Heinz H. Bauschke, Manish Krishan Lal, Xianfu Wang
{"title":"Projecting onto rectangular hyperbolic paraboloids in Hilbert space","authors":"Heinz H. Bauschke, Manish Krishan Lal, Xianfu Wang","doi":"10.23952/asvao.5.2023.2.04","DOIUrl":null,"url":null,"abstract":"In $\\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\\mathbb{R}^n$. Motivated by his work, we provide a rigorous analysis of the associated projection. In some cases, finding this projection amounts to finding a certain root of a quintic or cubic polynomial. We also observe when the projection is not a singleton and point out connections to graphical and set convergence.","PeriodicalId":362333,"journal":{"name":"Applied Set-Valued Analysis and Optimization","volume":"29 2","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Set-Valued Analysis and Optimization","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.23952/asvao.5.2023.2.04","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we provide a rigorous analysis of the associated projection. In some cases, finding this projection amounts to finding a certain root of a quintic or cubic polynomial. We also observe when the projection is not a singleton and point out connections to graphical and set convergence.