Title: Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions,
Abstract: By building up on the recent theory that established the connection between
implicit generative modeling 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, where the resulting
algorithm resembles the recent dynamics-based Markov Chain Monte Carlo
algorithms. We provide formal theoretical analysis where we prove finite-time
error guarantees for the proposed algorithm. Our experimental results support
our theory and shows that our algorithm is able to capture the structure of
challenging distributions.