Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent

Understand

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric.

• Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP solvers.
• This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension.
• In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate.