Abstract: The Wasserstein distance is the distance between two probability distributions, and has recently become popular in statistics and machine learning, owing to its attractive properties. One important approach to extending this distance is to use low-dimensional projections of the distributions, thus avoiding a high computational cost and the curse of dimensionality in empirical estimation; hare, examples include the sliced Wasserstein and max-sliced Wasserstein distances. Despite their practical success in machine learning tasks, statistical inferences for projection-based Wasserstein distances are limited, owing to the lack of distributional limit results. Thus, for probability distributions supported on finite points, we derive the limit distributions of the empirical versions of the projection-based Wasserstein distances. We examine the general class of distances defined by integrating or maximizing the Wasserstein distances between the low-dimesional projections of two distributions. After deriving the limit distributions, we propose a bootstrap procedure for estimating the quantiles of these distributions from the data. This facilitates asymptotically exact interval estimation and hypothesis testing for these distances. Our theoretical results are based on deriving the distributional limit of empirical Wasserstein distances on finite spaces and the theory of sensitivity analysis in nonlinear programming. Finally, we demonstrate the applicability of our inferential methods using a real-data analysis..

Key words and phrases: Bootstrap, distributional limit, projection-based Wasserstein distances, statistical inference.