{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T20:33:33Z","timestamp":1762979613081},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2008,3,21]],"date-time":"2008-03-21T00:00:00Z","timestamp":1206057600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput Manag Sci"],"published-print":{"date-parts":[[2009,2]]},"DOI":"10.1007\/s10287-008-0072-5","type":"journal-article","created":{"date-parts":[[2008,3,20]],"date-time":"2008-03-20T08:56:20Z","timestamp":1206003380000},"page":"53-79","source":"Crossref","is-referenced-by-count":30,"title":["The weight-decay technique in learning from data: an optimization point of view"],"prefix":"10.1007","volume":"6","author":[{"given":"Giorgio","family":"Gnecco","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcello","family":"Sanguineti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2008,3,21]]},"reference":[{"key":"72_CR1","unstructured":"Aarts E, Korst J (1989) Simulated annealing and Boltzmann machines: a stochastic approach to combinatorial optimization and neural computing. Wiley,"},{"key":"72_CR2","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","volume":"68","author":"N Aronszajn","year":"1950","unstructured":"Aronszajn N (1950) Theory of reproducing kernels. Trans AMS 68: 337\u2013404","journal-title":"Trans AMS"},{"issue":"2","key":"72_CR3","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1109\/18.661502","volume":"44","author":"PL Bartlett","year":"1998","unstructured":"Bartlett PL (1998) The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network. IEEE Trans Inf Theory 44(2): 525\u2013536","journal-title":"IEEE Trans Inf Theory"},{"key":"72_CR4","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1128-0","volume-title":"Harmonic analysis on semigroups","author":"C Berg","year":"1984","unstructured":"Berg C, Christensen JPR, Ressel P (1984) Harmonic analysis on semigroups. Springer, New York"},{"key":"72_CR5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0065-2539(08)60946-4","volume":"75","author":"M Bertero","year":"1989","unstructured":"Bertero M (1989) Linear inverse and ill-posed problems. Adv Electron Electron Phys 75: 1\u2013120","journal-title":"Adv Electron Electron Phys"},{"key":"72_CR6","volume-title":"Nonlinear programming","author":"DP Bertsekas","year":"1999","unstructured":"Bertsekas DP (1999) Nonlinear programming. Athena Scientific, Belmont"},{"key":"72_CR7","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198538493.001.0001","volume-title":"Neural networks for pattern recognition","author":"C Bishop","year":"1995","unstructured":"Bishop C (1995) Neural networks for pattern recognition. Oxford University Press, London"},{"key":"72_CR8","volume-title":"Pattern recognition and machine learning","author":"C Bishop","year":"2006","unstructured":"Bishop C (2006) Pattern recognition and machine learning. Springer, Heidelberg"},{"key":"72_CR9","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1023\/A:1016641629556","volume":"13","author":"M Burger","year":"2000","unstructured":"Burger M, Engl H (2000) Training neural networks with noisy data as an ill-posed problem. Adv Comput Math 13: 335\u2013354","journal-title":"Adv Comput Math"},{"key":"72_CR10","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0893-6080(02)00167-3","volume":"16","author":"M Burger","year":"2002","unstructured":"Burger M, Neubauer A (2002) Analysis of Tikhonov regularization for function approximation by neural networks. Neural Netw 16: 79\u201390","journal-title":"Neural Netw"},{"key":"72_CR11","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511801389","volume-title":"An introduction to support vector machines and other kernel-based learning methods","author":"N Cristianini","year":"2000","unstructured":"Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, London"},{"key":"72_CR12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-01-00923-5","volume":"39","author":"F Cucker","year":"2001","unstructured":"Cucker F, Smale S (2001) On the mathematical foundations of learning. Bull AMS 39: 1\u201349","journal-title":"Bull AMS"},{"key":"72_CR13","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s102080010030","volume":"2","author":"F Cucker","year":"2002","unstructured":"Cucker F, Smale S (2002) Best choices for regularization parameters in learning theory: on the bias-variance problem. Found Comput Math 2: 413\u2013428","journal-title":"Found Comput Math"},{"key":"72_CR14","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/S0885-064X(03)00010-4","volume":"19","author":"JA Cuesta-Albertos","year":"2003","unstructured":"Cuesta-Albertos JA, Wschebor M (2003) Some remarks on the condition number of a real random square matrix. J Compl 19: 548\u2013554","journal-title":"J Compl"},{"key":"72_CR15","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0885-064X(87)90027-6","volume":"3","author":"J Demmel","year":"1987","unstructured":"Demmel J (1987) The geometry of ill-conditioning. J Compl 3: 201\u2013229","journal-title":"J Compl"},{"key":"72_CR16","doi-asserted-by":"crossref","unstructured":"Dontchev AL (1983) Perturbations, approximations and sensitivity analysis of optimal control systems. Lecture Notes in Control and Information Sciences, vol 52. Springer, Berlin","DOI":"10.1007\/BFb0043612"},{"key":"72_CR17","volume-title":"Foundations of Modern Analysis","author":"A Friedman","year":"1970","unstructured":"Friedman A (1970) Foundations of Modern Analysis. Holt, Rinehart, and Winston, New York"},{"key":"72_CR18","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1162\/neco.1995.7.2.219","volume":"7","author":"F Girosi","year":"1995","unstructured":"Girosi F, Jones M, Poggio T (1995) Regularization theory and neural networks architectures. Neural Comput 7: 219\u2013269","journal-title":"Neural Comput"},{"key":"72_CR19","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1162\/089976698300017269","volume":"10","author":"F Girosi","year":"1998","unstructured":"Girosi F (1998) An equivalence between sparse approximation and support vector machines. Neural Comput 10: 1455\u20131480","journal-title":"Neural Comput"},{"key":"72_CR20","first-page":"166","volume-title":"From Statistics to Neural Networks. Theory and pattern recognition applications, ser. NATO ASI Series F, Computer and Systems Sciences","author":"F Girosi","year":"1994","unstructured":"Girosi F (1994) Regularization theory, radial basis functions and networks. In: Cherkassky JHFV, Wechsler H(eds) From Statistics to Neural Networks. Theory and pattern recognition applications, ser. NATO ASI Series F, Computer and Systems Sciences. Springer, Berlin, pp 166\u2013187"},{"key":"72_CR21","doi-asserted-by":"crossref","unstructured":"Gnecco G, Sanguineti M (2007) Accuracy of suboptimal solutions to kernel principal component analysis. Comput Optim Appl. doi: 10.1007\/s10589-007-9108-y","DOI":"10.1007\/s10589-007-9108-y"},{"key":"72_CR22","volume-title":"Genetic algorithms in search, optimization, and machine learning","author":"DE Goldberg","year":"1989","unstructured":"Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading"},{"key":"72_CR23","volume-title":"Matrix computations","author":"GH Golub","year":"1996","unstructured":"Golub GH, Loan CFV (1996) Matrix computations. John Hopkins University Press, London"},{"key":"72_CR24","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1023\/A:1018945915940","volume":"78","author":"A Gupta","year":"1998","unstructured":"Gupta A, Lam M (1998) The weight decay backpropagation for generalizations with missing values. Ann Oper Res 78: 165\u2013187","journal-title":"Ann Oper Res"},{"key":"72_CR25","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1016\/S0893-6080(98)00046-X","volume":"11","author":"A Gupta","year":"1998","unstructured":"Gupta A, Lam M (1998) Weight decay backpropagation for noisy data. Neural Netw 11: 1127\u20131138","journal-title":"Neural Netw"},{"key":"72_CR26","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.jat.2006.03.016","volume":"143","author":"A Hofinger","year":"2006","unstructured":"Hofinger A (2006) Nonlinear function approximation: computing smooth solutions with an adaptive greedy algorithm. J Approxim Theory 143: 159\u2013175","journal-title":"J Approxim Theory"},{"key":"72_CR27","unstructured":"Hofinger A, Pillichshammer F (2005) Learning a function from noisy samples at a finite sparse set of points. J. Kepler University, Linz, Technical Report, SFB F013"},{"key":"72_CR28","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1214\/aoms\/1177697089","volume":"41","author":"GS Kimeldorf","year":"1970","unstructured":"Kimeldorf GS, Wahba G (1970) A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann Math Stat 41: 495\u2013502","journal-title":"Ann Math Stat"},{"key":"72_CR29","unstructured":"Krogh A, Hertz JA (1992) A simple weight decay can improve generalization. In: Advances in neural information processing systems, vol. 4. Morgan Kaufmann Pub., pp 950\u2013957"},{"key":"72_CR30","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/978-1-4612-1996-5_16","volume-title":"Computer-intensive methods in control and signal processing. The curse of dimensionality.","author":"V K\u016frkov\u00e1","year":"1997","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.. Birkh\u00e4user, Boston, pp 261\u2013270"},{"key":"72_CR31","first-page":"1377","volume-title":"COMPSTAT 2004\u2014proceedings in computational statistics","author":"V K\u016frkov\u00e1","year":"2004","unstructured":"K\u016frkov\u00e1 V (2004) Learning from data as an inverse problem. In: Antoch J(eds) COMPSTAT 2004\u2014proceedings in computational statistics. Physica-Verlag\/Springer, Heidelberg, pp 1377\u20131384"},{"key":"72_CR32","doi-asserted-by":"crossref","first-page":"2659","DOI":"10.1109\/18.945285","volume":"47","author":"V K\u016frkov\u00e1","year":"2001","unstructured":"K\u016frkov\u00e1 V, Sanguineti M (2001) Bounds on rates of variable-basis and neural-network approximation. IEEE Trans Inf Theory 47: 2659\u20132665","journal-title":"IEEE Trans Inf Theory"},{"key":"72_CR33","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1137\/S1052623403426507","volume":"15","author":"V K\u016frkov\u00e1","year":"2005","unstructured":"K\u016frkov\u00e1 V, Sanguineti M (2005) Error estimates for approximate optimization by the extended Ritz method. SIAM J Optim 15: 461\u2013487","journal-title":"SIAM J Optim"},{"key":"72_CR34","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.jco.2004.11.002","volume":"21","author":"V K\u016frkov\u00e1","year":"2005","unstructured":"K\u016frkov\u00e1 V, Sanguineti M (2005) Learning with generalization capability by kernel methods of bounded complexity. J Compl 21: 350\u2013367","journal-title":"J Compl"},{"key":"72_CR35","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"},{"issue":"5","key":"72_CR36","first-page":"764","volume":"168","author":"ES Levitin","year":"1966","unstructured":"Levitin ES, Polyak BT (1966) Convergence of minimizing sequences in conditional extremum problems. Dokl Akad Nauk SSSR 168(5): 764\u2013767","journal-title":"Dokl Akad Nauk SSSR"},{"key":"72_CR37","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971323","volume-title":"Numerical analysis: a second course","author":"JM Ortega","year":"1990","unstructured":"Ortega JM (1990) Numerical analysis: a second course. SIAM, Philadelphia"},{"key":"72_CR38","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1109\/5.58326","volume":"78","author":"T Poggio","year":"1990","unstructured":"Poggio T, Girosi F (1990) Networks for approximation and learning. Proc IEEE 78: 1481\u20131497","journal-title":"Proc IEEE"},{"key":"72_CR39","first-page":"536","volume":"50","author":"T Poggio","year":"2003","unstructured":"Poggio T, Smale S (2003) The mathematics of learning: dealing with data. Notices AMS 50: 536\u2013544","journal-title":"Notices AMS"},{"key":"72_CR40","doi-asserted-by":"crossref","unstructured":"Poggio T, Mukherjee S, Rifkin R, Rakhlin A, Verri A (2002) \u201cb\u201d. In: Winkler J, Niranjan M (eds) Uncertainty in Geometric Computations. Kluwer, Dordrecht, pp 131\u2013141","DOI":"10.1007\/978-1-4615-0813-7_11"},{"key":"72_CR41","volume-title":"Learning with kernels\u2014support vector machines, regularization, optimization, and beyond","author":"B Sch\u00f6lkopf","year":"2002","unstructured":"Sch\u00f6lkopf B, Smola AJ (2002) Learning with kernels\u2014support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge"},{"key":"72_CR42","doi-asserted-by":"crossref","unstructured":"Sch\u00f6lkopf B, Herbrich R, Smola AJ, Williamson RC (2001) A generalized representer theorem. In: Proceedings of COLT\u201901, Lecture Notes in Artificial Intelligence. Springer, Heidelberg, pp 416\u2013424","DOI":"10.1007\/3-540-44581-1_27"},{"key":"72_CR43","volume-title":"Solutions of ill-posed problems","author":"AN Tikhonov","year":"1977","unstructured":"Tikhonov AN, Arsenin VY (1977) Solutions of ill-posed problems. W.H. Winston, Washington"},{"issue":"4","key":"72_CR44","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1109\/72.701179","volume":"9","author":"NK Treadgold","year":"1998","unstructured":"Treadgold NK, Gedeon TD (1998) Simulated annealing and weight decay in adaptive learning: the SARPROP algorithm. IEEE Trans Neural Netw 9(4): 662\u2013668","journal-title":"IEEE Trans Neural Netw"},{"key":"72_CR45","volume-title":"Statistical learning theory","author":"VN Vapnik","year":"1998","unstructured":"Vapnik VN (1998) Statistical learning theory. Wiley, New York"},{"key":"72_CR46","unstructured":"Vladimirov AA, Nesterov YE, Chekanov YN (1978) On uniformly convex functionals. Vestnik Moskovskogo Universiteta. Seriya 15\u2014Vychislitel\u2019naya Matematika i Kibernetika, vol 3, pp 12\u201323 (English translation: Moscow University Computational Mathematics and Cybernetics, pp 10\u201321, 1979)"}],"container-title":["Computational Management Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10287-008-0072-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10287-008-0072-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10287-008-0072-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T18:31:22Z","timestamp":1708713082000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10287-008-0072-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3,21]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,2]]}},"alternative-id":["72"],"URL":"https:\/\/doi.org\/10.1007\/s10287-008-0072-5","relation":{},"ISSN":["1619-697X","1619-6988"],"issn-type":[{"value":"1619-697X","type":"print"},{"value":"1619-6988","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3,21]]}}}