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The Wasserstein distance (WD), also known as Monge-Kantorovich-Rubinstein distance in the physical sciences, earth-mover's distance in computer science or optimal transport distance in optimization, is one of the most fundamental metrics on probability measures. It has perceived great interest recently in many applications ranging from computer vision to genetics as it measures the amount of 'work' required to transform one probability distribution into another.
In this talk we provide a statistical perspective on the WD estimated from data. We derive distributional limits for 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. We give an explicit form of the limiting distribution for ultra-metric spaces, e.g. rooted trees. Deviation bounds and bootstrap schemes will be discussed which offer tools for optimal transport based inference for large scale problems. Applications from nanoscale cell imaging and classification are given.
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