Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions
This work addresses the challenge of generative modeling for complicated datasets, offering a novel parameter-free approach with theoretical guarantees, which is incremental as it builds on existing connections between implicit generative modeling and optimal transport.
The authors tackled the problem of learning and sampling from complex data distributions by proposing a nonparametric generative modeling algorithm based on optimal transport and gradient flows, achieving finite-time error guarantees and successfully capturing data structures in experiments.
By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm 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 probability measures. The connections between gradient 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 algorithm is the first nonparametric IGM algorithm with explicit theoretical guarantees. Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.