About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 683053, 8 pages
http://dx.doi.org/10.1155/2013/683053
Research Article

Split-and-Combine Singular Value Decomposition for Large-Scale Matrix

Department of Mathematical Sciences, National Chengchi University, No. 64, Section 2, ZhiNan Road, Wenshan District, Taipei City 11605, Taiwan

Received 16 November 2012; Revised 17 January 2013; Accepted 22 January 2013

Academic Editor: Nicola Mastronardi

Copyright © 2013 Jengnan Tzeng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable.