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向量映射的非线性标量化函数的性质及应用 Title:PropertiesandApplicationsofNonlinearScalarizationFunctionsforVectorMapping Abstract: Invectormapping,theconversionofhigh-dimensionaldataintoasinglescalarvalueisacrucialstepinmanyapplications.Nonlinearscalarizationfunctionsplayasignificantroleinthisprocessbypreservingimportantcharacteristicsoftheoriginaldatawhilereducingitsdimensionality.Thispaperexploresthepropertiesandapplicationsofnonlinearscalarizationfunctions,highlightingtheiradvantages,challenges,andpotentialfuturedevelopments. 1.Introduction Vectormappinginvolvesthetransformationofhigh-dimensionaldataintoalower-dimensionalspaceforvariouspurposessuchasvisualization,classification,andregression.Nonlinearscalarizationfunctionsserveasabridgebetweenthehigh-dimensionalandlow-dimensionalrepresentations.Unlikelinearscalarizationfunctions,whichsimplycombinethevectorcomponentsintoaweightedsum,nonlinearscalarizationfunctionscapturecomplexrelationshipsandinteractionswithinthedata.Thispaperinvestigatesthepropertiesandapplicationsofsuchfunctions. 2.PropertiesofNonlinearScalarizationFunctions 2.1.PreservationofInformation Nonlinearscalarizationfunctionsaimtopreservekeyinformationfromtheoriginalhigh-dimensionaldatawhilecompressingitintoasinglescalarvalue.Thispropertyallowsformeaningfulcomparisonsandrankingsbasedonintrinsiccharacteristicsofthedata.Varioustypesofnonlinearscalarizationfunctions,suchasdistance-basedorsimilarity-basedmeasures,canbeemployedtopreservedifferentaspectsofthedata. 2.2.Non-Convexity Unlikelinearscalarizationfunctions,nonlinearscalarizationfunctionscanexhibitnon-convexbehavior.Thispropertyallowsthemtocapturecomplexrelationshipsanddependenciesthatmaybemissedbylinearapproaches.Theabilitytomodelnon-convexstructuresenhancesthediscriminativepowerofthescalarizationfunction,enablingmoreaccurateandrobustperformanceinvariousapplications. 2.3.Flexibility Nonlinearscalarizationfunctionsofferflexibilityintheirdesign,allowingforcustomizationbasedonthespecificrequirementsoftheapplication.Diff