Nonlinear Dimension Reduction Methods

在大数据时代，从天文学到基因组学再到医学，几乎所有学科的技术进步都带来了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、高维数据中寻找有意义的模式和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以及其他稀疏性或结构简单性技术的主要工具。降维方法将极大地帮助从数据中收集信息的过程。该项目涉及非线性降维方法，可视为标准主成分分析（PCA）的扩展-一种广泛使用的低秩近似数据的工具。该研究旨在将主成分分析的范围扩展到从二进制到有序计数响应的各种类型的数据，并揭示主成分分析的非线性扩展所带来的数据嵌入。加强对现有工具的理解和本研究中新工具的开发将在许多方面改进统计实践。本项目主要集中于研究PCA的两种非线性扩展：核PCA和广义PCA，用于包括指数族数据在内的各种数据类型。本研究有两个具体目的：(i)通过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非线性主成分的聚类的影响，以及（ii）开发PCA方法的统计原则扩展，用于使用广义线性模型框架从指数族分布分析和建模数据矩阵。在方法方面，研究将线性模型的相干扩展平行于广义线性模型框架，以获得数据的最佳低秩逼近。计算工具将被开发用于研究方法的广泛应用。

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.