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Binary separation and training support vector machines*

Published online by Cambridge University Press:  10 May 2010

Roger Fletcher  and
Gaetano Zanghirati
Roger Fletcher
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK, E-mail: fletcher@maths.dundee.ac.uk
Gaetano Zanghirati
Affiliation:
Department of Mathematics and Math4Tech Center, University of Ferrara, 44100 Ferrara, Italy, E-mail: g.zanghirati@unife.it
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Extract

We introduce basic ideas of binary separation by a linear hyperplane, which is a technique exploited in the support vector machine (SVM) concept. This is a decision-making tool for pattern recognition and related problems. We describe a fundamental standard problem (SP) and show how this is used in most existing research to develop a dual-based algorithm for its solution. This algorithm is shown to be deficient in certain aspects, and we develop a new primal-based SQP-like algorithm, which has some interesting features. Most practical SVM problems are not adequately handled by a linear hyperplane. We describe the nonlinear SVM technique, which enables a nonlinear separating surface to be computed, and we propose a new primal algorithm based on the use of low-rank Cholesky factors.

It may be, however, that exact separation is not desirable due to the presence of uncertain or mislabelled data. Dealing with this situation is the main challenge in developing suitable algorithms. Existing dual-based algorithms use the idea of L1 penalties, which has merit. We suggest how penalties can be incorporated into a primal-based algorithm. Another aspect of practical SVM problems is often the huge size of the data set, which poses severe challenges both for software package development and for control of ill-conditioning. We illustrate some of these issues with numerical experiments on a range of problems.

Type
Research Article
Information
Acta Numerica , Volume 19 , May 2010 , pp. 121 - 158
Copyright
Copyright © Cambridge University Press 2010

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