Keywords: Bures-Wasserstein barycenter, dimension-free convergence, entropic regularization, first-order optimization, geometric median, non-convex optimization, Riemannian optimization
TL;DR: We improve state-of-the-art convergence guarantees for Riemannian gradient descent for computing geometric averages of Gaussians.
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.
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