On The Convergence of Gradient Descent for Finding the Riemannian Center of Mass

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 2dimensional manifolds of nonnegative curvature and manifolds of constant nonnegative curvature (e.g., the sphere in R and the rotation group in R) 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.