Abstract. Let M be a complete Riemannian manifold and ν a probability measure on M . Assume 1 ≤ p ≤ ∞. We derive a new bound (in terms of p, the injectivity radius of M and an upper bound on the sectional curvatures of M ) on the radius of a ball containing the support of ν which ensures existence and uniqueness of the global Riemannian L p center of mass with respect to ν. A significant consequence of our result is that under the best available existence and uniqueness conditions for the so-called "local" L p center of mass, the global and local centers coincide. In our derivation we also give an alternative proof for a uniqueness result by W. S. Kendall. As another contribution, we show that for a discrete probability measure on M , under the existence and uniqueness conditions, the (global) L p center of mass belongs to the closure of the convex hull of the masses. We also give a refined result when M is of constant curvature.
Abstract. We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant stepsize gradient descent algorithms for solving this problem. The challenge is that often the underlying cost function is neither globally differentiable nor convex, and despite this one would like to have guaranteed convergence to the global minimizer. After some necessary preparations we state a conjecture which, we argue is the best convergence condition (in a specific described sense) that one can hope for. The conjecture specifies conditions on the spread of the data points, step-size range, and the location of the initial condition (i.e., the region of convergence) of the algorithm. These conditions depend on the topology and the curvature of the manifold and can be conveniently described in terms of the injectivity radius and the sectional curvatures of the manifold. For 2-dimensional manifolds of nonnegative curvature and manifolds of constant nonnegative curvature (e.g., the sphere in R n and the rotation group in R 3 ) we show that the conjecture holds true. For more general manifolds we prove convergence results which are weaker than the conjectured one (but still superior over the available results). We also briefly study the effect of the configuration of the data points on the speed of convergence. Finally, we study the global behavior of the algorithm on certain manifolds proving (generic) convergence of the algorithm to a local center of mass with an arbitrary initial condition. An important aspect of our presentation is our emphasize on the effect of curvature and topology of the manifold on the behavior of the algorithm.
Abstract. A class of simple Jacobi-type algorithms for non-orthogonal matrix joint diagonalization based on the LU or QR factorization is introduced. By appropriate parametrization of the underlying manifolds, i.e. using triangular and orthogonal Jacobi matrices we replace a high dimensional minimization problem by a sequence of simple one dimensional minimization problems. In addition, a new scale-invariant cost function for non-orthogonal joint diagonalization is employed. These algorithms are step-size free. Numerical simulations demonstrate the efficiency of the methods.
Abstract-In this paper we propose a discrete time protocol to align the states of a network of agents evolving in the space of rotations SO(3). The starting point of our work is Riemannian consensus, a general and intrinsic extension of classical consensus algorithms to Riemannian manifolds. Unfortunately, this algorithm is guaranteed to align the states only when the initial states are not too far apart. We show how to modify Riemannian consensus so that the states of the agents can be aligned, in practice, from almost any initial condition. While we focus on the specific case of SO(3), we hope that this work will represent the first step toward more general results.
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