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

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Abstract

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.

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Acknowledgments

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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Authors and Affiliations

  1. Department of Economic Information Management, Remin University of China, 100872, Beijing, China

    Xun Liang

  2. Department of Economics and Operations Research, Stanford University, Stanford, CA, 95035, USA

    Xun Liang

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  1. Xun Liang
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Correspondence to Xun Liang.

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Liang, X. Feature space versus empirical kernel map and row kernel space in SVMs. Neural Comput & Applic 19, 487–498 (2010). https://doi.org/10.1007/s00521-010-0337-0

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