非负矩阵分解及其在模式识别中的应用1

51320062**†(,710049.*.†,E-mail:[e-mail]).(non-negativematrixfactorization,NMF),..,,,,.[1],,;,,.[2],,.1999LeeSeung[3]Nature,.,,.,,,,,.,.,,.,:,,;,..,,..,??.,,,.(PCA),,.1,.,().,,,()[4]..mn,.,.,nmX×0XnmX×,(1)nmnrrmXBC×××≈,.,,nrB×rmC×rn[3].2,,.[4].[3,5],,,,.2.1[6]242(2)nmnrrmnmXBCE××××=+.,nmE×()ijijijXBCE=+.(3)B,C,[6]:(4),,{,}argmax(|,)argmin[log(|,)],BCBCBCpXBCpXBC==−,.,()2()1(|,)exp2π,2ijijijijijXBCpXBCσσ⎧⎫⎡⎤−⎪⎪=−⎢⎥⎨⎬⎢⎥⎪⎪⎣⎦⎩⎭(5).ijσ,(6)(|,)(|,)ijijpXBCpXBC=∏:221(,)()log(2π).2ijijijijijijLBCXBCσ⎡⎤=−+⎣⎦∑∑σ(7)1ijσ=12log(2π),ijijσ∑(8)2(,)().EDijijijLBCXBC⎡=−⎣∑⎤⎦,()()()()2,2,TTEDikikikTTEDkjkjkjLXCBCCBLBXBBCC∂⎡⎤=−−⎢⎥⎣⎦∂∂⎡⎤=−−⎢⎥∂⎣⎦(9)[6]:(10)()()()(),.TTikikikikikTTkjkjkjkjkjBBXCBCCCCBXBBCφϕ⎡⎤←+−⎢⎥⎣⎦⎡⎤←+−⎢⎥⎣⎦()(),kjikikkjTTikkjCBBCCBBCφϕ==,(10)[5]:()(),()()TTkjikikikkjkjTTikkjBXXCBBCCBCCBBC←←.(11),()(|,)exp[()]!ijXijijijijBCpXBCBCX=−.(12),:(13)(,)log()()log(!),ijijijijijLBCXBCBCX⎡⎤=−−⎣⎦∑KL:(,)log().()ijKLijijijijXLBCXXBCBC⎛⎞=−+⎜⎟⎜⎟⎝⎠∑ij(14),,,(,)LBC[6]:,().()ijikikikkjkjijjjijkjkjkjikikijiiXBBCCBCXCCBBBCφϕ⎡⎤←+−⎢⎥⎢⎥⎣⎦⎡⎤←+−⎢⎥⎢⎥⎣⎦∑∑∑∑(15),kjikikkjkjikjiCBCBφϕ==∑∑,[5]:(),().ikijijikjkjikikjijijjikikkjjBXBCCCBCXBCBBC←←∑∑∑∑(16),,.,51320062.,:()()(|,)exp2ijijijijijXBCpXBCσσ⎧⎫−⎪⎪=−⎨⎬⎪⎪⎩⎭,(17):1()(,)ijijijijXBCLBCσ−=∑.(18)(18),(18),[7]:app()(,)logcoshijijijijXBCLBCγσ⎧⎫⎡⎤⎛⎞−⎪⎪=⎢⎥⎜⎨⎬⎜⎢⎥⎪⎪⎝⎠⎣⎦⎩⎭∑⎟⎟,(19)1γ.,:(20)min(,)s.t.0,0.fBCBC.,,[3]1,,:(,)fBC1,ikiB=∀∑k(),().ikikkjijijjikiklklkjkjikijijiBBCXBCBBBCCBXBC←←←∑∑∑,(21),,p[8].2.2(LNMF),,.Li[9],3:,;;.,:LNMFmin(,)log()()s.t.0,0,1,,ijijijijijijijiiijiikiXLBCXXBCBCUVBCBkαβ⎛⎞=−+⎜⎟⎜⎟⎝⎠+−=∀∑∑∑∑(22).T,UBBVCC==T[9](),,(),kjijijjikikkjjikiklklkjkjikijijiCXBCBBCBBBCCBXBC←←←∑∑∑∑(23),,αβ.,.2.3(NNSC)[10],Hoyer[11](NNSC),.:(24)2NNSC2min(,)()s.t.0,0,1,.ijijkjijkjikiLBCXBCCBCBkλ⎡⎤=−+⎣⎦=∀∑∑∑:2(),0,0,,.*()./(),TikikikikikiTTBBBCXCBBBBBCCBXBBCηλ←−−←←←+∑(25)η.,,.Hoyer[12].2.4(SNMF),(SNMF)[13]:51320062SNMFmin(,)log()()s.t.0,0,1,.ijijijijijijkjkjikiXLBCXXBCBCCBCBkβ⎛=−+⎜⎜⎝+=∀∑∑∑⎞⎟⎟⎠(26).Ramanath[22].244(),,().1ikikkjijijjikiklklkjikijijikjBBCXBCBBBCBXBCCβ←←←+∑∑∑(27)Kawamo[23],(),.[24].Novak[25].,Xu[26].,.,SeppanenBrunet[27]1).NNSC,,.,[14].2),.2.5,.,,()[4,15],[16][17].Wang[18]Fisher,Ahn[19].3.1[28]..,.,UNIX,200.,.,.,,.3,,.,LeeSeung[20],.,,.,.Lee[21]PET,,UNMsendmail3)named4).,200.,,,,,dmX×ijXi..j1)SeppanenJK,HollménJE,BinghamE,etal.Nonnegativematrixfactorizationongeneexpressiondata.Bioinformatics2002,poster49,Ber-gen,20022)..,20053)~immsec/data/4)»immsec/data/live-named.html51320062sendmail[29],300,16.16,1861,254.1500,2250.,162250.,53.,SunOS()182,,182×2266,2250,16.30,222016.named9002,[30].2,969,831.1000,9230.,164.,164×9232,9230,2.100,91302(1).,23PCANMF1sendmail18230222016named164100913022PCANMFsendmailrεTDR/%FDR/%510−593.750.091110−51003.38510−581.2502110−593.753.06510−593.754.193110−51006.62510−593.757.43PCA10110−51007.75510−893.750.051110−81000.36510−81000.412110−81000.41510−81000.413110−81000.41510−81000.41SendmailNMF10110−81000.413PCANMFnamedrεTDR/%FDR/%210−41000.281110−41000.38210−41000.262110−41000.34210−41000.253110−41000.34210−4500.38PCA10110−41000.48210−710001110−71000.01210−71000.212110−71000.35210−71000.033110−71000210−71000NamedNMF10110−71000.21(TDR)(FDR).,NMF.3.2.,.1).1.,(,)(H,);,();,(),,.iWiW′1,()//.,,..,.,DCT[31].Lena,256256,[0,1],(1000)×[31].,1000(500)..2.,.2(a)DCT;(b)NMF3.8×8DCT,10.3,,NMFDCT.3(a)DCT;(b)NMF4.1.1.,NMFDCT.1)=shuiyin246(a)DCT;(b)NMF5,1000.,NMFDCT,NMF.5(a)DCT;(b)NMF3.3[32](braincomputerinterface,BCI).[33].1~20Hz,1~50Hz.[23],,.1)Keirn[34]2).6,250Hz,10s,(0.1100Hz).[35,36].:L,.,.,.,(/.6,().,()198.,/2OLL=2kL=L(/21)L+21)L+6(/21)L×+6k=[34].170,300.,5.:(math)(letter).[35],,..,,77,308..45.6264L==32OL=1))KeirnZA.Alternativemodesofcommunicationbetweenmanandmachine.Mastersthesis,ElectricalEngineering,PurdueUniversity,1988513200624(%)198.797.798.2253.295.574.4338.396.467.4464.994.279.5550.668.559.6674.479.276.8799.493.896.6248:5,198.2%,[35]90%;796.6%.,1,1,5.,385.,80.6%,[35]75%.Garrett[36],,.,,.39,156.125L=63OL=5NMF,,a,b,c,d,e5,f.Garrett6,(LDA)(NN)(SVM)[36],6.,,NMF5NMF(%)a75.684.684.652.646.268.7b67.994.961.548.748.764.4c58.390.494.951.951.369.4d63.548.128.263.532.147.1e58.382.792.343.668.669.1f64.190.491.053.851.370.16GarrettLDA(%)47.345.151.138.844.544.8NN(%)64.347.354.751.147.352.8SVM(%)59.444.552.757.047.952.3.,NMF.,,,.44.1,.,.,80%.,.200,1).,,,;,.,:,;,.,,,,.,,..,,“”“”2).1).?.)..,200451320062“”“”.,,,[9]1),.,,.,,.,[19].4.2[3],,.,,;.,,[37].,[38].,,,,..4.3,:(1)?,.,|()ijijijXBC−∑|(2).,[5].,[39]2).(3),?r(4)?,(,[40]).(5).,.,()3,4).(6),[41]Boltzmann[42].(7)[43,44].5,.,;,,;,,;,.,.,,,.,[45]5)..(:60021302)(:60205001).1HubertLJ,MeulmanJJ,HeiserWJ.Twopurposesformatrixfactorization:Ahistoricalappraisal.SIAMRe