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

arXiv:2609.03762 · cs.LG, math.OC, quant-ph · Submitted 2026-09-03 · Read on arXiv

cs.LG, math.OC, quant-ph

Submitted: 2026-09-03

Updated: 2026-09-03

Comments: 33 Pages, 3 figures. Comments welcome!

Code: https://github.com/afhamash/projected-rgd

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Fidelity — 7× in this paper · explained in 2 episodes
• Accuracy — 5× in this paper · explained in 1 episode
• Optimal Transport — 4× in this paper · explained in 5 episodes
• Entanglement — 3× in this paper · explained in 3 episodes
• Trace — 3× in this paper · explained in 2 episodes
• Consistency — 2× in this paper · explained in 3 episodes
• Gradient Flow — 2× in this paper · explained in 2 episodes
• Objective Function — 2× in this paper · explained in 2 episodes
• Riemannian optimization — 2× in this paper · explained in 2 episodes
• Wasserstein Distance — 2× in this paper · explained in 4 episodes
• Convergence Guarantees — 1× in this paper · explained in 3 episodes
• Generalization — 1× in this paper · explained in 13 episodes
• Large Language Models (LLMs) — 1× in this paper · explained in 6 episodes
• Non-asymptotic Analysis — 1× in this paper · explained in 2 episodes

Sources

• Quantum mean states are nicer than you think: fast algorithms to compute states maximizing average fidelity
• Projections with Respect to Bures Distance and Fidelity: Closed-Forms and Applications
• Anderson Mixing in Bures Wasserstein Space of Gaussian Measures
• Wasserstein Riemannian Geometry of Positive Definite Matrices
• A Framework for Non-Asymptotic Quantum Information Theory

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