Data Driven Conditional Optimal Transport

TL;DR

This paper introduces a data-driven method for computing conditional optimal transport maps that depend on covariates, demonstrated through synthetic and real-world image transfer applications.

Contribution

It presents a novel data-driven approach for conditional optimal transport, extending traditional methods to incorporate covariate dependence.

Findings

Successfully applied to synthetic data from ACIC Challenge 2017

Effective in non-uniform lightness transfer between images

Highlights differences with ordinary optimal transport

Abstract

A data driven procedure is developed to compute the optimal map between two conditional probabilities $\rho(x|z_{1},...,z_{L})$ and $\mu(y|z_{1},...,z_{L})$ depending on a set of covariates $z_{i}$. The procedure is tested on synthetic data from the ACIC Data Analysis Challenge 2017 and it is applied to non uniform lightness transfer between images. Exactly solvable examples and simulations are performed to highlight the differences with ordinary optimal transport.