Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications

By Carla Tameling, Max Sommerfeld, Axel Munk. The Annals of Applied Probability, 2017

67 citations recorded by the source corpus.

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 non-linear 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 ultra-metric spaces. Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.

Subfields: Applied Mathematics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/043de314327487a110dd02dacb051ed7e86e7cc3

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Citations in the corpus

Fréchet means and Procrustes analysis in Wasserstein space (2017, background): “[73] for challenges involved in the discrete setting).”
Plugin estimation of smooth optimal transport maps (2021, background): “Key exceptions are obtained when P and Q are one-dimensional (Munk and Czado, 1998; Freitag and Munk, 2005) or countable (Sommerfeld and Munk, 2018; Tameling et al., 2019), in which case non-degenerate limiting distributions for the process W2(Pn, Qm) − W2(P,Q) are known up to suitable scaling.”
Statistical Aspects of Wasserstein Distances (2018, background): “These results extend to countably supported measures, where one needs to impose an extra summability condition on r in order to ensure convergence of √ n ( r n − r ) to the Gaussian limit G (Tameling et al., 2017).”