Maximum-margin polyhedral separation for binary Multiple Instance Learning

Multiple Instance Learning (MIL) is a kind of weak supervised learning, where each sample is represented by a bag of instances. The main characteristic of such problems resides in the training phase, since the class labels are provided only for each bag, whereas the instance labels are unknown. We focus on binary MIL problems characterized by two types of instances (positive and negative): based on the standard MIL assumption, a bag is considered positive if at least one of its instances is positive and it is considered negative otherwise. Then our idea is to generate a maximum-margin polyhedral separation surface such that, for each positive bag, at least one of its instances is inside the polyhedron and all the instances of the negative bags are outside. The resulting optimization problem is a nonlinear, nonconvex and nonsmooth mixed integer program, that we heuristically solve by a Block Coordinate Descent type method, based on repeatedly applying the DC (Difference of Convex) Algorithm. Numerical results are presented on a set of benchmark datasets.