Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces

Wasserstein distances have a clear intuitive meaning: What is the minimum cost if we want to obtain ν by transporting the probability mass in µ? Here the cost is defined as the product of probability mass moved and the distance moved raised to the pth power. Therefore, the Wasserstein distance is also called "optimal transport distance" or "earth mover's distance". The problem of optimal transport can be traced back to [18] and [13]. 1 Since the introduction in [25], the Wasserstein distances have become an important tool in computer vision and statistical machine learning. In addition to the connection with optimal transport, Wasserstein distances have some desirable features. For example, they can be meaningfully defined for any two distributions without any requirement on the existence of density or absolute continuity. Other measurements, such as the Kullback-Leibler divergence, have more stringent requirements on µ and ν. See [21] for a more thorough historical review of Wasserstein distances and their applications in statistics, and [27] for further details about Wasserstein distances and optimal transport in a broader context. In statistics and machine learning, we often do not have access to µ but only its empirical version ˆµ, which puts 1/n mass at each one of n independent samples from µ.