{"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/architopes-an-architecture-modification-for","title":"A Canonical Transform for Strengthening the Local $L^p$-Type Universal Approximation Property","arxiv_id":"2006.14378","date":"2020-06-24","proceeding":null,"authors":["Anastasis Kratsios","Behnoosh Zamanlooy"],"abstract":"Most $L^p$-type universal approximation theorems guarantee that a given machine learning model class $\\mathscr{F}\\subseteq C(\\mathbb{R}^d,\\mathbb{R}^D)$ is dense in $L^p_{\\mu}(\\mathbb{R}^d,\\mathbb{R}^D)$ for any suitable finite Borel measure $\\mu$ on $\\mathbb{R}^d$. Unfortunately, this means that the model's approximation quality can rapidly degenerate outside some compact subset of $\\mathbb{R}^d$, as any such measure is largely concentrated on some bounded subset of $\\mathbb{R}^d$. This paper proposes a generic solution to this approximation theoretic problem by introducing a canonical transformation which \"upgrades $\\mathscr{F}$'s approximation property\" in the following sense. The transformed model class, denoted by $\\mathscr{F}\\text{-tope}$, is shown to be dense in $L^p_{\\mu,\\text{strict}}(\\mathbb{R}^d,\\mathbb{R}^D)$ which is a topological space whose elements are locally $p$-integrable functions and whose topology is much finer than usual norm topology on $L^p_{\\mu}(\\mathbb{R}^d,\\mathbb{R}^D)$; here $\\mu$ is any suitable $\\sigma$-finite Borel measure $\\mu$ on $\\mathbb{R}^d$. Next, we show that if $\\mathscr{F}$ is any family of analytic functions then there is always a strict \"gap\" between $\\mathscr{F}\\text{-tope}$'s expressibility and that of $\\mathscr{F}$, since we find that $\\mathscr{F}$ can never dense in $L^p_{\\mu,\\text{strict}}(\\mathbb{R}^d,\\mathbb{R}^D)$. In the general case, where $\\mathscr{F}$ may contain non-analytic functions, we provide an abstract form of these results guaranteeing that there always exists some function space in which $\\mathscr{F}\\text{-tope}$ is dense but $\\mathscr{F}$ is not, while, the converse is never possible. Applications to feedforward networks, convolutional neural networks, and polynomial bases are explored.","url_abs":"https://arxiv.org/abs/2006.14378v3","url_pdf":"https://arxiv.org/pdf/2006.14378v3.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":"architopes-an-architecture-modification-for","repo_url":"https://github.com/bzamanlooy/Architopes","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"architopes-an-architecture-modification-for","repo_url":"https://github.com/AnastasisKratsios/Random_Lipschitz_Partition","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"sigmoid-activation","method_name":"Sigmoid Activation"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}