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However, what is primarily called into question by people is its lack of theoretical foundation investigations, especially for structured neural networks. This paper theoretically studies the multichannel deep convolutional neural networks equipped with the downsampling operator, which is frequently used in applications. The results show that the proposed networks have outstanding approximation and generalization ability of functions from ridge class and Sobolev space. Not only does it answer an open and crucial question of why multichannel deep convolutional neural networks are universal in learning theory, but it also reveals the convergence rates.<\/jats:p>","DOI":"10.1155\/2023\/8208424","type":"journal-article","created":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T17:35:19Z","timestamp":1684431319000},"page":"1-12","source":"Crossref","is-referenced-by-count":0,"title":["Error Bounds for Approximations Using Multichannel Deep Convolutional Neural Networks with Downsampling"],"prefix":"10.1155","volume":"2023","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4524-6517","authenticated-orcid":true,"given":"Xinling","family":"Liu","sequence":"first","affiliation":[{"name":"Key Laboratory of Optimization Theory and Applications at China West Normal University of Sichuan Province, School of Mathematics and Information, China West Normal University, Nanchong 637009, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7539-8207","authenticated-orcid":true,"given":"Jingyao","family":"Hou","sequence":"additional","affiliation":[{"name":"Key Laboratory of Optimization Theory and Applications at China West Normal University of Sichuan Province, School of Mathematics and Information, China West Normal University, Nanchong 637009, China"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"article-title":"Very deep convolutional networks for large-scale image recognition","year":"2015","author":"K. 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