Vincent Divol (NYU)
Title: Measure estimation on manifolds through optimal transport
Abstract. The Wasserstein distances Wp are measures of similarity between probability distributions that have found numerous applications in machine learning. For example, Wasserstein Generative Adversarial Networks are able to generate realistic fake images by approximating the empirical distribution of a sample of images with respect to W1. From a statistical perspective, the question of the estimation of quantities related to the optimal transport problem is then raised. We will present two settings where one can bypass the curse of dimensionality. First, in the case where the target distribution is supported on a low-dimensional unknown submanifold. Second, in the case where the target distribution is obtained as the pushforward of the gaussian distribution through some map that is known to belong to a given functional class F. In the latter case, we are able to obtain fast rates of estimation that depend uniquely on the metric entropy of the class F.
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