Rigorous Methods for Dimensionality Reduction of High-Dimensional Data

提出了一项研究工作，旨在为高维环境中的数据分析和推理创建工具。这项工作使用了随机矩阵理论（RMT）、巴纳赫空间理论（BST）和微分几何（DG）的工具，揭示了高维统计推断和数据分析中的新现象，产生了在精心陈述的条件下具有严格建立属性的实用统计方法。研究结果将影响广泛的数据分析问题，包括线性模型的建立，多元数据复杂假设的检验，以及高维数据中微妙非线性结构的检测。在研究中，研究人员在RMT、BST和DG之间建立了进一步的桥梁，并解决了三个问题领域：(a)稀疏线性建模——如何从许多可用的预测因子中选择相对较少的预测因子来构建预测模型？(b)高维多变量分析——如何最好地估计和检验高维数据的结构，特别是当变量数量多而观测值数量少的时候？(c)流形学习——如何才能最好地在高维数据中找到非线性结构并最好地将该结构参数化？这些领域中的每一个都对高维数据的分析至关重要，研究者确定了一种使用RMT、BST和DG的策略，以对每个领域做出重大贡献。该策略建立在作者最近使用RMT， BST和DG的研究成果的基础上，将扩展到显示：(a)如何在不花费指数时间搜索模型空间的情况下找到最佳拟合的低维线性模型-扩展先前使用Basis Pursuit， LARS和Lasso的成功；(b)如何使用Tracy-Widom分布正确地检验多元分析中的一系列重要假设——扩展了之前将Tracy-Widom分布应用于主成分分析的结果；(c)如何正确估计高维空间中稀疏采样曲线数据的非线性参数化——扩展了先前在开发降维的Hessian特征映射技术方面的成功。这个项目的动机在于“数据洪流”现在吞没了科学和技术的每一个分支。在一个又一个领域，新的传感器正在创造无与伦比的广度和深度的数据流。因此，今天的科学和技术进步在很大程度上依赖于处理高维数据和降低其维数的能力，有时会大幅降低，通过使用原始测量的一些精心选择的组合来获得良好的近似值。虽然已经提出了许多降维方法，但该领域的许多现有研究活动都是启发式和推测性的；这些工具的可靠性通常是未知的，它们的性质在未知的普遍性条件下也能保持不变。该项目开发了基于仔细的数学分析的方法，以开发严格正确和/或最优的降维方法。这些方法向用户保证，重要的特征在保留的维度中被捕获，而在丢弃的维度中很少有重要的特征被丢弃。该项目在三个方面发展了这种严格的方法：(a)从许多可能的预测者的数据库中建立简洁但准确的预测模型；(b)在看似高维的“噪音”中测试隐藏的结构；(c)发现本质上是非线性的数据的正确表示。对这一项目取得成功的强烈期望可以建立在调查人员在这三个领域取得的坚实成就的基础上。

A research effort is proposed to create tools for data analysis and inference in high-dimensional settings. The effort uses tools from random matrix theory (RMT), Banach Space Theory (BST), and differential geometry (DG) to expose new phenomena in high-dimensional statistical inference and data analysis, yielding practical statistical methods with rigorously-established properties under carefully-stated conditions. The results will impact a wide range of data analysis problems, including the building of linear models, the testing of complex hypotheses about multivariate data, and the detection of subtle nonlinear structures in high-dimensional data. In the research, the investigators build further bridges between RMT, BST, and DG and three problem areas: (a) Sparse Linear Modelling -- How should one build a predictive model choosing relatively few predictors out of many available predictors?; (b) Multivariate Analysis in High Dimensions -- How should one best estimate and test for structure in high-dimensional data, particularly when the number of variables is large and the number of observations is small?; (c) Manifold Learning -- How can one best find nonlinear structure in high-dimensional data and best parametrize that structure? Each of these areas is of fundamental importance to the analysis of high-dimensional data, and the investigators identify a strategy to use RMT, BST, and DG to make substantial contributions to each. This strategy builds on the authors' recent research accomplishments using RMT, BST, and DG, which will be extended to show: (a) how to find the best-fitting low-dimensional linear model without spending exponential time searching through model space -- extending previous successes in using Basis Pursuit, LARS and Lasso; (b) how to correctly test a wide range of important hypotheses in multivariate analysis using the Tracy-Widom distribution -- extending previous results in applying the Tracy-Widom distribution to Principal Components Analysis; and (c) how to correctly estimate a nonlinear parametrization of sparsely sampled curved data in high dimensional space -- extending previous successes in developing the Hessian Eigenmap technique of dimensionality reduction.The motivation for this project lies in the `data deluge' now engulfing every branch of science and technology. In field after field, new sensors are creating data streams of unparalleled breadth and depth. As a result, today scientific and technological progress depends heavily on the ability to process high-dimensional data and reduce its dimensionality, sometimes drastically, obtaining a good approximation using a few well-chosen combinations of the original measurements. While many methods of dimensionality reduction have already been proposed, much existing research activity in this area is heuristic and speculative; the tools are often of unknown reliability and their properties hold under conditions of unknown generality. This project develops methods based on careful mathematical analysis to develop methods of dimensionality reduction which are rigorously correct and/or optimal. These methods give the user the assurance that important features are captured in the dimensions which remain and that little of importance is discarded in the dimensions that are thrown away. The project develops such rigorous methods in three areas: (a) building parsimonious but accurate predictive models out of a database of many possible predictors; (b) testing for hidden structure in what otherwise seems to be high dimensional `noise'; (c) discovering the correct representation for data which are intrinsically nonlinear. Strong expectations for success of this project can be based on existing solid achievements by the investigators in each of these three areas.