Recursive Computation of the Fréchet Mean on Non-positively Curved Riemannian Manifolds with Applications

Computing the Riemannian center of mass or the finite sample Frechet mean has attracted enormous attention lately due to the easy availability of data that are manifold valued. Manifold-valued data are encountered in numerous domains including but not limited to medical image computing, mechanics, statistics, machine learning. It is common practice to estimate the finite sample Frechet mean by using a gradient descent technique to find the minimum of the Frechet function when it exists. The convergence rate of this gradient descent method depends on many factors including the step size and the variance of the given manifold-valued data. As an alternative to the gradient descent technique, we propose a recursive (incremental) algorithm for estimating the Frechet mean/expectation (iFEE) of the distribution from which the sample data are drawn. The proposed algorithm can be regarded as a geometric generalization of the well-known incremental algorithm for computing arithmetic mean, since it reinterprets this algebraic formula in terms of geometric operations on geodesics in the more general manifold setting. In particular, given known formulas for geodesics, iFEE does not require any optimization in contrast to the non-incremental counterparts and offers significant improvement in efficiency and flexibility. For the case of simply connected, complete and nonpositively curved Riemannian manifolds, we prove that iFEE converges to the true expectation in the limit. We present several experiments demonstrating the efficiency and accuracy of iFEE in comparison to the non-incremental counterpart for computing the finite sample Frechet mean of symmetric positive definite matrices as well as applications of iFEE to K-means clustering and diffusion tensor image segmentation.