Multi-instance classification through spherical separation and VNS

A two-class classification problem is considered where the objects to be classified are bags of instances in d-space. The classification rule is defined in terms of an open d-ball. A bag is labeled positive if it meets the ball and labeled negative otherwise. Determining the center and radius of the ball is modeled as a SVM-like margin optimization problem. Necessary optimality conditions are derived leading to a polynomial algorithm in fixed dimension. A VNS type heuristic is developed and experimentally tested. The methodology is extended to classification by several balls and to more than two classes.