{"title":"具有 \"整流线性单元 \"的双层神经网络的全局通用性","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":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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. 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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\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-01-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1155/2024/3262798\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/3262798","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Global Universality of the Two-Layer Neural Network with the -Rectified Linear Unit
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.
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Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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