Properties of Support Vector Machines

Support Vector Machines (SVMs) perform pattern recognition between two point classes by finding a decision surface determined by certain points of the training set, termed Support Vectors (SV). This surface, which in some feature space of possibly infinite dimension can be regarded as a hyperplane, is obtained from the solution of a problem of quadratic programming that depends on a regularization parameter. In this paper we study some mathematical properties of support vectors and show that the decision surface can be written as the sum of two orthogonal terms, the first depending only on the margin vectors (which are SVs lying on the margin), the second proportional to the regularization parameter. For almost all values of the parameter, this enables us to predict how the decision surface varies for small parameter changes. In the special but important case of feature space of finite dimension m, we also show that there are at most m+1 margin vectors and observe that m+1 SVs are usually sufficient to fully determine the decision surface. For relatively small m this latter result leads to a consistent reduction of the SV number.

A Note on Support Vector Machines Degeneracy

When training Support Vector Machines (SVMs) over non-separable 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 ${f w} = {f 0}$, 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 unboundedspolyhedron, which we characterize in terms of its extreme points and rays.