Adversarial Optimal Transport Through The Convolution Of Kernels With Evolving Measures

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.