7 - Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications

Abstract

Finding the Riemannian barycenter (center of mass) or the Fréchet mean (FM) of manifold-valued data sets is a commonly encountered problem in a variety of fields of science and engineering, including, but not limited to, medical image computing, machine learning, and computer vision. For example, it is encountered in tasks such as atlas construction, clustering, principal geodesic analysis, and so on. Traditionally, algorithms for computing the FM of the manifold-valued data require that the entire data pool be available a priori and not incrementally. Thus, when encountered with new data, the FM needs to be recomputed over the entire pool, which can be computationally and storage inefficient. A computational and storage efficient alternative is to consider a recursive algorithm for computing the FM, which simply updates the previously computed FM when presented with a new data sample. In this chapter we present such an alternative called the inductive/incremental Fréchet mean estimator (iFME) for data residing on two well-known Riemannian manifolds, namely the hypersphere S(d) and the special orthogonal group SO(d). We prove the asymptotic convergence of iFME to the true FM of the underlying distribution from which the data samples were drawn. Further we present several experiments demonstrating the performance iFME on synthetic and real data sets.