{"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/an-adaptive-transport-framework-for-joint-and","title":"On the representation and learning of monotone triangular transport maps","arxiv_id":"2009.10303","date":"2020-09-22","proceeding":null,"authors":["Ricardo Baptista","Youssef Marzouk","Olivier Zahm"],"abstract":"Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps$\\unicode{x2014}$approximations of the Knothe$\\unicode{x2013}$Rosenblatt (KR) rearrangement$\\unicode{x2014}$are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.","url_abs":"https://arxiv.org/abs/2009.10303v3","url_pdf":"https://arxiv.org/pdf/2009.10303v3.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":"abstracts"},"code_links":[{"paper_slug":"an-adaptive-transport-framework-for-joint-and","repo_url":"https://github.com/baptistar/atm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[{"task_slug":"bayesian-inference","task_name":"Bayesian Inference"},{"task_slug":"density-estimation","task_name":"Density Estimation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2009.10303","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}