Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 864540, 15 pages
Research Article

Relationship Matrix Nonnegative Decomposition for Clustering

Faculty of Science and State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an Shaanxi Province, Xi'an 710049, China

Received 18 January 2011; Revised 28 February 2011; Accepted 9 March 2011

Academic Editor: Angelo Luongo

Copyright © 2011 Ji-Yuan Pan and Jiang-She Zhang. 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.


Nonnegative matrix factorization (NMF) is a popular tool for analyzing the latent structure of nonnegative data. For a positive pairwise similarity matrix, symmetric NMF (SNMF) and weighted NMF (WNMF) can be used to cluster the data. However, both of them are not very efficient for the ill-structured pairwise similarity matrix. In this paper, a novel model, called relationship matrix nonnegative decomposition (RMND), is proposed to discover the latent clustering structure from the pairwise similarity matrix. The RMND model is derived from the nonlinear NMF algorithm. RMND decomposes a pairwise similarity matrix into a product of three low rank nonnegative matrices. The pairwise similarity matrix is represented as a transformation of a positive semidefinite matrix which pops out the latent clustering structure. We develop a learning procedure based on multiplicative update rules and steepest descent method to calculate the nonnegative solution of RMND. Experimental results in four different databases show that the proposed RMND approach achieves higher clustering accuracy.