  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.
