Computational Methods for Nonlinear Dimension Reduction

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

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.