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Title: Gradient descent algorithms for Bures-Wasserstein barycenters
We study first order methods to compute the barycenter of a probability distribution $P$ over the space of probability measures with finite second moment. We develop a framework to derive global rates of convergence for both gradient descent and stochastic gradient descent despite the fact that the barycenter functional is not geodesically convex. Our analysis overcomes this technical hurdle by employing a Polyak-Ł{}ojasiewicz (PL) inequality and relies on tools from optimal transport and metric geometry. In turn, we establish a PL inequality when $P$ is supported on the Bures-Wasserstein manifold of Gaussian probability measures. It leads to the first global rates of convergence for first order methods in this context.  more » « less
Award ID(s):
1712596
NSF-PAR ID:
10219148
Author(s) / Creator(s):
; ; ;
Editor(s):
Abernethy, Jacob; Agarwal, Shivani
Date Published:
Journal Name:
Proceedings of Machine Learning Research
Volume:
125
ISSN:
2640-3498
Page Range / eLocation ID:
1276--1304
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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