Distributed optimization on manifolds for consensus algorithms and camera network localization
René Vidal, Roberto Tron · 2012
In recent years, there has been a surge of interest in solving optimization problems defined on non-linear manifolds. Such kind of problems appears in a variety of fields, ranging from medical imaging, to computer vision and machine learning. In parallel there has also been a rising interest in developing distributed algorithms for networked systems, where a set of nodes or agents uses a communication network to interact and achieve a common goal. For instance, these algorithms could be used by a group of vehicles to coordinate their motion, or to find their relative positions. The goal of this thesis is to bring together these two fields by studying distributed optimization algorithms on manifolds. We advance the state of the art in three major directions. First, we develop general theoretical tools for designing and analyzing distributed optimization on manifolds. Second, we propose an extension of an existing class of distributed coordination algorithms, called consensus algorithms, to the case where the data lies on a Riemannian manifold with bounded curvature. Third, we propose a distributed algorithm for estimating the location of cameras in a network by using only correspondences between pairs of images.