Adaptive Optimal Transport

TL;DR

This paper introduces an adaptive, adversarial approach for optimal transport that learns data-specific features and constructs complex maps by solving local problems, demonstrated through synthetic 1D and 2D examples.

Contribution

It develops a novel adaptive, adversarial method for optimal transport that avoids prior distribution knowledge and constructs complex maps via local problem solutions.

Findings

Successfully constructs complex transport maps from local solutions.

Avoids reliance on predefined features or distributions.

Demonstrates effectiveness through synthetic 1D and 2D examples.

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.