On the computation of means on Grassmann manifolds

摘要

Given a set of data points on a Grassmann manifold sufficiently close to each other, one way to define their centroid or geometric mean is via the minimizer of a certain cost function. If one chooses the cost as the sum of squared geodesic distances between a given point and all the data points we end up with the definition of the Karcher mean. In this paper we analyze the critical points for this cost function. I. INTRODUCTION In recent years numerical methods for the computation of means on differentiable manifolds have awoken increased interest. Since the seminal paper by Karcher, cf. (13), several papers have been published, emphasizing mainly differen- tial geometric aspects as well as statistical applications in connection with convexity concepts, cf. (8), (14), (15), to cite just a few. The reason for new interest into the topic of computing averages, means or the like on differentiable manifolds, lies in the fact, that in applications, noise is often an intrinsic obstacle during the process of measurement. In modern engineering applications state variables as well as other observables do not lie necessarily in a vector space. Instead, they might be elements of a nonlinear differentiable manifold. This often requires the generalization of the con- cept of means and other statistical tools, now also from a computational point of view. There has been done already a lot in this direction. We mention computer graphics, cf. (7), pose estimation in robotics, DNA-modeling or geology, where averaging over rotations play an important role, cf. (9), (19), linear algebra and matrix analysis, cf. (3), (4), (5), (6), (20), signal process- ing, cf. (18), variational problems, cf. (16), and more recently also in medical imaging, computer vision and distributed consensus problems (21), (22), (23). In this paper we discuss the computation of the Karcher mean on the real Grassmann manifold. Only a little has been done in this direction for this important compact symmetric space, see however (10) or (2). The Riemannian geometry of this manifold is exploited to derive explicit formulas for the exponential map and its inverse. The critical point set of the distance function related to the Karcher mean is computed. To derive a gradient or Newton-like algorithm is then straight forward.