Adaptive Optimal Transport

Abstract: An adaptive, adversarial methodology is developed for the optimal transport problem between two distributions $\mu$ and $\nu$, known only through a finite set of independent samples $(x_i){i=1..N}$ and $(y_j){j=1..M}$. The methodology automatically creates features that adapt to the data, thus avoiding reliance on a priori knowledge of data distribution. Specifically, instead of a discrete point-bypoint assignment, the new procedure seeks an optimal map $T(x)$ defined for all $x$, minimizing the Kullback-Leibler divergence between $(T(xi))$ and the target $(y_j)$. The relative entropy is given a sample-based, variational characterization, thereby creating an adversarial setting: as one player seeks to push forward one distribution to the other, the second player develops features that focus on those areas where the two distributions fail to match. The procedure solves local problems matching consecutive, intermediate distributions between $\mu$ and $\nu$. As a result, maps of arbitrary complexity can be built by composing the simple maps used for each local problem. Displaced interpolation is used to guarantee global from local optimality. The procedure is illustrated through synthetic examples in one and two dimensions.