{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T19:10:17Z","timestamp":1754161817870,"version":"3.41.2"},"reference-count":40,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"name":"MUR Excellence Department","award":["MatMod@TOV"],"award-info":[{"award-number":["MatMod@TOV"]}]},{"name":"Department of Mathematics"},{"name":"University of Rome Tor Vergata","award":["CUP E83C18000100006"],"award-info":[{"award-number":["CUP E83C18000100006"]}]},{"name":"GRAFIA","award":["202284Z9E4"],"award-info":[{"award-number":["202284Z9E4"]}]},{"name":"INdAM group GNAMPA"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Journal on Mathematics of Data Science"],"published-print":{"date-parts":[[2025,9,30]]},"DOI":"10.1137\/24m1675746","type":"journal-article","created":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T09:58:14Z","timestamp":1753783094000},"page":"1154-1183","source":"Crossref","is-referenced-by-count":0,"title":["Spectral Complexity of Deep Neural Networks"],"prefix":"10.1137","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-6391-3734","authenticated-orcid":true,"given":"Simmaco","family":"Di Lillo","sequence":"first","affiliation":[{"name":"RoMaDS, Department of Mathematics, University of Rome Tor Vergata, Rome, Italy."}]},{"given":"Domenico","family":"Marinucci","sequence":"additional","affiliation":[{"name":"RoMaDS, Department of Mathematics, University of Rome Tor Vergata, Rome, Italy."}]},{"given":"Michele","family":"Salvi","sequence":"additional","affiliation":[{"name":"RoMaDS, Department of Mathematics, University of Rome Tor Vergata, Rome, Italy."}]},{"given":"Stefano","family":"Vigogna","sequence":"additional","affiliation":[{"name":"RoMaDS, Department of Mathematics, University of Rome Tor Vergata, Rome, Italy."}]}],"member":"351","published-online":{"date-parts":[[2025,7,29]]},"reference":[{"key":"ref1","volume-title":"Random Fields and Geometry","author":"Adler R. J.","year":"2007"},{"key":"ref2","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","volume":"68","author":"Aronszajn N.","year":"1950","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref3","series-title":"Lecture Notes in Math. 2044","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-25983-8","volume-title":"Spherical Harmonics and Approximations on the Unit Sphere: An Introduction","author":"Atkinson K.","year":"2012"},{"key":"ref4","doi-asserted-by":"crossref","DOI":"10.1002\/9780470434642","volume-title":"Level Sets and Extrema of Random Processes and Fields","author":"Azais J.-M.","year":"2009"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/22M147517X"},{"key":"ref6","doi-asserted-by":"crossref","first-page":"101668,","DOI":"10.1016\/j.acha.2024.101668","volume":"72","author":"Balasubramanian K.","year":"2024","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref7","volume-title":"in Advances in Neural Information Processing Systems (NeurIPS)","volume":"30","author":"Bartlett P.","year":"2017"},{"key":"ref8","first-page":"1","volume":"20","author":"Bartlett P.","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref9","volume-title":"in Advances in Neural Information Processing Systems (NeurIPS)","volume":"11","author":"Bartlett P.","year":"1998"},{"key":"ref10","volume-title":"in Advances in Neural Information Processing Systems (NeurIPS)","volume":"17","author":"Bertschinger N.","year":"2004"},{"key":"ref11","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1109\/TNNLS.2013.2293637","volume":"25","author":"Bianchini M.","year":"2014","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref14","first-page":"1","volume":"20","author":"Bohn B.","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref16","doi-asserted-by":"crossref","first-page":"85","DOI":"10.15559\/23-VMSTA238","volume":"11","author":"Cammarota V.","year":"2024","journal-title":"Modern Stoch. Theory Appl."},{"key":"ref17","volume-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume":"22","author":"Cho Y.","year":"2009"},{"key":"ref19","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","volume":"2","author":"Cybenko G.","year":"1989","journal-title":"Math. Control Signals Systems"},{"key":"ref21","volume-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume":"29","author":"Daniely A.","year":"2016"},{"key":"ref22","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s00365-021-09548-z","volume":"55","author":"Daubechies I.","year":"2022","journal-title":"Constr. Approx."},{"key":"ref26","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1007\/s00440-025-01360-1","volume":"191","author":"Favaro S.","year":"2025","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref29","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1086\/427976","volume":"622","author":"Gorski K. M.","year":"2005","journal-title":"Astrophys. J."},{"key":"ref30","doi-asserted-by":"crossref","first-page":"115667","DOI":"10.1016\/j.cam.2023.115667","volume":"441","author":"Goujon A.","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"ref31","doi-asserted-by":"crossref","first-page":"4798","DOI":"10.1214\/23-AAP1933","volume":"33","author":"Hanin B.","year":"2023","journal-title":"Ann. Appl. Probab."},{"key":"ref33","volume-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume":"32","author":"Hanin B.","year":"2019"},{"key":"ref35","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0893-6080(91)90009-T","volume":"4","author":"Hornik K.","year":"1991","journal-title":"Neural Netw."},{"key":"ref36","first-page":"1","volume":"24","author":"Huang W.","year":"2023","journal-title":"J. Mach. Learn. Res."},{"key":"ref37","series-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume-title":"Convergence and Generalization in Neural Networks","volume":"31","author":"Jacot A.","year":"2018"},{"key":"ref38","doi-asserted-by":"crossref","first-page":"3047","DOI":"10.1214\/14-AAP1067","volume":"25","author":"Lang A.","year":"2015","journal-title":"Ann. Appl. Probab."},{"key":"ref40","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/S0893-6080(05)80131-5","volume":"6","author":"Leshno M.","year":"1993","journal-title":"Neural Netw."},{"key":"ref41","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511751677","volume-title":"Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications","author":"Marinucci D.","year":"2011"},{"key":"ref42","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1142\/S0219530516400042","volume":"14","author":"Mhaskar H. N.","year":"2016","journal-title":"Anal. Appl."},{"key":"ref43","volume-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume":"27","author":"Montufar G. F.","year":"2014"},{"key":"ref44","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0745-0","volume-title":"Bayesian Learning for Neural Networks","author":"Neal R. M.","year":"1996"},{"key":"ref45","first-page":"1","volume":"247","author":"Parkinson S.","year":"2024","journal-title":"J. Mach. Learn. Res."},{"key":"ref46","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1017\/S0962492900002919","volume":"8","author":"Pinkus A.","year":"1999","journal-title":"Acta Numer."},{"key":"ref47","volume-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume":"29","author":"Poole B.","year":"2016"},{"key":"ref48","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/3206.001.0001","volume-title":"Gaussian Processes for Machine Learning","author":"Rasmussen C. E.","year":"2005"},{"key":"ref50","first-page":"1875","volume":"48","author":"Schmidt-Hieber J.","year":"2020","journal-title":"Ann. Statist."},{"key":"ref51","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1215\/S0012-7094-42-00908-6","volume":"9","author":"Schoenberg I. J.","year":"1942","journal-title":"Duke Math. J."},{"key":"ref54","volume-title":"Orthogonal Polynomials","author":"Szeg\u0151 G.","year":"1975"},{"key":"ref56","first-page":"1","volume":"23","author":"Venturi L.","year":"2022","journal-title":"J. Mach. Learn. Res."},{"key":"ref57","volume-title":"Advances in Neural Information Processing Systems (NeurIPS)","volume":"9","author":"Williams C.","year":"1996"}],"container-title":["SIAM Journal on Mathematics of Data Science"],"original-title":[],"language":"en","deposited":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T09:58:31Z","timestamp":1753783111000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/24M1675746"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,29]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9,30]]}},"alternative-id":["10.1137\/24M1675746"],"URL":"https:\/\/doi.org\/10.1137\/24m1675746","relation":{},"ISSN":["2577-0187"],"issn-type":[{"value":"2577-0187","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,29]]}}}