Wasserstein barycenters : statistics and optimization
Author(s)Stromme, Austin J.(Austin James)
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.
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We study a geometric notion of average, the barycenter, over 2-Wasserstein space. We significantly advance the state of the art by introducing extendible geodesics, a simple synthetic geometric condition which implies non-asymptotic convergence of the empirical barycenter in non-negatively curved spaces such as Wasserstein space. We further establish convergence of first-order methods in the Gaussian case, overcoming the nonconvexity of the barycenter functional. These results are accomplished by various novel geometrically inspired estimates for the barycenter functional including a variance inequality, new so-called quantitative stability estimates, and a Polyak-Łojasiewicz (PL) inequality. These inequalities may be of independent interest.
Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 67-71).
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Electrical Engineering and Computer Science.