Complexity Penalization in High Dimensional Matrix Estimation Problems

研究人员将研究基于大型矩阵的线性泛函的噪声测量的估计的各种问题。主要集中在目标矩阵要么是低秩阵，要么可以用低秩阵很好地逼近的问题。所提出的估计方法是基于带复杂性惩罚的经验风险最小化方法，其目标是获得估计误差的锐界(特别是低阶Oracle不等式)，这些界表明估计误差如何依赖于问题的重要参数，如噪声水平、样本大小、目标矩阵的大小和秩等。要研究的问题包括：(A)用于迹回归的新的低阶Oracle不等式，它在无噪声和有噪声的更复杂的设计分布模型的已知结果之间提供了一座桥梁；(B)量子状态层析成像中的密度矩阵估计，目的是研究矩阵回归设置中的最小二乘方法和具有适当复杂性惩罚的更一般测量模型的最大似然方法；(C)图和流形上低阶核的估计(学习)，目的是开发新的方法来预测嵌入在欧几里德空间中的未知流形中的图的顶点或点之间的相似性。该项目是高维矩阵估计的一个非常活跃的交叉学科领域，该领域介于统计学、数学和计算机科学之间，它将促进这些领域之间的研究合作。它将更好地理解高维矩阵估计问题的微妙方面以及这些问题中的复杂性正则化问题。大型矩阵的估计问题在高维统计中是非常基本的问题，在信号和图像处理、压缩传感、生物信息学、量子信息和量子统计、高维数据可视化和视觉分析等领域中有着广泛的应用。在这些问题中，重要的是在高维数据中找到反映描述复杂高维系统的变量之间的基本关系的低维结构。在矩阵问题的情况下，寻找这种结构可以归结为对低阶矩阵的估计，所提出的项目将导致对现有方法的更好的理解以及低阶矩阵恢复的新方法的发展。

The investigator will study a variety of problems of estimation of large matrices based on noisy measurements of linear functionals of these matrices. The main focus is on the problems where the target matrix is either low rank, or it can be well approximated by low rank matrices. The proposed estimation methods are based on empirical risk minimization with complexity penalties that favor low rank solutions and the objective is to obtain sharp bounds on the estimation error (in particular, low rank oracle inequalities) that show how it depends on the important parameters of the problem such as the level of the noise, the sample size, the size and the rank of the target matrix. The problems to be studied include: (a) new low rank oracle inequalities for trace regression providing a bridge between known results in the noiseless case and in the noisy case for more complex models of design distribution; (b) estimation of density matrix in quantum state tomography with a goal of studying both the least squares method in matrix regression setting and the maximum likelihood method for more general measurement models with proper complexity penalization; (c) estimation (learning) of low rank kernels on graphs and manifolds with a goal of developing new methods of predicting similarities between vertices of a graph or points in an unknown manifold embedded in a Euclidean space.This project is in a very active interdisciplinary area of high-dimensional matrix estimation that is borderline between statistics, mathematics and computer science, and it will facilitate research collaborations between these areas. It will provide a better understanding of subtle aspects of high-dimensional problems of matrix estimation and of complexity regularization in these problems. The problem of estimation of large matrices is very basic in high-dimensional statistics and in a variety of its applications in such areas as signal and image processing, compressed sensing, bioinformatics, quantum information and quantum statistics, high-dimensional data visualization and visual analytics. In these problems, it is of importance to find low-dimensional structures in high-dimensional data that reflect basic relationships between the variables describing complex high-dimensional systems. In the case of matrix problems, finding such structures can be reduced to estimation of low rank matrices and the proposed project will result in a better understanding of the existing methods as well as in the development of new methods of low rank matrix recovery.