Binary classification via spherical separator by DC programming and DCA

In this paper, we consider a binary supervised classification problem, called spherical separation, that consists of finding, in the input space or in the feature space, a minimal volume sphere separating the set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\mathcal{A}}$$</EquationSource> </InlineEquation> from the set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\mathcal{B}}$$</EquationSource> </InlineEquation> (i.e. a sphere enclosing all points of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${ \mathcal{A}}$$</EquationSource> </InlineEquation> and no points of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\mathcal{B}}$$</EquationSource> </InlineEquation>). The problem can be cast into the DC (Difference of Convex functions) programming framework and solved by DCA (DC Algorithm) as shown in the works of Astorino et al. (J Glob Optim 48(4):657–669, <CitationRef CitationID="CR3">2010</CitationRef>). The aim of this paper is to investigate more attractive DCA based algorithms for this problem. We consider a new optimization model and propose two interesting DCA schemes. In the first scheme we have to solve a quadratic program at each iteration, while in the second one all calculations are explicit. Numerical simulations show the efficiency of our customized DCA with respect to the methods developed in Astorino et al. Copyright Springer Science+Business Media, LLC. 2013