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Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size

arXiv · AI, language, vision and robotics · article · Sep 3, 2026 · UTC

The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information. Riemannian gradient descent (RGD) at unit step size -- the fixed-point iteration used in practice -- converges rapidly, yet existing analyses present a dichotomy: unit-step guarantees carry worst-case exponential dependence on the dimension, while dimension-independent guarantees require small step sizes that forfeit the empirical speed. We resolve this dichotomy, not by improving the guarantees for unit-step RGD,

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