{"title":"Global Universality of the Two-Layer Neural Network with the -Rectified Linear Unit","authors":"Naoya Hatano, Masahiro Ikeda, Isao Ishikawa, Yoshihiro Sawano","doi":"10.1155/2024/3262798","DOIUrl":null,"url":null,"abstract":"This paper concerns the universality of the two-layer neural network with the <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 6.66314 9.49473\" width=\"6.66314pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-108\"></use></g></svg>-</span>rectified linear unit activation function with <span><svg height=\"10.8649pt\" style=\"vertical-align:-1.57657pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 17.802 10.8649\" width=\"17.802pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,10.171,0)\"></path></g></svg><span></span><svg height=\"10.8649pt\" style=\"vertical-align:-1.57657pt\" version=\"1.1\" viewbox=\"21.384183800000002 -9.28833 9.204 10.8649\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.434,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,27.674,0)\"></path></g></svg><span></span><svg height=\"10.8649pt\" style=\"vertical-align:-1.57657pt\" version=\"1.1\" viewbox=\"32.7671838 -9.28833 9.204 10.8649\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,32.817,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,39.057,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"10.8649pt\" style=\"vertical-align:-1.57657pt\" version=\"1.1\" viewbox=\"44.1501838 -9.28833 13.505 10.8649\" width=\"13.505pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,44.2,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,49.343,0)\"><use xlink:href=\"#g113-47\"></use></g><g transform=\"matrix(.013,0,0,-0.013,54.486,0)\"><use xlink:href=\"#g113-47\"></use></g></svg></span> with a suitable norm without any restriction on the shape of the domain in the real line. This type of result is called global universality, which extends the previous result for <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 17.802 9.49473\" width=\"17.802pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,10.171,0)\"><use xlink:href=\"#g117-34\"></use></g></svg><span></span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"21.384183800000002 -9.28833 6.416 9.49473\" width=\"6.416pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.434,0)\"><use xlink:href=\"#g113-50\"></use></g></svg></span> by the present authors. This paper covers <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 6.66314 9.49473\" width=\"6.66314pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-108\"></use></g></svg>-</span>sigmoidal functions as an application of the fundamental result on <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 6.66314 9.49473\" width=\"6.66314pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-108\"></use></g></svg>-</span>rectified linear unit functions.","PeriodicalId":15840,"journal":{"name":"Journal of Function Spaces","volume":"28 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Function Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/3262798","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper concerns the universality of the two-layer neural network with the -rectified linear unit activation function with with a suitable norm without any restriction on the shape of the domain in the real line. This type of result is called global universality, which extends the previous result for by the present authors. This paper covers -sigmoidal functions as an application of the fundamental result on -rectified linear unit functions.
期刊介绍：
Journal of Function Spaces (formerly titled Journal of Function Spaces and Applications) publishes papers on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. As well as original research, Journal of Function Spaces also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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