Nonlinear Dimension Reduction Methods

非线性降维方法

• 批准号: 1513566
• 负责人: Yoonkyung Lee
• 金额: $ 15万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1513566&HistoricalAwards=false
• 关键词: Nonlinear; Dimension; Reduction; Methods

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.

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每个学科产生的数据的数量和复杂性的重大变化。它已经成为一个重要的组成部分，智力奋进找到有意义的模式，并提取相关的信息，从大规模，高维数据在一个可靠和有效的方式。理解和捕获数据背后的规则结构对于后续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沿着其他技术的稀疏性或结构简单性的主要工具。降维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，可以被看作是标准主成分分析（PCA）的扩展-一个广泛使用的工具，低秩近似的数据。该研究旨在将PCA的范围扩展到各种类型的数据，从二进制到有序的计数响应，并解开PCA的非线性扩展所给出的数据嵌入。在本研究中，加强对现有工具的理解和新工具的开发将在许多方面改善统计实践，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，适用于各种数据类型，包括指数族数据。这项研究有两个具体目标：（i）通过核算子的谱分析来理解由核PCA给出的非线性数据嵌入的几何结构，以及核和定中心核对用于与数据分布相关的聚类的那些非线性主成分的影响，以及（ii）发展PCA方法的统计学原理扩展，用于使用广义线性模型框架对指数族分布的数据矩阵进行分析和建模。在方法论方面，研究平行的线性模型的一致性扩展到广义线性模型框架的最佳低秩逼近的数据。计算工具将被开发为广泛的应用所研究的方法。

项目成果

The Geometry of Nonlinear Embeddings in Kernel Discriminant Analysis

• DOI: 10.1109/tpami.2022.3192726
• 发表时间: 2020-05
• 期刊: IEEE Transactions on Pattern Analysis and Machine Intelligence
• 影响因子: 23.6
• 作者: Jiae Kim;Yoonkyung Lee;Zhiyu Liang