{"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/universal-regular-conditional-distributions","title":"Universal Regular Conditional Distributions","arxiv_id":"2105.07743","date":"2021-05-17","proceeding":null,"authors":["Anastasis Kratsios"],"abstract":"We introduce a deep learning model that can universally approximate regular conditional distributions (RCDs). The proposed model operates in three phases: first, it linearizes inputs from a given metric space $\\mathcal{X}$ to $\\mathbb{R}^d$ via a feature map, then a deep feedforward neural network processes these linearized features, and then the network's outputs are then transformed to the $1$-Wasserstein space $\\mathcal{P}_1(\\mathbb{R}^D)$ via a probabilistic extension of the attention mechanism of Bahdanau et al.\\ (2014). Our model, called the \\textit{probabilistic transformer (PT)}, can approximate any continuous function from $\\mathbb{R}^d $ to $\\mathcal{P}_1(\\mathbb{R}^D)$ uniformly on compact sets, quantitatively. We identify two ways in which the PT avoids the curse of dimensionality when approximating $\\mathcal{P}_1(\\mathbb{R}^D)$-valued functions. The first strategy builds functions in $C(\\mathbb{R}^d,\\mathcal{P}_1(\\mathbb{R}^D))$ which can be efficiently approximated by a PT, uniformly on any given compact subset of $\\mathbb{R}^d$. In the second approach, given any function $f$ in $C(\\mathbb{R}^d,\\mathcal{P}_1(\\mathbb{R}^D))$, we build compact subsets of $\\mathbb{R}^d$ whereon $f$ can be efficiently approximated by a PT.","url_abs":"https://arxiv.org/abs/2105.07743v5","url_pdf":"https://arxiv.org/pdf/2105.07743v5.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":"universal-regular-conditional-distributions","repo_url":"https://github.com/AnastasisKratsios/Universal_Regular_Conditional_Distributions_Kratsios_2021","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2105.07743","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}