2021/06/15 by Jason M. Altschuler, Sinho Chewi, Altschuler, Jason M. +5 · 4 citations
Mathematics · Medicine · #Acupuncture Treatment Research Studies #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Machine Learning (cs.LG) #Morphological variations and asymmetry #Optimization and Control (math.OC) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2106.08502
openalex publication_date 2021/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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 Riemannian GD for these problems.