Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal\n Transport and Diffusions

Abstract

By building upon the recent theory that established the connection between\nimplicit generative modeling (IGM) and optimal transport, in this study, we\npropose a novel parameter-free algorithm for learning the underlying\ndistributions of complicated datasets and sampling from them. The proposed\nalgorithm is based on a functional optimization problem, which aims at finding\na measure that is close to the data distribution as much as possible and also\nexpressive enough for generative modeling purposes. We formulate the problem as\na gradient flow in the space of probability measures. The connections between\ngradient flows and stochastic differential equations let us develop a\ncomputationally efficient algorithm for solving the optimization problem. We\nprovide formal theoretical analysis where we prove finite-time error guarantees\nfor the proposed algorithm. To the best of our knowledge, the proposed\nalgorithm is the first nonparametric IGM algorithm with explicit theoretical\nguarantees. Our experimental results support our theory and show that our\nalgorithm is able to successfully capture the structure of different types of\ndata distributions.\n

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