Learning Algorithms for Enclosing Points in Bregmanian Spheres

We discuss the problem of finding a generalized sphere that encloses points originating from a single source. The points contained in such a sphere are within a maximal divergence from a center point. The divergences we study are known as the Bregman divergences which include as a special case both the Euclidean distance and the relative entropy. We cast the learning task as an optimization problem and show that it results in a simple dual form which has interesting algebraic properties. We then discuss a general algorithmic framework to solve the optimization problem. Our training algorithm employs an auxiliary function that bounds the dual’s objective function and can be used with a broad class of Bregman functions. As a specific application of the algorithm we give a detailed derivation for the relative entropy. We analyze the generalization ability of the algorithm by adopting margin-style proof techniques. We also describe and analyze two schemes of online algorithms for the case when the radius of the sphere is set in advance.