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

By building upon the recent theory that estab- lished the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algo- rithm for learning the underlying distributions of complicated datasets and sampling from them. The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes. We formulate the problem as a gradient flow in the space of proba- bility measures. The connections between gradi- ent flows and stochastic differential equations let us develop a computationally efficient algorithm for solving the optimization problem. We provide formal theoretical analysis where we prove finite- time error guarantees for the proposed algorithm. To the best of our knowledge, the proposed algo- rithm is the first nonparametric IGM algorithm with explicit theoretical guarantees. Our experi- mental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.