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1、NeuralProcessingLetters11:39–49,2000.39©2000KluwerAcademicPublishers.PrintedintheNetherlands.AMinimaxMethodforLearningFunctionalNetworksE.CASTILLO1;?,J.M.GUTIÉRREZ1,A.COBO1andC.CASTILLO21DepartmentofAppliedMathematicsandComputationalSciences,E-mail:castie@c
2、caix3.unican.es;2OceanandCoastalResearchGroup,UniversityofCantabria,SpainAbstract.Inthispaper,aminimaxmethodforlearningfunctionalnetworksispresented.Theideaofthemethodistominimizethemaximumabsoluteerrorbetweenpredictedandobservedvalues.Inaddition,theinverti
3、blefunctionsappearinginthemodelareassumedtobelinearconvexcom-binationsofinvertiblefunctions.Thisguaranteestheinvertibilityoftheresultingapproximations.Thelearningmethodleadstoalinearprogrammingproblemandthen:(a)thesolutionisobtainedinafinitenumberofiteration
4、s,and(b)theglobaloptimumisattained.Themethodisillustratedwithseveralexamplesofapplications,includingtheHénonandLoziseries.Theresultsshowthatthemethodoutperformsstandardleastsquaresdirectmethods.Keywords:functionalequations,functionalnetworks,learning,neural
5、networks1.IntroductionFunctionalnetworkshavebeenintroducedbyCastillo[3]andCastillo,Cobo,Gutiér-rezandPruneda[5].Themainadvantageoffunctionalnetworksisthattheycanusedomainknowledgetogetherwithdataknowledge.Infact,theinitialtopologyofthenetworkisderivedfromth
6、epropertiestherealworldisassumedtohave.Next,functionalequations(seeAczél[1]andCastilloandRuizCobo[2])allowsimplifyingofthenetworktoobtainamuchsimplertopology,wheretheinitiallymultivariateneuralfunctionscanbewrittenintermsofunidimensionalneuronfunctions.Once
7、theuniquenessofrepresentationofthisnetworkhasbeenanalyzedandsetsoflinearlyindependentfunctionshavebeenselectedforapproximatingtheresultingneuronfunctions,leastsquaresmethodsallowforestimatingtheparamet-ersofthemodel,thus,guaranteeingtheglobaloptimumvalueisa
8、ttained.DetailsofthisprocessisgiveninCastillo,Cobo,GutiérrezandPruneda[5].Inthispaperwepresentanewminimaxmethodforlearningfunctionalnet-works.Theideaofthemethodistominimizethemaximumabsoluteerrorbetwee
