In this work, we consider a classification problem where the objects to be classified are bags of instances which are vectors measuring d different attributes. The classification rule is defined in terms of a ball, whose center and radius are the parameters to be computed. Given a bag, it is assigned to the positive class if at least one element is strictly included inside the ball, and it is labelled as negative otherwise. We model this question as a margin optimization problem. Several necessary optimality conditions are derived leading to a polynomial algorithm in fixed dimension. A VNS type heuristic is proposed and experimentally tested.

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View Building separating concentric balls to solve a multi-instance classification problem