Inference for Projection-Based Wasserstein Distances on Finite Spaces

The Wasserstein distance is a distance between two probability distributions and has recently gained increasing popularity in statistics and machine learning, owing to its attractive properties. One important approach to extending this distance is using low-dimensional projections of distributions to avoid a high computational cost and the curse of dimensionality in empirical estimation, such as the sliced Wasserstein or max-sliced Wasserstein distances. Despite their practical success in machine learning tasks, the availability of statistical inferences for projection-based Wasserstein distances is limited owing to the lack of distributional limit results. In this paper, we consider distances defined by integrating or maximizing Wasserstein distances between low-dimensional projections of two probability distributions. Then we derive limit distributions regarding these distances when the two distributions are supported on finite points. We also propose a bootstrap procedure to estimate quantiles of limit distributions from data. This facilitates asymptotically exact interval estimation and hypothesis testing for these distances. Our theoretical results are based on the arguments of Sommerfeld and Munk (2018) for deriving distributional limits regarding the original Wasserstein distance on finite spaces and the theory of sensitivity analysis in nonlinear programming. Finally, we conduct numerical experiments to illustrate the theoretical results and demonstrate the applicability of our inferential methods to real data analysis.