The Solution Path of the Slab Support Vector Machine

Given a set of points in a Hilbert space that can be separated from the origin. The slab support vector machine (slab SVM) is an optimization problem that aims at finding a slab (two parallel hyperplanes whose distance—the slab width—is essentially fixed) that encloses the points and is maximally separated from the origin. Extreme cases of the slab SVM include the smallest enclosing ball problem and an interpolation problem that was used (as the slab SVM itself) in surface reconstruction with radial basis functions. Here we show that the path of solutions of the slab SVM, i.e., the solution parametrized by the slab width is piecewise linear.