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1、TikhonovregularizationFromWikipedia,thefreeencyclopediaTikhonovregularization isthemostcommonlyusedmethodof of named for.In,themethodisalsoknownasridgeregression .Itisrelatedtothe forproblems.Thestandardapproachtosolveanof givenasAx - b,isknownas andseekstominimizetheAx 一 b 2where isthe.However,them
2、atrix maybe or yieldinganon-uniquesolution.Inordertogivepreferencetoaparticularsolutionwithdesirableproperties,theregularizationtermisincludedinthisminimization:Ax - b 2 + lirxll2forsomesuitablychosen Tikhonovmatrix, r .Inmanycases,thismatrixis chosenasther = givingpreferencetosolutionswithsmallerno
3、rms.Inothercases, operators.,a oraweighted)maybeusedtoenforcesmoothnessifthe underlyingvectorisbelievedtobemostlycontinuous.Thisregularization improvestheconditioningoftheproblem,thusenablinganumericalsolution.An explicitsolution,denotedby r,isgivenby:x AtA + r T)1 ATbTheeffectofregularizationmaybev
4、ariedviathescaleofmatrix .For aI,when a=Othisreducestotheunregularizedleastsquaressolutionprovided that(A TA)-iexists.ContentsBayesianinterpretationAlthoughatfirstthechoiceofthesolutiontothisregularizedproblemmaylook artificial,andindeedthematrix r seemsratherarbitrary,theprocesscanbe justifiedfroma
5、.Notethatforanill-posedproblemonemustnecessarily introducesomeadditionalassumptionsinordertogetastablesolution.Statisticallywemightassumethat weknowthat isarandomvariablewitha.Forsimplicitywetakethemeantobezeroandassumethateachcomponentis independentwithx.Ourdataisalsosubjecttoerrors,andwetaketheerr
6、orsin 力tobealsowithzeromeanandstandarddeviation.UndertheseassumptionstheTikhonov-regularizedsolutionisthe solutiongiventhedataandtheaprioridistributionof ,accordingto.TheTikhonovmatrixisthenr = a/forTikhonovfactor a= b / xIftheassumptionof isreplacedbyassumptionsof anduncorrelatednessof, andstillass
7、umezeromean,thenthe entailsthatthesolutionisminimal.GeneralizedTikhonovregularizationForgeneralmultivariatenormaldistributionsfor xandthedataerror,onecan applyatransformationofthevariablestoreducetothecaseabove.Equivalently, onecanseekan xtominimizeAx - b2 +X 一 Xp02wherewehaveused|x|2 tostandforthew
8、eightednormBayesianinterpretation Pistheinverseof b, XistheXTFx(cf.the).Inthe ofx,and Qistheinversecovariancematrixof X.TheTikhonovmatrixisthengivenasa factorizationofthematrix Q= r t.the),andisconsidereda.AtPA +Q)1 AtP(b - Ax0)ThisgeneralizedproblemcanbesolvedexplicitlyusingtheformulaQRegularizatio
9、ninHilbertspaceTypicallydiscretelinearill-conditionedproblemsresultasdiscretizationof,and onecanformulateTikhonovregularizationintheoriginalinfinitedimensional context.Intheabovewecaninterpret &sa on,and xand baselementsinthedomainandrangeof .Theoperator A*A + rtr isthenabounded invertibleoperator.R
10、elationtosingularvaluedecompositionandWienerfilterWith r =a thisleastsquaressolutioncanbeanalyzedinaspecialwayvia the.GiventhesingularvaluedecompositionofAA = UYVtwithsingularvalues theTikhonovregularizedsolutioncanbeexpressedas = VDUrbwhere hasdiagonalvaluesandiszeroelsewhere.Thisdemonstratestheeff
11、ectoftheTikhonovparameter onthe oftheregularizedproblem.Forthegeneralizedcaseasimilar representationcanbederivedusinga.Finally,itisrelatedtothe:wheretheWienerweightsaref = !and Qistheof A.iDetermination of the Tikhonov factorTheoptimalregularizationparameter aisusuallyunknownandofteninpracticalprobl
12、emsisdeterminedbyanadhocmethod.Apossibleapproach reliesontheBayesianinterpretationdescribedabove.Otherapproachesinclude the, and. provedthattheoptimalparameter,inthesenseof minimizes: where H S Sisthe and Tistheeffectivenumber.八 RSSG =T 2XfcXTTXTUsingthepreviousSVDdecomposition,wecansimplifytheabove
13、expression:andRSS =i=1RSS=RSS0i=1b 2 + a 2 i ii=1it= m-E, E a 2 .b2 +a2 = m q + ZjbTZOIi=1 ii=1 iRelationtoprobabilisticformulationTheprobabilisticformulationofan introduces(whenalluncertaintiesareGaussian)acovariancematrix representingtheaprioriuncertaintiesonthemodelparameters,andacovariancematrix
14、 Drepresentingtheuncertaintieson theobservedparameters(see,forinstance,Tarantola,2004).Inthespecialcase whenthesetwomatricesarediagonalandisotropic, .：/and ：，and,inthiscase,theequationsofinversetheoryreducetothe equationsabove,with a= D / M.HistoryTikhonovregularizationhasbeeninventedindependentlyin
15、manydifferent contexts.Itbecamewidelyknownfromitsapplicationtointegralequationsfrom theworkof andD.L.Phillips.Someauthorsusetheterm Tikhonov-Phillips regularization.ThefinitedimensionalcasewasexpoundedbyA.E.Hoerl,who tookastatisticalapproach,andbyM.Foster,whointerpretedthismethodasa- filter.FollowingHoerl,itisknowninthestatisticalliteratureas ridge regression.QReferences (1943).O6ycTO访quBOCTuo6paTHbix3agaqOnthestabilityofinverse problems. 39(5):195 -198. Tychonoff,A.N.(1963). OpemeHuuHeKoppeKTHonocraB刀eHHbix3agaq uMeTogepery刀月pu3auuuSolutionofincorre
