Nonlinear Dimension Reduction Methods

在大数据时代，技术进步带来了从天文学到基因组学再到医学的几乎所有学科产生的数据量和复杂性的重大变化。它已经成为以可靠和高效的方式从大规模、高维数据中发现有意义的模式并提取相关信息的智力努力的重要组成部分。理解和捕获数据背后的规则结构对于后续的建模和预测至关重要。数据的低维投影通常是揭示结构和处理高维的主要工具，以及其他稀疏性或结构简单性的技术。降维的方法将大大有助于从数据中收集信息的过程。这个项目涉及非线性降维方法，它可以被视为标准主成分分析(PCA)的扩展，主成分分析是一种广泛使用的数据低阶逼近工具。本研究的目的是将主成分分析的范围扩展到各种类型的数据，从对计数的二进制响应到顺序响应，并揭示主成分分析的非线性扩展所给出的数据嵌入。本课题主要研究包括指数族数据在内的各种数据类型的主成分分析的两种非线性扩展：核主成分分析和广义主成分分析。这项研究有两个具体的目的：(I)通过对核算子的谱分析，了解核主成分分析给出的非线性数据嵌入的几何形状，以及核和中心核对与数据分布相关的用于聚类的非线性主成分的影响；(Ii)发展基于广义线性模型框架的主成分分析方法的统计原则性扩展，用于从指数族分布中分析和建模数据矩阵。在方法论方面，本研究将线性模型的一致性扩展到广义线性模型框架，以获得数据的最佳低阶逼近。将为所研究方法的广泛应用开发计算工具。

In the era of Big Data, technological advances have brought significant changes in the amount and the complexity of data generated in almost every discipline from astronomy to genomics to medicine. It has become an essential component of the intellectual endeavor to find meaningful patterns and extract relevant information from large scale, high-dimensional data in a reliable and efficient fashion. Understanding and capturing the regular structures underlying the data is crucial for subsequent modeling and prediction. Low-dimensional projections of data are often primary tools for uncovering the structure and coping with high-dimensionality along with other techniques for sparsity or structural simplicity. Methods for dimension reduction will help the process of gathering information from data significantly. This project concerns nonlinear dimension reduction methods which can be viewed as an extension of standard principal component analysis (PCA) - a widely used tool for low-rank approximation of data. The research aims to expand the scope of PCA to various types of data from binary to ordinal responses to counts, and unravel the data embeddings given by nonlinear extensions of PCA. Enhanced understanding of the existing tools and the development of new tools in this research will improve statistical practice in many ways.This project is primarily focused on investigation of two nonlinear extensions of PCA: kernel PCA and generalized PCA, for various data types including the exponential family data. This research has two specific aims: (i) to understand the geometry of the nonlinear data embeddings given by the kernel PCA through the spectral analysis of the kernel operator, and the effect of a kernel and centering kernels on those nonlinear principal components for clustering in relation to the data distribution, and (ii) to develop statistically principled extensions of the PCA methodology for analysis and modeling of data matrices from the exponential family distributions using generalized linear model framework. On the methodological aspect, the research parallels the coherent extension of linear model to generalized linear model framework for the best low-rank approximation of data. Computational tools will be developed for a wide range of applications of the studied methods.