Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space
By Thibaut Le Gouic, Quentin Paris, Philippe Rigollet, Austin J. Stromme. Journal of the European Mathematical Society (Print), 2019
63 citations recorded by the source corpus.
This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural conditions that characterize the bi-extendibility of geodesics emanating from a barycenter. These results largely advance the state-of-the-art on the subject both in terms of rates of convergence and the variety of spaces covered. In particular, our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Radiology, Nuclear Medicine and Imaging
View Paper: https://www.semanticscholar.org/paper/e1b1ad9520bedcd360fbce83ebfa5b472be0cc12
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Citations in the corpus
- Statistical inference for Bures–Wasserstein barycenters (2019, methodology): “The original Bures metric appears in quantum mechanics in relation to the fidelity measure between two quantum states and is used for the measurement of quantum entanglement Marian and Marian [2008], Dajka et al.”
- Distribution-on-Distribution Regression via Optimal Transport Maps (2021, background): “[9] who make use of this assumption in the context of obtaining dimension-free rates for Fréchet mean estimation.”