Computational Methods for Nonlinear Dimension Reduction

非线性降维的流形学习方法引起了机器学习和应用数学界的极大兴趣。基本思想是将数据视为嵌入在高维空间中的低维非线性流形的样本。Hongyuan Zha和Haesun Park提出开发高效的非线性降维计算算法和分析和更好理解其行为的理论工具，并将其应用于视频序列标注和ad-hoc传感器网络定位问题，重点关注流形学习中的以下两个重要问题：1)通过分析排列矩阵的谱性质，深入理解局部切空间排列等流形学习方法的行为，并开发专门的预处理方法，有效地处理病态问题；2)探索和适应领域分解等方法，以开发更有效和可扩展的流形学习计算算法。作为所提出算法的应用，将开发半监督流形学习背景下的视频序列标注和非平坦地形下自组织传感器网络定位问题的算法。提取复杂和高维数据的紧凑表示是许多科学和工程努力的核心。流形学习已经成为一个非常活跃的研究领域，它旨在从许多复杂高维数据中固有的统计和几何规则中发现隐藏的结构。研究人员研究的方法有望显著扩展现有和新的流形学习方法的适用性和功能，从而推动流形学习研究的最新进展。本研究是科学计算与机器学习应用的交叉领域，为跨学科研究提供了一个理想的研究环境，同时也为跨学科研究的研究生培养提供了一个理想的环境。视频序列标注在国土安全监控分析中的应用具有重要意义，传感器网络的定位方法将有助于新一代网络系统的发展。

Manifold learning approach for nonlinear dimension reduction has drawn considerable interests from the machine learning as well as applied mathematics communities. The basic idea is to consider data as samples from a low-dimensional nonlinear manifold embedded in a high-dimensional space. Hongyuan Zha and Haesun Park propose to develop efficient computational algorithms for nonlinear dimension reduction and theoretical tools for analyzing and better understanding their behaviors as well as applications to video sequence annotation and ad-hoc sensor network localization problems, focusing on the following two important issues in manifold learning: 1) deeper understanding of the behaviors of manifold learning methods such as local tangent space alignment through analysis of the spectral properties of the alignment matrix, and developing specialized pre-conditioning methods for effectively handling ill-conditioned problems; and 2) exploring and adapting methods such as domain decomposition to develop more efficient and scalable computational algorithms for manifold learning. As applications of the proposed algorithms, video sequence annotation in the context of semi-supervised manifold learning and algorithms for ad-hoc sensor network localization problems especially for the case when the terrain is nonflat will be developed. Extracting compact representations of complex and high-dimensional data are at the core of many scientific and engineering endeavors. Manifold learning has become a very active research field aiming at discovering hidden structures from the statistical and geometric regularity inherent in many complex high-dimensional data. The investigators study methods that have the promise of significantly expanding the applicability and functionality of existing and new manifold learning methods and thus advancing the state of the art in manifold learning research. The proposed research lies at the interface between scientific computing and machine learning applications and provides an ideal setting for research cross-fertilization and collaboration as well as training of graduate students in interdisciplinary research. The applications in Video sequence annotation is important in surveillance analysis for homeland security and localization methods for sensor networks will contribute to the development of new generation of networking systems.