Abstract. The geometric median as well as the Fréchet mean of points in an Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a split version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of D. Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only does it yield algorithms for the median and the mean, but it also applies to various other optimization problems. We moreover show that another algorithm for computing the Fréchet mean can be derived from the law of large numbers due to K.-T. Sturm (2002).In applications, computing medians and means is probably most needed in tree space, which is an instance of an Hadamard space, invented by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic trees. It turns out, however, that it can be also used to model numerous other tree-like structures. Since there now exists a polynomial-time algorithm for computing geodesics in tree space due to M. Owen and S. Provan (2011), we obtain efficient algorithms for computing medians and means, which can be directly used in practice.
The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of nonpositive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.
We introduce a new nonsmooth variational model for the restoration of manifoldvalued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization, and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices. important image structures as edges but tends to produce staircasing: instead of reconstructing smooth areas as such, the reconstruction consists of constant plateaus with small jumps. An approach for avoiding this effect incorporates higher order differences, respectively, derivatives, in a continuous setting. The pioneering work [15] couples the TV term with higher order terms by infimal convolution. Since then, various techniques with higher order differences/derivatives were proposed in the literature, among them [13,17,21,22,36,43,45,46,50,61,62,63]. We further note that the second order total generalized variation was extended for tensor fields in [72].In various applications in image processing and computer vision the functions of interest take values in a Riemannian manifold. One example is diffusion tensor imaging where the data lives in the Riemannian manifold of positive definite matrices; see, e.g., [7,14,53,65,75,78]. Other examples are color images based on nonflat color models [16,40,41,73] where the data lives on spheres. Motion group and SO(3)valued data play a role in tracking, robotics, and (scene) motion analysis and were considered, e.g., in [28,52,55,58,70]. Because of the natural appearance of such nonlinear data spaces, processing manifold-valued data has gained a lot of interest in applied mathematics in recent years. As examples, we mention wavelet-type multiscale transforms [35,54,76], robust principal component pursuit on manifolds [37], and partial differential equations [19,32,69] for manifold-valued functions. Although statistics on Riemannian manifolds is not in the focus of this work, we w...
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