Data Driven Conditional Optimal Transport

Abstract

A data-driven procedure is developed to compute the optimal map between two conditional probabilities ρ(x|z1,…,zL)\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\rho (x|z_{1},\ldots ,z_{L})$\end{document} and μ(y|z1,…,zL)\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\mu (y|z_{1},\ldots ,z_{L})$\end{document}, known only through samples and depending on a set of covariates zl\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$z_{l}$\end{document}. 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.