{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/vector-quantile-regression-an-optimal","title":"Vector Quantile Regression: An Optimal Transport Approach","arxiv_id":"1406.4643","date":"2014-06-18","proceeding":null,"authors":["Guillaume Carlier","Victor Chernozhukov","Alfred Galichon"],"abstract":"We propose a notion of conditional vector quantile function and a vector quantile regression. A \\emph{conditional vector quantile function} (CVQF) of a random vector $Y$, taking values in $\\mathbb{R}^d$ given covariates $Z=z$, taking values in $\\mathbb{R}% ^k$, is a map $u \\longmapsto Q_{Y\\mid Z}(u,z)$, which is monotone, in the sense of being a gradient of a convex function, and such that given that vector $U$ follows a reference non-atomic distribution $F_U$, for instance uniform distribution on a unit cube in $\\mathbb{R}^d$, the random vector $Q_{Y\\mid Z}(U,z)$ has the distribution of $Y$ conditional on $Z=z$. Moreover, we have a strong representation, $Y = Q_{Y\\mid Z}(U,Z)$ almost surely, for some version of $U$. The \\emph{vector quantile regression} (VQR) is a linear model for CVQF of $Y$ given $Z$. Under correct specification, the notion produces strong representation, $Y=\\beta \\left(U\\right) ^\\top f(Z)$, for $f(Z)$ denoting a known set of transformations of $Z$, where $u \\longmapsto \\beta(u)^\\top f(Z)$ is a monotone map, the gradient of a convex function, and the quantile regression coefficients $u \\longmapsto \\beta(u)$ have the interpretations analogous to that of the standard scalar quantile regression. As $f(Z)$ becomes a richer class of transformations of $Z$, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where $Y$ is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.","url_abs":"https://arxiv.org/abs/1406.4643v4","url_pdf":"https://arxiv.org/pdf/1406.4643v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"vector-quantile-regression-an-optimal","repo_url":"https://github.com/DatenBiene/Vector_Quantile_Regression","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}