Duality and Geometry in SVM Classifiers

We develop an intuitive geometric interpretation of the standard support vector machine (SVM) for classification of both linearly separable and inseparable data and provide a rigorous derivation of the concepts behind the geometry. For the separable case finding the maximum margin between the two sets is equivalent to finding the closest points in the smallest convex sets that contain each class (the convex hulls). We now extend this argument to the inseparable case by using a reduced convex hull reduced away from outliers. We prove that solving the reduced convex hull formulation is exactly equivalent to solving the standard inseparable SVM for appropriate choices of parameters. Some additional advantages of the new formulation are that the effect of the choice of parameters becomes geometrically clear and that the formulation may be solved by fast nearest point algorithms. By changing norms these arguments hold for both the standard 2-norm and 1-norm SVM.