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Mathematical Problems in Engineering
Volume 2015, Article ID 241436, 18 pages
http://dx.doi.org/10.1155/2015/241436
Research Article

Enhancing Both Efficiency and Representational Capability of Isomap by Extensive Landmark Selection

School of Mathematics and Statistics and Institute for Information and System Science, Xi’an Jiaotong University, Xi’an 710049, China

Received 24 November 2014; Accepted 20 February 2015

Academic Editor: Wanquan Liu

Copyright © 2015 Dong Liang et al. 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 problems of improving computational efficiency and extending representational capability are the two hottest topics in approaches of global manifold learning. In this paper, a new method called extensive landmark Isomap (EL-Isomap) is presented, addressing both topics simultaneously. On one hand, originated from landmark Isomap (L-Isomap), which is known for its high computational efficiency property, EL-Isomap also possesses high computational efficiency through utilizing a small set of landmarks to embed all data points. On the other hand, EL-Isomap significantly extends the representational capability of L-Isomap and other global manifold learning approaches by utilizing only an available subset from the whole landmark set instead of all to embed each point. Particularly, compared with other manifold learning approaches, the data manifolds with intrinsic low-dimensional concave topologies and essential loops can be unwrapped by the new method more successfully, which are shown by simulation results on a series of synthetic and real-world data sets. Moreover, the accuracy, robustness, and computational complexity of EL-Isomap are analyzed in this paper, and the relation between EL-Isomap and L-Isomap is also discussed theoretically.