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1、JGeod(2012),86,35-52,DOI10.1007/s00190-011-0490-y1ParallelCholesky-basedreductionfortheweightedintegerleastsquaresproblemPeiliangXuDisasterPreventionResearchInstituteKyotoUniversityUji,Kyoto611-0011,Japanemail:pxu@rcep.dpri.kyoto-u.ac.jpAbstract:TheLLLreductionoflatticevectorsanditsvariantshavebeen
2、widelyusedtosolvetheweightedintegerleastsquares(ILS)problem,orequivalently,theweightedclosestpointproblem.Insteadofreducinglatticevectors,weproposeaparallelCholesky-basedreductionmethodforpositivedefinitequadraticforms.Thenewreductionmethoddirectlyworksonthepositivedefinitematrixassociatedwiththeweig
3、htedILSproblemandisshowntosatisfypartoftheinequalitiesrequiredbyMinkowski’sreductionofpositivedefinitequadraticforms.Thecomplexityofthealgorithmcanbefixedaprioribylimitingthenumberofiterations.ThesimulationshaveclearlyshownthattheparallelCholesky-basedreductionmethodissignificantlybetterthantheLLLalgo
4、rithmtoreducetheconditionnumberofthepositivedefinitematrix,andasaresult,cansignificantlyreducethesearchingspacefortheglobaloptimal,weightedILSormaximumlikelihoodestimate.Keywords:globalpositioningsystem(GPS),integerlinearmodel,integerleastsquares,closestpointproblem,reductionofquadraticforms,LLLreduc
5、tion,multiple-input-multiple-output.1IntroductionConsiderthefollowingintegerlinearmodel:y=Bz+ǫ,(1)whereyisann-dimensionalvectorofreal-valueddata,Bisan(n×m)real-valuedmatrixoffullcolumnrank,andǫistherandomerrorvectorofthemeasurementsywiththevariance-covariancematrixW−1σ2.ThematrixWhasoftenbeencalled
6、aweightmatrixofthemeasurementsyintheengineeringliterature(see,e.g.,Koch1999;Xu2006),andσ2isgenerallyassumedtobeanunknownpositivescalar.Abasicprobleminconnectionwiththemodel(1)istoestimatetheunknownintegervectorzfromthereal-valueddatay,i.e.,z∈Zm,andZmisdefinedasanm-dimensionalintegerspace.Theintegerl
7、inearmodel(1)hasbeenaninterdisciplinarysubjectofstudy,ranging,forexample,fromthegeometryofnumbersandintegerprogrammingtomultiple-input-multiple-output(MIMO)commu-nicationsystems,learningwitherrors,cryptogra
