Non-Orthogonal Tensor Diagonalization Based on Successive Rotations and LU Decomposition

Transcription

1 o-othogoal Teo Diagoalizatio Baed o Succeive Rotatio ad LU Decompoitio Yig-Liag Liu Xiao-eg Gog ad Qiu-ua Li School of Ifomatio ad ommuicatio Egieeig Dalia Uiveity of Techology Dalia hia lyl13@maildluteduc btact aoical polyadic decompoitio (PD) ha bee exteively tudied ad ued i olvig blid ouce epaatio (BSS) poblem maily due to it ice idetifiability popety i mild coditio I ove-detemied BSS ad joit BSS (J-BSS) PD i how to be equivalet to teo diagoalizatio (TD) I thi tudy we popoe a algoithm fo o-othogoal TD (TD) baed o LU decompoitio ad ucceive otatio ad examie it applicatio i BSS ad J-BSS We ue LU decompoitio to covet the oveall optimizatio ito L ad U tage ad the the facto matice i thee tage ca be appopiately paameteized by a equece of imple elemetay tiagula matice which ca be olved aalytically We compaed the popoed algoithm with othogoal TDteo DIgoalizatio (TEDI) ad PD with imulatio the eult how that i the ove-detemied cae TD geeate impoved accuacy ove TEDI PD ad othogoal TD ad fate covegece tha TEDI Keywod - Teo diagoalizatio; LU decompoitio; caoical polyadic decompoitio; blid ouce epaatio I ITRODUTIO Uig teo tool fo the aalyi of multi-dimeioal igal ha attacted wide iteet i the lat few decade We IJ K defie a thid-ode ak-1 teo a the oute poduct of thee o-zeo vecto a I J K b c : a b c whee deote teo oute-poduct defied a: tijk aibjck The ak of a teo i equal to the miimal umbe of ak-1 teo that yield i a liea combiatio ume that the ak of i R ad the it ca be witte a: R a b c (1) 1 which i efeed to a the caoical polyadic decompoitio (PD) of We defie the facto matice of PD a: a 1 a R B b 1 b R c 1 c R We ote hee that Thi wok wa uppoted by Doctoal ud of Miity of Educatio of hia ude gat atioal atual Sciece oudatio of hia ude gat Scietific Reeach ud of Liaoig Povicial Educatio Depatmet ude gat L ad hia Scholahip oucil PD i oe of the mot impotat teo decompoitio tool ad ha foud ucceful applicatio i blid ouce epaatio (BSS) [1] maily due to it ice idetifiability i mild coditio We ca efomulate (1) a follow: 1 B3 () RRR whee i a diagoal cubic teo with o-zeo elemet o the upe-diagoal The mode- poduct of thid-ode teo ca be deoted a: mode-1 poduct = : Y L mode- poduct = B: Y BL mode-3 poduct = : Y L I (3) Y 13 deote the ufig of teo i mode-1 mode- ad mode-3 a follow: mode-1: m j k 1 J mode- : m i k 1 I mode-3: m i j 1 I We aume that facto matice B ae full colum ak (ove-detemied) ad i thi cae () i labelled a teo diagoalizatio (TD) I thi pape we coide the cubic cae that I J K R without lo of geeality I fact i pactice thi aumptio of cubic ceaio teo ca alway be atified via a compeig pocedue [6] if the full colum ak equiemet i met The above fomulatio ugget that PD ca be coveted ito TD i ove-detemied coditio which make TD a teoial tool of paticula iteet to ove-detemied BSS poblem The cocept of TD wa fit itoduced i [7] [8] I thee pape TD eek othogoal matice that tafom the give teo ito a diagoal oe though Give otatio [8] peeted a BSS algoithm by fouth-ode cumulat ad imultaeou thid-ode teo diagoalizatio (STOTD) I thee wok the facto matice ae aumed to be othogoal ad idetical fo all the thee mode ad thu ae oly uitable fo poblem of igle-et BSS of pewhiteed mixtue Recetly the teo DIgoalizatio (TEDI) method wa tudied i [10] oweve TEDI algoithm may take (3) (4)

2 huded of iteatio util covegece thu limitig it computatioal efficiecy I thi pape we popoe a efficiet TD algoithm of thid-ode cubic teo i the geeal cae that all the thee loadig matice ae ditict ad o-othogoal by uig ucceive otatio ad LU decompoitio [11] with potetial applicatio i BSS ad J-BSS The popoed TD algoithm ca povide impoved accuacy tha PD ad othogoal TD ad moe efficiet computatio tha PD i ove-detemied BSS ad J-BSS applicatio I the et of the pape poblem fomulatio i give i Sectio II ad Sectio III peet the popoed algoithm Simulatio eult ae how i Sectio IV ad Sectio V coclude thi pape II PROBLEM ORMULTIO The mai idea of TD i to fid o-othogoal facto matice that tafom the give teo ito a appoximately diagoal coe teo I thi ectio we peet the thid-ode o-othogoal teo diagoalizatio fomulatio explicit demotatio of TD i give i igue 1 B = igue 1 Model of the teo digoalizatio: thid-ode teo though TD algoithm tafom teo to a diagoal teo To olve the above TD poblem i () we eek the etimate fo B B ad uch that 1 B 3 ae a diagoal a poible We miimize the um of off-diagoal quaed om fo the etimatio of facto matice a follow: 1 3 B ag mi off B (5) B whee deote the obeiu om of a teo ad off l 1 i j k ijk The above TD fomulatio ca be obtaied fom igle-et data model fo BSS o double-et data model fo J-BSS TD with igle-et data model fo BSS poblem of igle-et data model the itataeou mixig model i aumed a: whee t t t x (6) x t deote the obevatio mutually idepedet ouce ad mixig matix epectively I additio we ca calculate the followig taget matice: k E x k x k E k k (7) E{ [ ] } i diagoal if we aume the ouce ae mutually ucoelated We ca tack alog the thid dimeio ito a teo: Moeove we ote hee that k k k k k k :: diag E (8) To obtai the cubic teo fit teo hould be T ufed i mode-3 by (5) ito a matix a T ftewad igula value decompoitio (SVD) i coducted to the matix We chooe picipal eigevecto to cotitute a ew matix T ially the compeig poge i fiihed by ehapig the matix T to a teo B TD with double-et data model o J-BSS ove multi-et igal J-BSS poblem ca be modeled by calculatig ecod-ode tatitic i [1] o the co 4th-ode cumulat i [13] The itataeou mixig model fo double-et data i aumed a: whee t t t x (9) x t deote the obevatio ouce ad mixig matix i the th dataet epectively 1 Theefoe we could calculate the followig taget matice: 1 k 1 k k 1 1 k k E x x E (10) the ouce i each data et ae idepedet ad the coepodig compoet of two data et ae mutually depedet E{ 1 k[ k] } i diagoal The the licewie fom of taget teo ca be cotucted a: 1 k 1 1 k whee :: D (11) D1 diag(e{ 1 [ ] k k k }) I additio we hould compe teo thid dimeio to atify the TD model i the ame way a befoe III PROPOSED LGORITM The cot fuctio popoed i (5) i a highly o-liea multi-paamete optimizatio poblem Theefoe LU decompoitio ad ucceive otatio ae adopted i TD algoithm which covet the optimizatio poblem ito the followig thee ubpoblem: 1 ag mi off (1a) Bag mi off B (1b) B ag mi off 3 (1c) The facto matice B ad update the ufig of teo i mode-1 mode- mode-3 epectively I (1b) ad (1c) the ad deote the updated teo i the

3 peviou tep ow we adde how to olve (13) by uig LU decompoitio ad ucceive otatio Without lo of geeality we adde the update of (The updatig pocedue fo B ad ca be imilaly deived) The matix ca be witte ito a LU decompoitio fom a follow: det 1 L U (1) whee L i a lowe-tiagula matix with oe at it diagoal U i uppe-tiagula with oe at it diagoal uch i (13) ca be olved i the followig alteatig mae: U ag mi off UX 1 (14a) U L ag mi off LX 1 (14b) L whee L ad U ae the updated L ad U epectively The etimated matix ca be obtaied by LU ad Y 1 L Y 1 Y1 i the ufig of teo i mode-1 I the optimizatio of the algoithm the U-tage ad L-tage i fomula (14) alteatively update util the algoithm i coveged The L-tage ad U-tage ae vey imila i that both of them update facto matice by olvig the tiagle matix etimatio Theefoe i what follow we will maily cocetate o the U-tage I ode to obtai the optimal uppe tiagula matix U i the U-tage we deote U a the poduct of a et of elemetay otatio matix T i j : 1 U T (13) ( i j) i1 ji1 whee T i j i defied a follow: I j i j T i j 0 Iij Ii The elemetay otatio matix T i j ha oe ad oly oe o-zeo ad o-diagoal elemet which i i the ith ( i j) ow ad jth colum The update poce of U theefoe ca be completed by calculatig the poduct of ome ucceive otatio I othe wod all poible (ij) idice ( 1 j i ) ae taveed to olve the ubpoblem of etimate which ca be defied a: T( i j) ag mi off( T( i j) Y1 T Y 1 ew T( i j) Y1 U ew T( i j) U (14) whee Y 1 ad U ae the peviouly obtaied eult ad Y 1ew U deote the update of elated facto i ew the cuet iteatio I each iteatio ca be calculated by miimizig ( i j) the fomula: ( i j) ew off( T( i j) Y1 ) I Y1 ew T ( i j) Y 1 the matix Y 1 pe-multiplied by T oly chage the ith ow elemet of matix ( i j) Y 1 The ub-cot fuctio the ca be defied a follow: ( i j) ew ad the ith ow of matix Y (15) p1 pi 1 ew ip Y 1ew ca be defied a: Y : Y : Y : (16) 1 ew i ( i j) 1 j 1 i The followed by (18) (17) ca be calculated a: ( i j) ew p1 pi p1 pi Y 1 ew i p ( i j) m ( i j) ( i j) m m ( i j) m whee m j p i p (17) Y1 Y uch the optimal 1 paamete ca be obtaied by ettig the deivative of ( i j) to zeo with egad to ( i j) ew ( i j) : which yield the followig: ( i j) ew ( i j) 0 Y 1 j: Y 1 i: ( i j) Y 1 j: (0a) The we ca update the matix ad uppe-tiagula U accodig to (0a) It eed to go though all idice ( i j ) 1 i j i U-tage With egad to matice B ad the update pocee ae imila to the pocedue decibed above The diffeece lie i that matice B ad update the ufig teo i mode- mode-3 epectively: j: i: B i j Y Y Y j: 3 j: 3 i: i j Y Y Y 3 j: (0b) (0c) Whe all idice of the matice B ad i the U-tage ae updated (which i called oe weep) the algoithm ete the L-tage oted ealie L-tage i imila to the U-tage but the idex hould atify 1 j i Whe the L-tage fiihe the weep the TD algoithm come back ito the U-tage The U-tage ad L-tage update alteatively util the iteatio i topped outlie of the popoed algoithm i peeted a Table I

4 time() leat quae eo TBLE I SUMMRIZTIO O TE PROPOSED LGORITM Iput: cubic teo ad a theh Output: The etimated facto matice B ad the diagoal teo Implemetatio: I B I I 1 0 While do The U-tage: fo all 1 i j U I do - o U-tage: obtai optimal elemetay uppe-tiagula matix T with l i j l B it i j th elemet detemied by (0) - Update matice: Y ew Tl ( i j) Y Ul ew Tl ( i j) Ul 13 l B ed fo The L-tage: L I fo all 1 j i do - o L-tage: obtai optimal elemetay lowe-tiagula matix T l i j l B with it i j th elemet detemied by (0) - Update matice: ed fo Y ew Tl ( i j) Y Ll ew Tl ( i j) Ll 13 l B L U B L U B L U B B ew mi LU l l I ew ew l B ed while teo i the oie-fee ceaio The LSE i deoted a: LSE iput etimated iput () whee iput deote the iput teo ad etimated deote the etimated teo The cuve of 10 idepedet u ae daw i igue fixig = 4 Moeove we fix the SR to 0 db let vay betwee 5 ad 5 ad u 0 idepedet expeimet to compae the aveage utime of the TD ad TEDI algoithm a how i igue umbe of weep igue leat quae eo of TD ad TEDI algoithm veu the umbe of weep TD TEDI TD_time TEDI_time IV SIMULTIOS 10 1 I thi ectio umeical imulatio ae coducted to demotate the pefomace of the popoed TD algoithm compaed with STOTD TEDI ad PD algoithm omputig cofiguatio fo pefomig the imulatio ae ummaized a follow PU: Itel oe i7 93Gz; Memoy: 16GB; Sytem: 64bit Widow 7; Matlab R010b Simulatio 1: We compae the ate of covegece of the TD ad TEDI algoithm We cotuct a 3th-ode cubic teo by () whee i the taget teo B ad ae the thee facto matice I additio a Gauia oie tem i added ito the taget teo i the followig way: (1) whee ad deote the igal ad oie powe epectively Meawhile the igal-to-oie atio (SR) ued i thi pape i defied a: SR 10log it we compae the etimatio accuacy by evaluatig the leat quae eo (LSE) of the iput teo ad etimated igue 3 veage time veu teo dimeio om igue we ca ee the iteatio umbe of TD algoithm ae le tha TEDI I igue 3 it i how that the utime of popoed TD algoithm ae hote tha TEDI lagely epecially i the cae of i lage So a cocluio ca be daw that the TD ha bette covegece tha TEDI algoithm Simulatio : We meaue the pefomace of the popoed TD algoithm i the applicatio of BSS by compaio with STOTD TEDI ad PD with oliea leat quae (PD_LS) [14] We adopt the BSS model i T (6) ad aume that the ouce t ae BPSK igal with hot-time tatioaity With the peece of the T white oie t the obeved igal ca be cotucted a: X t t t t (3)

5 PI PI PI whee i the mixig matix with zeo mea ad uit vaiace ccodig to (7) ad (8) the covaiace matix of the obevatio igal cotitute the taget teo which will be ued i the above metioed algoithm We evaluate the pefomace of all the compaed algoithm by mai pefomace idex (PI) By takig 00 idepedet u the PI cuve of all the compaed algoithm ae plotted i igue 4 with = 4 T = 000 ad SR vaied fom 0 to 0 db SR(dB) igue 4 veage PI of veu SR om the eult we obeve that the popoed TD algoithm yield bette pefomace followed by TEDI ad PD_LS Moeove the STOTD udepefom the above thee algoithm becaue it aume that the thee facto matice ae equal while the facto matice of teo cotucted by BSS poblem ae diffeet Simulatio 3: We apply the popoed algoithm i J-BSS poblem Moe exactly we cotuct the obeved igal fo double-et data a follow: X t t t t 1 (4) T whee i the mixig matix ad t i ouce igal fo the th dataet epectively Theefoe the taget teo of the J-BSS model ca be calculated accodig to (10) ad (11) We let SR vay fom 0 to 30dB fix = 4 T = 000 take 00 idepedet u fo each fixed SR poit ad daw the aveage PI cuve fo both etimate of 1 ad fo all the compaed algoithm The eult ae how i igue TD STOTD PD_LS TEDI TD PD_LS TEDI SR(dB) (a) veage PI of 1 veu SR SR(dB) (b) veage PI of veu SR igue 5 veage PI of veu SR om the eult we ca obeve that TD PD ad TEDI algoithm all ca tackle the J-BSS poblem fo double-et data oweve the PI cuve of TD declie fate tha PD ad TEDI which mea that the pefomace of the TD i upeio to the othe Theefoe TD i a effective method to olve the J-BSS poblem with impoved accuacy tha PD ad TEDI algoithm V OLUTIO I thi tudy a o-othogoal teo diagoalizatio (TD) algoithm baed o LU decompoitio ad ucceive otatio i popoed which ca be well applied i BSS ad J-BSS poblem Simulatio have how that the upeioity of the popoed TD algoithm i the much impoved covegece peed ove the TEDI algoithm a well a impoved accuacy ove PD TEDI ad STOTD i dealig with the BSS poblem I additio TD i moe accuate tha othe algoithm i joit blid ouce epaatio (J-BSS) fo double-et data REEREES TD PD_LS TEDI [1] G Zhou ichocki aoical Polyadic Decompoitio Baed o a Sigle Mode Blid Souce Sepaatio IEEE Sigal Poceig Lette vol 19 o 8 pp ug 01 [] ichocki D Madic Pha Teo decompoitio fo igal poceig applicatio fom two-way to multiway compoet aalyi axiv pepit axiv: [3] T G Ka B W Bade Teo decompoitio ad applicatio SIM eview vol 51 o 3 pp ov 009 [4] L D Lathauwe B D Moo J Vadewalle O the bet ak-1 ad ak- 1 appoximatio of highe-ode teo SIM Joual o Matix alyi ad pplicatio vol 1 o 4 pp Oct 000 [5] L Xig Q Ma M Zhu Teo ematic model fo a audio claificatio ytem Sciece hia Ifomatio Sciece vol 56 o 4 pp Ma 013 [6] ichocki R Zduek Pha oegative matix ad teo factoizatio: applicatio to exploatoy multi-way data aalyi ad blid ouce epaatio Joh Wiley & So 009 [7] P omo Teo Diagoalizatio ueful Tool i Sigal Poceig 10th I Sympoium o Sytem Idetificatio vol 1 pp 77 8 July 1994 [8] M Soee P omo S Icat ad L Deeie ppoximate teo diagoalizatio by ivetible tafom Sigal Poceig ofeece ug 009

6 [9] L D Lathauwe B D Moo ad J Vadewalle Idepedet compoet aalyi ad (imultaeou) thid-ode teo diagoalizatio Sigal Poceig IEEE Taactio o vol 49 o 10 pp 6-71 Oct 001 [10] P Tichavky Pha ichocki Teo diagoalizatio - a ew tool fo PR ad block-tem decompoitio Epit xiv 014 [11] B fai Simple LU ad QR baed o-othogoal matix joit diagoalizatio Idepedet ompoet alyi ad Blid Sigal Sepaatio vol 3889 pp [1] X Gog X L Wag Q Li Geealized o-othogoal Joit Diagoalizatio with LU Decompoitio ad Succeive Rotatio Sigal Poceig IEEE Taactio o vol 63 o 5 pp Dec 013 [13] X L Li T da M deo Joit blid ouce epaatio by geealized joit diagoalizatio of cumulat matice Sigal Poceig IEEE Taactio o vol 91 o 10 pp Oct 011 [14] L Sobe M V Bael ad L D Lathauwe Teolab v0 vailable olie Jauay 014 URL: