Neural Computing and Applications

, Volume 19, Issue 3, pp 487–498 | Cite as

Feature space versus empirical kernel map and row kernel space in SVMs

  • Xun Liang
Original Article


In machine-learning technologies, the support vector machine (SV machine, SVM) is a brilliant invention with many merits, such as freedom from local minima, the widest possible margins separating different clusters, and a solid theoretical foundation. In this paper, we first explore the linear separability relationships between the high-dimensional feature space H and the empirical kernel map U as well as between H and the space of kernel outputs K. Second, we investigate the relations of the distances between separating hyperplanes and SVs in H and U, and derive an upper bound for the margin width in K. Third, as an application, we show experimentally that the separating hyperplane in H can be slightly adjusted through U. The experiments reveal that existing SVM training can linearly separate the data in H with considerable success. The results in this paper allow us to visualize the geometry of H by studying U and K.


Separating hyperplane Linear separability High-dimensional feature space Support vector machine 



The authors thank the anonymous reviewers for their valuable comments and suggestions, which helped improve the paper greatly. The project was sponsored by the NSF of China under grants 70571003 and 70871001, and the 863 Project of China under grant 2007AA01Z437.


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Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Department of Economic Information ManagementRemin University of ChinaBeijingChina
  2. 2.Department of Economics and Operations ResearchStanford UniversityStanfordUSA

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