A Note on Support Vector Machine Degeneracy

When training Support Vector Machines (SVMs) over non-separable data sets, one sets the threshold b using any dual cost coeecient that is strictly between the bounds of 0 and C. We show that there exist SVM training problems with dual optimal solutions with all coeecients 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 suucient 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.