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

Part of Advances in Neural Information Processing Systems 34 pre-proceedings (NeurIPS 2021)

Paper Supplemental

Bibtek download is not available in the pre-proceeding


Authors

Jason Altschuler, Sinho Chewi, Patrik Gerber, Austin Stromme

Abstract

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 gradient descent empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean gradient descent and SDP solvers. This stands in stark contrast to the best-known theoretical results, 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. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for these problems.