{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T10:05:56Z","timestamp":1790503556829,"version":"4.1.0"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101401"],"award-info":[{"award-number":["12101401"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003399","name":"Science and Technology Commission of Shanghai Municipality","doi-asserted-by":"publisher","award":["2021SHZDZX0102"],"award-info":[{"award-number":["2021SHZDZX0102"]}],"id":[{"id":"10.13039\/501100003399","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Journal on Mathematics of Data Science"],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:p>In an attempt to better understand structural benefits and generalization power of deep neural networks, we first present a novel graph theoretical formulation of neural network models, including fully connected, residual network (ResNet) and densely connected networks (DenseNet). Second, we extend the error analysis of the population risk for a two-layer network [W. E., C. Ma, and L. Wu, Commun. Math. Sci., 17 (2019), pp. 1407--1425] and ResNet [W. E., C. Ma, and Q. Wang, Commun. Math. Sci., 18 (2020), pp. 1755--1774] to DenseNet, and show further that for neural networks satisfying certain mild conditions, similar estimates can be obtained. These estimates are a priori in nature since they depend solely on the information prior to the training process, in particular, the bounds for the estimation errors do not suffer from the curse of dimensionality.<\/jats:p>","DOI":"10.1137\/21m140955x","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T10:33:37Z","timestamp":1655116417000},"page":"694-720","source":"Crossref","is-referenced-by-count":3,"title":["Nonlinear Weighted Directed Acyclic Graph and A Priori Estimates for Neural Networks"],"prefix":"10.1137","volume":"4","author":[{"given":"Yuqing","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2029-0362","authenticated-orcid":true,"given":"Tao","family":"Luo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chao","family":"Ma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,6,13]]},"reference":[{"key":"atypb1","volume-title":"Advances in Neural Information Processing Systems, H. 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