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October 2019 Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications
Carla Tameling, Max Sommerfeld, Axel Munk
Ann. Appl. Probab. 29(5): 2744-2781 (October 2019). DOI: 10.1214/19-AAP1463

Abstract

We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for nonlinear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this, we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distributional limit to hold. We give an explicit form of the limiting distribution for tree spaces.

Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.

Citation

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Carla Tameling. Max Sommerfeld. Axel Munk. "Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications." Ann. Appl. Probab. 29 (5) 2744 - 2781, October 2019. https://doi.org/10.1214/19-AAP1463

Information

Received: 1 September 2018; Revised: 1 January 2019; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155058
MathSciNet: MR4019874
Digital Object Identifier: 10.1214/19-AAP1463

Subjects:
Primary: 60B12, 60F05, 62E20
Secondary: 62G10, 90C08, 90C31

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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