Machine Learning
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
A Tutorial on Support Vector Machines for Pattern Recognition
Data Mining and Knowledge Discovery
Support vector machines: training and applications
Support vector machines: training and applications
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When training Support Vector Machines (SVMs) over nonseparable data sets, one sets the threshold b using any dual cost coefficient that is strictly between the bounds of 0 and C. We show that there exist SVM training problems with dual optimal solutions with all coefficients at bounds, but that all such problems are degenerate in the sense that the "optimal separating hyperplane" is given by w = 0, and the resulting (degenerate) SVM will classify all future points identically (to the class that supplies more training data). We also derive necessary and sufficient conditions on the input data for this to occur. Finally, we show that an SVM training problem can always be made degenerate by the addition of a single data point belonging to a certain unbounded polyhedron, which we characterize in terms of its extreme points and rays.