{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T22:18:04Z","timestamp":1773094684783,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2017,4,26]],"date-time":"2017-04-26T00:00:00Z","timestamp":1493164800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Grantov\u00c3\u00a1 Agentura \u00c4\u0152esk\u00c3\u00a9 Republiky","doi-asserted-by":"publisher","award":["GA15-18108S"],"award-info":[{"award-number":["GA15-18108S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput &amp; Applic"],"published-print":{"date-parts":[[2018,4]]},"DOI":"10.1007\/s00521-017-2965-0","type":"journal-article","created":{"date-parts":[[2017,4,26]],"date-time":"2017-04-26T07:42:52Z","timestamp":1493192572000},"page":"305-315","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Constructive lower bounds on model complexity of shallow perceptron networks"],"prefix":"10.1007","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8181-2128","authenticated-orcid":false,"given":"V\u011bra","family":"K\u016frkov\u00e1","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,4,26]]},"reference":[{"key":"2965_CR1","unstructured":"Ba LJ, Caruana R (2014) Do deep networks really need to be deep? In: Ghahrani Z et al (eds) Advances in neural information processing systems, vol 27, pp 1\u20139"},{"key":"2965_CR2","unstructured":"Barron AR (1992) Neural net approximation. In: Narendra K (ed) Proceedings 7th Yale workshop on adaptive and learning systems, pp 69\u201372. Yale University Press"},{"key":"2965_CR3","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1109\/18.256500","volume":"39","author":"AR Barron","year":"1993","unstructured":"Barron AR (1993) Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans Inf Theory 39:930\u2013945","journal-title":"IEEE Trans Inf Theory"},{"key":"2965_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000006","volume":"2","author":"Y Bengio","year":"2009","unstructured":"Bengio Y (2009) Learning deep architectures for AI. Foundations and Trends in Machine Learning 2:1\u2013127","journal-title":"Foundations and Trends in Machine Learning"},{"key":"2965_CR5","unstructured":"Bengio Y, Delalleau O, Roux NL (2006) The curse of highly variable functions for local kernel machines. In: Advances in neural information processing systems 18, pp 107\u2013114. MIT Press"},{"key":"2965_CR6","doi-asserted-by":"crossref","unstructured":"Bengio Y, LeCun Y (2007) Scaling learning algorithms towards AI. In: Bottou LO, Chapelle D, DeCoste, Weston J (eds) Large-Scale Kernel Machines. MIT Press","DOI":"10.7551\/mitpress\/7496.003.0016"},{"issue":"8","key":"2965_CR7","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1109\/TNNLS.2013.2293637","volume":"25","author":"M Bianchini","year":"2014","unstructured":"Bianchini M, Scarselli F (2014) On the complexity of neural network classifiers: a comparison between shallow and deep architectures. IEEE Trans Neural Netw Learning Syst 25(8):1553\u20131565","journal-title":"IEEE Trans Neural Netw Learning Syst"},{"key":"2965_CR8","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/j.crma.2008.03.014","volume":"346","author":"EJ Cand\u00e8s","year":"2008","unstructured":"Cand\u00e8s EJ (2008) The restricted isometric property and its implications for compressed sensing. C R Acad Sci Paris I 346:589\u2013592","journal-title":"C R Acad Sci Paris I"},{"key":"2965_CR9","doi-asserted-by":"crossref","first-page":"4203","DOI":"10.1109\/TIT.2005.858979","volume":"51","author":"EJ Cand\u00e8s","year":"2005","unstructured":"Cand\u00e8s EJ, Tao T (2005) Decoding by linear programming. IEEE Trans Inf Process 51:4203\u20134215","journal-title":"IEEE Trans Inf Process"},{"key":"2965_CR10","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1109\/PGEC.1965.264137","volume":"14","author":"T Cover","year":"1965","unstructured":"Cover T (1965) Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Trans Electron Comput 14:326\u2013334","journal-title":"IEEE Trans Electron Comput"},{"key":"2965_CR11","unstructured":"Erd\u00f6s P, Spencer JH (1974) Probabilistic methods in combinatorics. Academic Press"},{"key":"2965_CR12","volume-title":"Feedforward neural network methodology","author":"TL Fine","year":"1999","unstructured":"Fine TL (1999) Feedforward neural network methodology. Springer, Berlin Heidelberg"},{"key":"2965_CR13","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1109\/TIT.2010.2090198","volume":"57","author":"G Gnecco","year":"2011","unstructured":"Gnecco G, Sanguineti M (2011) On a variational norm tailored to variable-basis approximation schemes. IEEE Trans Inf Theory 57:549\u2013558","journal-title":"IEEE Trans Inf Theory"},{"key":"2965_CR14","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1162\/neco.2006.18.7.1527","volume":"18","author":"GE Hinton","year":"2006","unstructured":"Hinton GE, Osindero S, Teh YW (2006) A fast learning algorithm for deep belief nets. Neural Comput 18:1527\u20131554","journal-title":"Neural Comput"},{"key":"2965_CR15","first-page":"69","volume":"17","author":"Y Ito","year":"1992","unstructured":"Ito Y (1992) Finite mapping by neural networks and truth functions. Mathematical Scientist 17:69\u201377","journal-title":"Mathematical Scientist"},{"key":"2965_CR16","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1109\/TIT.2011.2169531","volume":"58","author":"PC Kainen","year":"2012","unstructured":"Kainen PC, K\u016frkov\u00e1 V, Sanguineti M (2012) Dependence of computational models on input dimension: tractability of approximation and optimization tasks. IEEE Trans Inf Theory 58:1203\u20131214","journal-title":"IEEE Trans Inf Theory"},{"key":"2965_CR17","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/S0925-2312(99)00111-3","volume":"29","author":"PC Kainen","year":"1999","unstructured":"Kainen PC, K\u016frkov\u00e1 V, Vogt A (1999) Approximation by neural networks is not continuous. Neurocomputing 29:47\u201356","journal-title":"Neurocomputing"},{"key":"2965_CR18","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1006\/jath.2000.3467","volume":"105","author":"PC Kainen","year":"2000","unstructured":"Kainen PC, K\u016frkov\u00e1 V, Vogt A (2000) Geometry and topology of continuous best and near best approximations. J Approx Theory 105:252\u2013262","journal-title":"J Approx Theory"},{"key":"2965_CR19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jat.2006.12.009","volume":"147","author":"PC Kainen","year":"2007","unstructured":"Kainen PC, K\u016frkov\u00e1 V, Vogt A (2007) A Sobolev-type upper bound for rates of approximation by linear combinations of Heaviside plane waves. J Approx Theory 147:1\u201310","journal-title":"J Approx Theory"},{"key":"2965_CR20","volume-title":"Learning and soft computing","author":"V Kecman","year":"2001","unstructured":"Kecman V (2001) Learning and soft computing. MIT Press, Cambridge"},{"key":"2965_CR21","doi-asserted-by":"crossref","unstructured":"K\u016frkov\u00e1 V (1997) Dimension-independent rates of approximation by neural networks. In: Warwick K, K\u00e1rn\u00fd M (eds) Computer-intensive methods in control and signal processing. The curse of dimensionality, pp 261\u2013270. Birkh\u00e4user, Boston","DOI":"10.1007\/978-1-4612-1996-5_16"},{"key":"2965_CR22","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1162\/neco.2008.20.1.252","volume":"20","author":"V K\u016frkov\u00e1","year":"2008","unstructured":"K\u016frkov\u00e1 V (2008) Minimization of error functionals over perceptron networks. Neural Comput 20:250\u2013270","journal-title":"Neural Comput"},{"key":"2965_CR23","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.neunet.2012.05.002","volume":"33","author":"V K\u016frkov\u00e1","year":"2012","unstructured":"K\u016frkov\u00e1 V (2012) Complexity estimates based on integral transforms induced by computational units. Neural Netw 33:160\u2013167","journal-title":"Neural Netw"},{"key":"2965_CR24","doi-asserted-by":"crossref","unstructured":"K\u016frkov\u00e1 V (2016) Lower bounds on complexity of shallow perceptron networks. In: Jayne C, Iliadis L (eds) Engineering applications of neural networks. Communications in computer and information sciences, vol 629, pp 283\u2013294. Springer","DOI":"10.1007\/978-3-319-44188-7_21"},{"issue":"3","key":"2965_CR25","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1162\/neco.1994.6.3.543","volume":"6","author":"V K\u016frkov\u00e1","year":"1994","unstructured":"K\u016frkov\u00e1 V, Kainen PC (1994) Functionally equivalent feedforward neural networks. Neural Comput 6 (3):543\u2013558","journal-title":"Neural Comput"},{"issue":"2","key":"2965_CR26","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/0893-9659(96)00008-0","volume":"9","author":"V K\u016frkov\u00e1","year":"1996","unstructured":"K\u016frkov\u00e1 V, Kainen PC (1996) Singularities of finite scaling functions. Appl Math Lett 9(2):33\u201337","journal-title":"Appl Math Lett"},{"key":"2965_CR27","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.neunet.2014.05.005","volume":"57","author":"V K\u016frkov\u00e1","year":"2014","unstructured":"K\u016frkov\u00e1 V, Kainen PC (2014) Comparing fixed and variable-width Gaussian kernel networks. Neural Netw 57:23\u201328","journal-title":"Neural Netw"},{"key":"2965_CR28","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/18.971754","volume":"48","author":"V K\u016frkov\u00e1","year":"2002","unstructured":"K\u016frkov\u00e1 V, Sanguineti M (2002) Comparison of worst-case errors in linear and neural network approximation. IEEE Trans Inf Theory 48:264\u2013275","journal-title":"IEEE Trans Inf Theory"},{"key":"2965_CR29","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1287\/moor.1080.0317","volume":"33","author":"V K\u016frkov\u00e1","year":"2008","unstructured":"K\u016frkov\u00e1 V, Sanguineti M (2008) Approximate minimization of the regularized expected error over kernel models. Math Oper Res 33:747\u2013756","journal-title":"Math Oper Res"},{"key":"2965_CR30","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1016\/j.neucom.2015.07.014","volume":"171","author":"V K\u016frkov\u00e1","year":"2016","unstructured":"K\u016frkov\u00e1 V, Sanguineti M (2016) Model complexities of shallow networks representing highly varying functions. Neurocomputing 171:598\u2013604","journal-title":"Neurocomputing"},{"key":"2965_CR31","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/S0893-6080(98)00039-2","volume":"11","author":"V K\u016frkov\u00e1","year":"1998","unstructured":"K\u016frkov\u00e1 V, Savick\u00fd P, Hlav\u00e1\u010dkov\u00e1 K (1998) Representations and rates of approximation of real-valued Boolean functions by neural networks. Neural Netw 11:651\u2013659","journal-title":"Neural Netw"},{"key":"2965_CR32","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","volume":"521","author":"Y LeCunn","year":"2015","unstructured":"LeCunn Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521:436\u2013444","journal-title":"Nature"},{"key":"2965_CR33","unstructured":"MacWilliams F, Sloane NJA (1977) The theory of error-correcting codes. North-Holland, Amsterdam"},{"key":"2965_CR34","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0925-2312(98)00111-8","volume":"25","author":"V Maiorov","year":"1999","unstructured":"Maiorov V, Pinkus A (1999) Lower bounds for approximation by MLP neural networks. Neurocomputing 25:81\u201391","journal-title":"Neurocomputing"},{"key":"2965_CR35","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1023\/A:1018993908478","volume":"13","author":"VE Maiorov","year":"2000","unstructured":"Maiorov VE, Meir R (2000) On the near optimality of the stochastic approximation of smooth functions by neural networks. Adv Comput Math 13:79\u2013103","journal-title":"Adv Comput Math"},{"key":"2965_CR36","unstructured":"Mhaskar HN, Liao Q, Poggio T (2016) Learning functions: when is deep better than shallow. Center for brains, minds & machines CBMM Memo No. 045v3, pp 1\u201312"},{"key":"2965_CR37","unstructured":"Mhaskar HN, Liao Q, Poggio T (2016) Learning functions: when is deep better than shallow. Center for brains, minds & machines CBMM Memo No. 045v4, pp 1\u201312"},{"key":"2965_CR38","unstructured":"Sloane NJA A library of Hadamard matrices. http:\/\/www.research.att.com\/njas\/hadamard\/"},{"issue":"4","key":"2965_CR39","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/S0893-6080(05)80037-1","volume":"5","author":"HJ Sussman","year":"1992","unstructured":"Sussman HJ (1992) Uniqueness of the weights for minimal feedforward nets with a given input-output map. Neural Netw 5(4):589\u2013593","journal-title":"Neural Netw"},{"key":"2965_CR40","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1080\/14786446708639914","volume":"34","author":"J Sylvester","year":"1867","unstructured":"Sylvester J (1867) Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton\u2019s rule, ornamental tile-work, and the theory of numbers. Phil Mag 34:461\u2013 475","journal-title":"Phil Mag"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00521-017-2965-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-017-2965-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-017-2965-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T08:36:06Z","timestamp":1692779766000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00521-017-2965-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,26]]},"references-count":40,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2018,4]]}},"alternative-id":["2965"],"URL":"https:\/\/doi.org\/10.1007\/s00521-017-2965-0","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,4,26]]}}}