这些问题的解决依赖于高维概率方法的进一步发展，如浓度不等式，高斯，经验和相关类的随机过程，随机矩阵的非渐近界的一般链接的界限。

Estimation of large matrices and their spectral characteristics is crucial in a variety of problems of science and engineering that deal with large high-dimensional data sets. Spectral methods are of the utmost importance in kernel machine learning, manifold learning, functional data analysis, community detection in large networks, quantum statistics and quantum information, and many other applications. The purpose of the project is to develop new mathematical tools needed in analysis of high-dimensional and infinite-dimensional matrix estimation problems that could potentially lead to more powerful methods of statistical inference for complex, high-dimensional data. The project also includes a number of activities with an impact on graduate education and on research collaborations between statistics, computer science and other areas.The focus of the project is on concentration properties and asymptotics of spectral characteristics (eigenvalues, eigenvectors, spectral projectors) of several important classes of random matrices and operators playing crucial role in high dimensional and infinite dimensional statistical inference and in machine learning. They include sample covariance operators, kernel matrices in machine learning, empirical heat kernels and Laplacians for manifold data, matrices involved in spectral clustering problems on graphs, matrix estimators in trace regression problems (such as matrix completion and quantum state tomography). The main goal is to study the problems where there exists an operator norm consistent estimator of the target matrix (operator), but it converges at a slow rate and to develop a broad range of concentration bounds and asymptotic results for specific functionals of the underlying random matrix (operator) such as its eigenvalues, bilinear forms of its spectral projection operators, norms of deviations of empirical spectral projection operators from their true counterparts. The solution of these problems relies on further development of the methods of high-dimensional probability, such as concentration inequalities, generic chaining bounds for Gaussian, empirical and related classes of stochastic processes, non-asymptotic bounds for random matrices.