  A novel algorithm is proposed to solve the sample-based optimal transport
problem. An adversarial formulation of the push-forward condition uses a test
function built as a convolution between an adaptive kernel and an evolving
probability distribution $\nu$ over a latent variable $b$. Approximating this
convolution by its simulation over evolving samples $b^i(t)$ of $\nu$, the
parameterization of the test function reduces to determining the flow of these
samples. This flow, discretized over discrete time steps $t_n$, is built from
the composition of elementary maps. The optimal transport also follows a flow
that, by duality, must follow the gradient of the test function. The
representation of the test function as the Monte Carlo simulation of a
distribution makes the algorithm robust to dimensionality, and its evolution
under a memory-less flow produces rich, complex maps from simple parametric
transformations. The algorithm is illustrated with numerical examples.
