A-Global-Geometric-Framework-for-Nonlinear-Dimensi

23;right36,13,and27);superiorfrontalgyrus(left29,31,and45;right17,35,and37).17.AlthoughtheimprovementinWMperformancewithcholinergicenhancementwasanonsigniÞcanttrendinthecurrentstudy(P50.07),inapreviousstudy(9)withalargersample(n513)theeffectwashighlysigniÞcant(P,0.001).Inthecurrentstudy,weanalyzedRTdataforsixofoursevensubjectsbecausethebehavioraldataforonesubjectwereunavailableduetoacomputerfailure.ThedifferenceinthesigniÞcanceofthetwoÞndingsissimplyaresultofthedifferenceinsamplesizes.ApoweranalysisshowsthatthesizeoftheRTdifferenceandvariabilityinthecurrentsamplewouldyieldasignif-icantresult(P50.01)withasamplesizeof13.Duringthememorytrials,meanRTwas1180msduringplaceboand1119msduringphysostigmine.Duringthecontroltrials,meanRTwas735msduringplaceboand709msduringphysostigmine,adiffer-encethatdidnotapproachsigniÞcance(P50.24),suggestingthattheeffectofcholinergicenhance-mentonWMperformanceisnotduetoanonspeciÞcincreaseinarousal.18.Matched-pairttests(two-tailed)wereusedtotestthesigniÞcanceofdrug-relatedchangesinthevol-umeofregionsofinterestthatshowedsigniÞcantresponsecontrasts.19.H.Sato,Y.Hata,H.Masui,T.Tsumoto,J.Neuro-physiol.55,765(1987).20.M.E.Hasselmo,Behav.BrainRes.67,1(1995).21.M.G.Baxter,A.A.Chiba,Curr.Opin.Neurobiol.9,178(1999).22.B.J.Everitt,T.W.Robbins,Annu.Rev.Psychol.48,649(1997).23.R.Desimone,J.Duncan,Annu.Rev.Neurosci.18,193(1995).24.P.C.Murphy,A.M.Sillito,Neuroscience40,13(1991).25.M.Corbetta,F.M.Miezin,S.Dobmeyer,G.L.Shul-man,S.E.Peterson,J.Neurosci.11,2383(1991).26.J.V.Haxbyetal.,J.Neurosci.14,6336(1994).27.A.Rosier,L.Cornette,G.A.Orban,Neuropsychobiol-ogy37,98(1998).28.M.E.Hasselmo,B.P.Wyble,G.V.Wallenstein,Hip-pocampus6,693(1996).29.S.P.Mewaldt,M.M.Ghoneim,Pharmacol.Biochem.Behav.10,1205(1979).30.M.Petrides,Philos.Trans.R.Soc.LondonSer.B351,1455(1996).31.M.E.Hasselmo,E.Fransen,C.Dickson,A.A.Alonso,Ann.N.Y.Acad.Sci.911,418(2000).32.M.M.Mesulam,Prog.BrainRes.109,285(1996).33.R.T.Bartus,R.L.DeanIII,B.Beer,A.S.Lippa,Science217,408(1985).34.N.Qizilbashetal.,JAMA280,1777(1998).35.J.V.Haxby,J.Ma.Maisog,S.M.Courtney,inMappingandModelingtheHumanBrain,P.Fox,J.Lancaster,K.Friston,Eds.(Wiley,NewYork,inpress).36.WeexpressourappreciationtoS.Courtney,R.Desi-mone,Y.Jiang,S.Kastner,L.Latour,A.Martin,L.Pessoa,andL.Ungerleiderforcarefulandcriticalreviewofthemanuscript.WealsothankM.B.Scha-piroandS.I.Rapoportforinputduringearlystagesofthisproject.ThisresearchwassupportedbytheNationalInstituteonMentalHealthandNationalInstituteonAgingIntramuralResearchPrograms.7August2000;accepted15November2000AGlobalGeometricFrameworkforNonlinearDimensionalityReductionJoshuaB.Tenenbaum,1*VindeSilva,2JohnC.Langford3Scientistsworkingwithlargevolumesofhigh-dimensionaldata,suchasglobalclimatepatterns,stellarspectra,orhumangenedistributions,regularlycon-fronttheproblemofdimensionalityreduction:Þndingmeaningfullow-dimen-sionalstructureshiddenintheirhigh-dimensionalobservations.Thehumanbrainconfrontsthesameproblemineverydayperception,extractingfromitshigh-dimensionalsensoryinputsÑ30,000auditorynerveÞbersor106opticnerveÞbersÑamanageablysmallnumberofperceptuallyrelevantfeatures.Herewedescribeanapproachtosolvingdimensionalityreductionproblemsthatuseseasilymeasuredlocalmetricinformationtolearntheunderlyingglobalgeometryofadataset.Unlikeclassicaltechniquessuchasprincipalcomponentanalysis(PCA)andmultidimensionalscaling(MDS),ourapproachiscapableofdiscoveringthenonlineardegreesoffreedomthatunderliecom-plexnaturalobservations,suchashumanhandwritingorimagesofafaceunderdifferentviewingconditions.Incontrasttopreviousalgorithmsfornonlineardimensionalityreduction,oursefÞcientlycomputesagloballyoptimalsolution,and,foranimportantclassofdatamanifolds,isguaranteedtoconvergeasymptoticallytothetruestructure.Acanonicalproblemindimensionalityre-ductionfromthedomainofvisualperceptionisillustratedinFig.1A.Theinputconsistsofmanyimagesofaperson’sfaceobservedunderdifferentposeandlightingconditions,innoparticularorder.Theseimagescanbethoughtofaspointsinahigh-dimensionalvectorspace,witheachinputdimensioncor-respondingtothebrightnessofonepixelintheimageorthefiringrateofoneretinalganglioncell.Althoughtheinputdimension-alitymaybequitehigh(e.g.,4096forthese64pixelby64pixelimages),theperceptuallymeaningfulstructureoftheseimageshasmanyfewerindependentdegreesoffreedom.Withinthe4096-dimensionalinputspace,alloftheimageslieonanintrinsicallythree-dimensionalmanifold,orconstraintsurface,thatcanbeparameterizedbytwoposevari-ablesplusanazimuthallightingangle.Ourgoalistodiscover,givenonlytheunorderedhigh-dimensionalinputs,low-dimensionalrepresentationssuchasFig.1Awithcoordi-natesthatcapturetheintrinsicdegreesoffreedomofadataset.Thisproblemisofcentralimportancenotonlyinstudiesofvi-sion(1–5),butalsoinspeech(6,7),motorcontrol(8,9),andarangeofotherphysicalandbiologicalsciences(10–12).Theclassicaltechniquesfordimensional-ityreduction,PCAandMDS,aresimpletoimplement,efficientlycomputable,andguar-anteedtodiscoverthetruestructureofdatalyingonornearalinearsubspaceofthehigh-dimensionalinputspace(