  We propose a \textbf{uni}fied \textbf{f}ramework for \textbf{i}mplicit
\textbf{ge}nerative \textbf{m}odeling (UnifiGem) with theoretical guarantees by
integrating approaches from optimal transport, numerical ODE, density-ratio
(density-difference) estimation and deep neural networks. First, the problem of
implicit generative learning is formulated as that of finding the optimal
transport map between the reference distribution and the target distribution,
which is characterized by a totally nonlinear Monge-Amp\`{e}re equation.
Interpreting the infinitesimal linearization of the Monge-Amp\`{e}re equation
from the perspective of gradient flows in measure spaces leads to the
continuity equation or the McKean-Vlasov equation. We then solve the
McKean-Vlasov equation numerically using the forward Euler iteration, where the
forward Euler map depends on the density ratio (density difference) between the
distribution at current iteration and the underlying target distribution. We
further estimate the density ratio (density difference) via deep density-ratio
(density-difference) fitting and derive explicit upper bounds on the estimation
error. Experimental results on both synthetic datasets and real benchmark
datasets support our theoretical findings and demonstrate the effectiveness of
UnifiGem.
