Simultaneous Optimization of Geodesics and Fréchet Means

A central part of geometric statistics is to compute the Fr\'echet mean. This is a well-known intrinsic mean on a Riemannian manifold that minimizes the sum of squared Riemannian distances from the mean point to all other data points. The Fr\'echet mean is simple to define and generalizes the Euclidean mean, but for most manifolds even minimizing the Riemannian distance involves solving an optimization problem. Therefore, numerical computations of the Fr\'echet mean require solving an embedded optimization problem in each iteration. We introduce the GEORCE-FM algorithm to simultaneously compute the Fr\'echet mean and Riemannian distances in each iteration in a local chart, making it faster than previous methods. We extend the algorithm to Finsler manifolds and introduce an adaptive extension such that GEORCE-FM scales to a large number of data points. Theoretically, we show that GEORCE-FM has global convergence and local quadratic convergence and prove that the adaptive extension converges in expectation to the Fr\'echet mean. We further empirically demonstrate that GEORCE-FM outperforms existing baseline methods to estimate the Fr\'echet mean in terms of both accuracy and runtime.