Spherical approaches for Multiple Instance Learning

In a Multiple Instance Learning (MIL) problem the objective is to classify sets of items. Using the MIL terminology, such sets are called bags and the items inside them are called instances. Differently from the classical supervised learning, in a MIL problem only the label of each bag is known in the learning phase, whereas the labels of the instances inside the bags remain unknown. For solving these problems, in the literature there are three type of approaches: the instance-space approaches, the bag-space approaches and the embedding-space approaches, depending on the space where the bag separation is initially performed. We focus on the binary case, characterized by two types of bags and two types of instances, using the so-called standard MIL assumption, stating that a bag is positive if it contains at least a positive instance and it is negative otherwise. For solving this problem, we present a multi-sphere instance-space approach, which generates a finite and variable number of separating spheres such that, for each positive bag, at least one of its instances is inside at least a sphere and all the instances of each negative bag are outside every sphere. Numerical results are presented on some test problem drawn from the literature.