Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory

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1 Ln Bkly Nonl Lbooy Ln Bkly Nonl Lbooy Tl: To- Fok fo Topoloy-Bd Pojon nd Vulzon of Clfd Doun Collon Auho: Oln, Pk Publon D: Publon Info: Ln Bkly Nonl Lbooy Plnk: hp://holhp.o/u//0v73jx Lol Idnf: LBNL Pp LBNL-4074E Pfd Con: IEEE Confn on Vul Anly Sn nd Thnoloy (IEEE VAST) Sholhp povd opn, holly publhn v o h Unvy of Clfon nd dlv dyn h plfo o hol oldd.

2 To- Fok fo Topoloy-Bd Pojon nd Vulzon of Clfd Doun Collon Pk Oln Gk Shunn Svn Tnk Ghd Hy Unvy of Lpz Sffn Koh Tho El Unvy of Su Gunh H. Wb Ln Bkly Nonl Lbooy Fu 1: Ilnd-lk vulzon of doun pon loud opolol uu. By hn l dnon, doun uul n ubp of h hh dnonl nfoon p. Condn dnon od, lu ud o db op,.., lnd, n h fnl vulzon. ABSTRACT Dun h l dd, lon xul nfoon h bo h old l nd o pon nfoon ou vlbl. Popl hv ddd vy of dly npp, book, nf nd ovnnl publon, blo nd pv o h llpn of ndl nfoon nd knold. Sn nh h xn no h n nfoon n b d n ny, opu ud o x nd vulz nnful o nn op nd doun fo h hu nfoon lu. In h pp, xnd, pov nd obn xn ndvdul ppoh no n ovll fok h uppo opolol nly of hh dnonl doun pon loud vn by h ll-knon f-df doun- hn hod. W ho h donl dn-bd ppoh fl n vy hh dnonl p, nd db n povd o- hod fo opoloy-bd pojon fo h onl hh dnonl nfoon p o boh o dnonl (2-D) nd h dnonl (3-D) vulzon. To ho h uy nd ubly of h fok, op o hod nodud nly nd pply o oplx doun nd pn ollon. Indx T: H.5.2 [INFORMATION INTERFACES AND PRE- SENTATION]: U Inf Thoy nd hod; I.5.3 [Pn Ronon]: Clun Aloh; -l: {oln, hunn}@nfok.un-lpz.d -l: {nk, hy}@nfok.un-lpz.d -l: {Sffn.Koh, Tho.El}@v.un-u.d -l: hb@lbl.ov 1 INTRODUCTION Du o h xponnl n of lon xul d, opu qud o onbly p-po, lfy nd vulz ohn p of v d o pvn h u fo lly ndn houh nfoon lu. Fo h pupo, h hv bn onnly hn fo opl odl o uly pn oplx lnu lonhp. On of h odl h vo p odl h pn doun hh dnonl vo. In h pp, popo fok h k pobl o nv nd vulz uh doun pon loud un opolol ppoh. W fou on lfd doun, nd of fndn h lfon lf, bu lfon dpln n on h h ny dffn ppoh. Fuho, o doun ollon, uh npp o pn ollon, hv h on lfon h hh nully nd. Hov, h lfon lf do no uppo nnful d lyou n lo dnon. W, hfo, ju u lfon nfoon o bk up opoloy-bd pojon fo vy hh dnon no h dnon n od o vulzon hh db h nn uu nd nhbohood lonhp bn h p of h npu pon loud. In od o vod onfuon dn opolol u, pon ou h do no y o dnfy x opolol fu, opolol l o hoooph p. Tho hn ondd n h fld of lb o pon- opoloy ho loh nd upon y o oplx o b pplbl n vy hh dnon. A ood uvy houh h fld vn n [2]. Ind of dly ppoxn pon o nfold opoloy, ndly onu nd udy l funon opoloy by n of h jon-, hh h ld o db opolol oplx fu. W, hfo, do no xp h u o b opolo nd undnd oulv lun opo, n,

3 ll l on, nh h jon- no h 3-D lndp vulzon pbl of vulzn vou nd oplx hh dnonl fu. Rdn vul nly p, pn h u,.. jounl, fok o obn 2-D/ 3-D lyou of of doun. To hv h, u opoloy-bd pojon hh lly dpnd on nl p hh h o b found by nn h h vulzon. Th nv nly [17] uppo h u o vly fnd h dd p, nd hfo h dd nfoon. In h nd, n lun nfoon ld o only o nh, d ou d lyou n nl pon fo fuh xploon. 2 RELATED WORK Tx lfon, xu of nfoon vl, hn lnn, nd (l) lnu pon, onnd h buldn y h pon n unuud ollon of doun no nnful oup [20]. Th o n vn lun,.., fndn ln uu of pvouly dnd nub of oup, nd ozon,.., uun h d odn o oup uu knon n dvn. Fo h l upvd ppoh, dffn yp of ln hv bn ud, nludn pobbl nv Byn hod, don, nul nok nd xpl-bd hod. Hov, n h onx of opuonl lnn hoy nd opv xpn, uppo vo hn [12] nd boon [19] hv bn h o donn lnn hod n x lfon n h n y. Fo uul nh of onbly hh dnonl d, on n dly vulz h pon loud un x-bd hod lk,.., plo [1] o plll oodn [10]. Bu h hnqu dpnd on h d dnonly, ofn bnfl o pvouly poj h d o lo dnonl p. By dfnn nnful, lk,.., xl vn o xl dn bn lu nod, pojon y o pv h uu n h pojd dnon. So of h h upvd, uh LDA [6] o OCM [11], o unupvd, uh PCA [13] o MDS [16]. Choo l. [4] obnd h dvn of vl pojon o nz h lo of nfoon dun h nfoon. Bd h ln pojon, lo non-ln bddn x hh u ddonl uul nfoon hn dnn lyou of h d n lo dnon: Tkhh l. [22] popod nfold lnn ppoh o obn lyou n 3-D hh fl h opoloy of hh dnonl npu l funon. Thy u h k-nnhbohood ph o k h nfold poxy nd hy dfn n on l-bd dn u o dn h lon of pon. In od o u h hod fo pon loud nly, on h o nu h dfnon of ubl l funon. Fo,.., lun pupo, nnful l funon hould lo b dfnd n h vod p of d (h h no v) n od o ppoply p dn on fo on of lo dny. Thfo, Oln l. [17] fou on h ppop onuon of pon l funon, uppod by nhbohood dpon by n of h Gbl ph [7], nd of dvn nfold fo h funon. In h nd, h 3-D d lyou fl h opoloy of h d ppoxd dny funon, lzd by h opolol lndp pho [24], 3-D n hh h h opoloy h npu d. Spflly fo doun ollon, ThSp [25] lo u n pho o vulz h d, hov, ulzn dffn undlyn odl. 3 BACKGROUND Th onl onbuon of h ok l n xndn, povn nd obnn vl ndvdul ppoh no n ovll fok fo nlyzn doun ollon. To k o h d o follo h dl n h l p of h pp, nodu h o pon onp n h on. 3.1 Ln Dnn Anly (LDA) Fo lfd, hh dnonl d, Choo l. [4] dbd h dnon duon opzon pobl. Follon h nonlu, h lud -dnonl d pon R vn d x A = [A 1 A 2 A k ], h A R n k nd n = n. =1 Th oup of olun vo n A opond o h k oup of n lud -dnonl npu vo. Fo h oup N of olun nd of vo blonn o lu, h lu nod () nd h lobl nod vn by () = 1 n j nd = 1 n j N n j j=1 Un h lu vo oh h h nod nd h lobl nod, h hn-lu x S nd h bn-lu x S b dfnd S = S b = k ( j () )( j () ) T =1 j N k n ( () )( () ) T =1 By luln h of h (S ) = (S b ) = k j () 2 2 (1) =1 j N k n () 2 2 (2) =1 Choo l. [4] pfy lu quly u by ondn h dn bn h k lu nod nd h vn hn h lu, pvly. Th, ll-pd lun uully ll hv l (S b ) nd ll (S ). Eq.(1) db (S ) h qud u of p dn bn lu pon nd nod. Lk, Eq.(2) db h p dn bn h lu nod nd h lobl nod. Th fundnl d of h ppoh o ond dnon duon opzon h xz (G T S b G) nd nz (G T S G) n h dud dnonl p, un dnon dun ln nfoon G T R l : x R 1 z = G T x R lx1 pojn n -dnonl npu vo o n l-dnonl p ( > l). I un ou h h oluon, G LDA, of h LDA on J b/ = x((g T S G) 1 (G T S b G)) on of h olun vo hh h ldn nlzd nvo u of h nlzd nvlu pobl S b u = λs u (3) nd h LDA pv h onl lu uu f pojn h -dnonl npu vo no h l-dnonl p, uh h l = k 1. W f h nd d o fn [4] fo fuh nfoon, xpln ll h lonhp lly nd n uh o dl. In uy, LDA u h ddonl lun nfoon of h npu d o do upvd pojon fo h onl hh dnonl p no n opl lo dnonl p,.., (k 1)-dnonl, xzn h nlu dn nd nzn h n-lu dn n h dud dnonl p.

4 3.2 Appoxn Pon Cloud Topoloy Alhouh h LDA pv h lun uu n h nd p n of, h dnonly h ll b nfnly l hn o o h. A onqun, ubqun pojon ll u lo of nfoon du o pojv ovplon n onvnonl vulzon. In od o vod h ond pojon o, h nd p ould b nlyzd dly, nd of ondn h pon loud n h h onl -dnonl p o n h 2-D/ 3-D p. Oln l. [17] db hod o nlyz pon loud uu n hh dnon follod by opoloy-bd pojon o obn 3-D lyou of h d hh db h uu n h onl p. Th b d o db h pon loud uu ndly by onun l funon hh ppoply fl h d dbuon n of dny. In h onx of dny-bd lun, hy hv o vlu h nhbohood of h vn pon n od o dnuh on of boh hh nd lo dny. Afd, hy pfo opolol nly on h l fld, ulzn h jon- [3] hh nod h (jonn-) voluon of onou,.., on of qul dny houhou h l funon. Th fnl vulzon, n h nd, fl h jon hhl uu, hh, n un, fl h uu of h pon loud dny dbuon. In pul, hy pfo h follon p: 1) o fl h nvon of pon nhbohood, h npu pon loud onnd by h Gbl ph [7] hh pl nhbohood ph 2) ulz h nhbohood ph o pfo knl dny on nnful poon n p. In h, h ph v,.., h d pon (h dn), nd on h d-pon of h ph d,.., n h vod bn o nhbod pon (h lkly no dn) 3) h jon- opuon pfod on h ppoxd dny dbuon o nlyz h oun, z nd (jonn-) bhvo of h onou. Bu h onou db h dn on, h qul h dnon of nub, z nd hhy of lu 4) k u of h opolol lndp pho, popod by Wb l. n [24], o 3-D n hh h h opoloy h jon-. In h lndp, h uu of hll opond o h of h lu n h d W ho opolol lndp nd von of h pho houhou h nx on. Du p lon, f o [17] nd [24] fo fuh nfoon, n hy nodu nd xpln nub of oh onp unld o h onbuon of h ok. 3.3 Volly In h vo p odl (v) [18], x doun pnd vo. Eh dnon opond o p 1 nd dno h lvn n h pul doun. Mny dffn hv bn popod o x only nnful oh h h ndvdul nfn. A oon ppoh h f-df doun- hn: fo h nl doun, h fquny hd lvly o h nub of oh doun onnn h. Ind of only ondn fqun n nl doun o h hol opu, ddonl n nly n onbu o vo fnl xpvn. Tnk l. [23, 9] popod n ppoh o dfn 1 W k o n h nfld yp of od, h od ud o n n quvln l of nfld fo of b fo nnfuln o opl lvn by lookn h pol fluuon of lobl onx (.., ho nhbon hn ov ). Ulzn ok-k nonlu, h uho ll h fluuon of onx h volly. Thu, nlyzn h von of onx fo dffn l n b ulzd o d hhly dud op nd h pon ov. To povd ouh ouln of h hod, h jo lulon p : 1) Copu h nfn o-oun C() fo h n h hol opu 2) Copu h nfn o-oun C T () fo h n vy l T 3) Fo vy o-oun, j C() opu nk, j (), h of nk of, j n h onx of n vy l T 4) Copu h offn of von (.., h o of h ndd dvon o h n) CV (nk, j ()) fo vy ooun, j C() 5) Copu h volly h v of h vn: Vol() = 1 C() j CV (nk, j ()) Whn plon fquny nd volly ov, boh qun do no nly ol, h b d of volly o d op ( hn of onx) nd no dn ju hvy u of hh-fqun dbn. Alhouh oh hod y b ud o n h xpvn of, ho h odl o ddonlly uppo h f-df u nd o hh no voll nouh. Fo o dl nd xpl opn fquny nd volly, f o [23, 9]. 4 PROBLEMS IN VERY HIGH DIMENSIONS I Rhd Blln ho f d lo ffy y o h ldon h plud h n fo h l dy. Whl h ldon, blly, onn h pobl ud by nn h nub of ndpndn vbl n dffn fld of pplon, plly fo () p, h n n xponnl n of volu h h ddonl dnon. A onqun, pully fo dn-bd ppoh, h b hon [21, 14, 8] h dpndn on h hon, h dn bn pon h dpnd on h dnonly (L 1 no) o ppoh onn (L 2 no) o zo (L d 3 no). Th, h lv dffn bn h dn o pon fh nd lo pon ppoh zo. A onqun, dn bo lvly unfo n hh dnon nd o dnbd lonhp uh n nhbo bo nnl n ho p. Of ou, f dn bo unfo, vy dn-bd ppoh ffd by h phnonon. To llu h pobl fo lun loh, ond h MEDLINE 2 d. Th d on of 1,250 vo n 22, 095-dnonl p, dvdd no fv qully zd lu. Th blk ph n Fu 2() ho h nub of ndvdul dn bn ny o pon. I lly vbl 3 h h xu of ll dn l ound 2.0 nd 98,79% of h dn hn Th ky u h boh h nlu dn (n) nd h n-lu dn (d), hh obnd by ondn h vn lun nfoon, ho h bhvo. If d onn vl lu, hov, 2 W u h p kndly povdd n [4] 3 ll d n b bly nfd n h lon von of h pp

5 dn oun ll dn n lu dn n lu dn () dn oun ll dn n lu dn n lu dn Fu 2: Ponn of h MEDLINE dn bn ny (blk) o pon no n-lu (n) nd n-lu (d) dn, don () n h onl dnonly nd f pplyn LDA. No, h d nd h n pk pd. h ph yplly ho o pk: on fo h n-lu dn, nd pk pnn h v dn bn pon [21]. Conqunly, bu only on pk pn, ny puly dn-bd ppoh ll hv u h fndn h undlyn lun of h d. 4.1 Connn Hh Dnonl Doun In ddon o o u, oh n pobl onbu o h ud dn bn doun. If doun pnd by vo, nnful od uully v h vo dnon. Alhouh boh od nd nfn hvly dpnd on h hon loh, h ll ly b o od n h vo hh dub h b upon h (d)ly bn doun fld by h dn bn h opondn vo. Suh od, lk oon nd fqun od, n k o doun fo dffn op pp o l hn ully dd (o ndd). Th uon bo o f h dn bn o doun dond by h onbuon of non-dnn od. I pobl h o doun vo bou h op on of qul nub of dnn nd oon od. If h oon od vn dn, hy n u h vo o unnnonlly dp, hu nvly ffn h lun. In uy, hn ond doun ollon pon loud, fd h o n pobl. F, h onuon of pon loud h dn bn pon fl (d)l bn doun. Sond, du o u of dnonly, o lkly no bl o dnuh bn ly nd dly, bu o of h n- nd n-lu dn unfo. To llv h f pobl, lnu pon ny o vod hoon nnl od hh fo doun o unnnonlly ppoh o dp. Fo h ond pobl h upvd (dnndpndn n ou ) lun loh o upvd pojon o lo dnonl p, bn l ffld by u of dnonly, ndd. 5 TWO-STAGE PROJECTION Ou o- ppoh ld o h popod n [4]. Choo l. db vl o- obnon of upvd LDA o OCM nd unupvd PCA o poj h onl npu pon loud no n nd p, follod by ond pojon don o 2-D. Th upvd f poj h pon loud pvn lu uu nd h ol of h ond pojon o nz nfoon lo du o h dnon duon o () V V V9 Fu 3: () Rnk-2 LDA of h REUTERS d. So lu ll xd n h pojon p of h 9-D plo x, hon h nn pon of h ovplod lu. 2-D. On of h n onbuon of ou ok o pov h oupu of h o- ppoh by ubun h ond h d opolol nly of h nd p. To ov h, f ond h o- LDA-LDA pojon. 5.1 Exnn h Rnk-2 LDA pojon Th pojon popod by Choo l. [4] on of o ubqun LDA pojon. Th f fo h onl -dnonl p no n nd (k 1)-dnonl p. Th ond LDA poj o (k 1)-dnonl d don o h 2-D p. A uly xplnd n [4], h fnl pojon x V, V T R 2 : x R 1 z = V T x R 2 1 = [u 1 u 2 ] opon h o nl pojon, on of h ldn nlzd nvo of Eq. (3). Sn h Rnk-2 LDA nd LDA+PCA ld o podu h b dnn nd lo dnl ul, ond h Rnk-2 LDA oupu, un h REUTERS 2 d. Th doun ollon on of 800 vo n 11,941-dnonl p, nd qully o h follon k = 10 lu (h l ud n h d): n ( ), quon ( ), ony-fx ( ), n ( ), ud ( ), d ( ), n ( ), hp ( ), h ( ), nd on ( ) Th ul of h Rnk-2 LDA hon n Fu 3(). A n b n 3, h lun pvd ll by h Rnk-2 LDA. Hov, h lu on h h-hnd d nd n h op lf-hnd on of h plo ovplod. Th pvol quon hy. Ovplon ould h b du o d lonhp,.., lu ndd xd n h onl p, o du o ovplon n h ond. Sn LDA pv h lu lonhp n h l-dnonl p, n nlyz h 9-dnonl nd p by lookn plo x. Fu 3 ho u h h ond upon u. Exnn h pon loud no only fo h f o pnpl oponn, bu ondn h 7 h 9 h dnon n h plo x, n b n h boh ovplod lu on of ully pd lu n h nd vo p. Th no oplly upn, h ond- dnon duon only u o x o dn h l hh onbu o o h ond. Bu nvhl, du o h lk of ny nfoon bou h nd p h u ll o lkly nd o (knly) hnk of xd lu. 5.2 Subun h Sond S To ln h dbk of ovplon lu, ubu h pojon n h ond h opoloy-bd pojon fo h l-dnonl nd p o 3-D opolol lndp (hh n b ly dud o 2-D vulzon

6 () hp bn oup of pon. A dond n Fu 4, h hll (of h hhy lvl) poond n pl lyou ound h n hll, hh opond o h lobl xu. Th, h lobl xu l nd h / -lu, h h dn lu (.., pon p ). In h nhbohood o h dny xu h o oh o, opondn o (lol) uulon of -l nd -l pon. Nx o h lu, h yllo -lu nd h yn -lu lo o h oh hn o h y found / -lu. Th hold fo h, nd lu, hh ll hv h on dny o, bu hv h ddl poon n h nhbohood of h lu h hv bn found y. Th hlly nn uu of h / / uulon opond o lolly nhbod uon of h pon, bn pd fo h nn dny x (lu) by lo dny on. A opon of h lndp h Fu 3() ld o ouhly h nhbohood dpon, xp fo ll h nfoon h lo by h ond pojon n Fu 3(). 6 Fu 4: () opolol lndp of h REUTERS d n h nd 9-dnonl p. Th opolol nly vl h pd lu hh ovplod by h Rnk-2 LDA hou blnn h bnh dopoon, h pl lonhp bn dny o n b d off h lndp. dbd n h nx on). Condn h LDA-pojd (l = k 1)-dnonl pon loud, f h Gbl ph luld o obn nhbohood nfoon, nd Gun-lk fl knl ppld on h ph v nd d. Hvn h ppoxd dny dbuon, dn onou, o o ply h jon- pnv. Subqunly, h v h npu fo h ppn po o hv h fnl opolol lndp vulzon. To jufy h ubuon of h ond, v h REUTERS xpl fo bov. Alhouh h doun blonn o ()on, ()h nd ()n nlly ld, h plo x ho u h h opondn lu ll pd. Th pd pon uulon ld o vl dny x nd h dny dbuon, uln n vl hll n h opolol lndp. Fu 4() ho h opolol lndp of h REUTERS d. A n b n, no bl o dnuh dn p of h pon loud hh pd by on h lo dny. Th hll of h lndp db h opoloy of h pon loud dny dbuon,.., hll opond o onou volvn bn h ppn dny xu nd h n ddl pon. Th olo of h hll hv no pl nn nd hon ndoly. Th ll ph opond o h ul d pon nd pld on h hll h opond o h lu. Th olo of h ph fl h l oon, hu opondn o h olon n Fu 3. A dbd n [17], h vul nly po pfod by fndn n ppop fl du. Fo h pupo, h u xn h lndp n h on nd dn h fl du fo h nx v p. Th nly po fnhd hn h u h xd h dd lun nfoon o hn h lndp dno h h fl du n oo ll. In h l, no o podu dny o o vl x found nd on lu. Connn h blnn, pon ou h lhouh h p onlly popod n [24] o pov p ulzon, fo lun, h non-blnd lndp uly fl h pl lon- E XTENDING THE V ISUALIZATION Alhouh pbl of vulzn by, hh dnonl d, h onl opolol lndp pho only ppld o vulz h opoloy of 3-D nf l fld, vn on ully pld d. Un h pho o vulz d on oplly unuud d, pld oly nd h lu nd hdly n bn, ld o o ppul pobl: D o 2-D F of ll, h lndp ll h dnonl, uff fo ovlppn of h hll nd hfo h bnf of h vulzon v-dpndn (plly hn d pon poond h bk of hll). To llv h ovlppn of hll nd d pon, popo flnn of h onl opolol lndp. Un h onuon h lk n 3-D, fl 2-D lndp by un h jon (npold) ovlu ddonl vx nfoon nd of ondn h 3-D hh vlu. On h 2-D l fld, hh h h opoloy lk h npu jon, pply nol olo ppn nd oln xon o nod h bolu dn. In od o uppo h dvn of pho, l h vulzon o n oll by pplyn olo odn fo blu () o yllo (bh), hn fdn fo n () no bon (ounn) nd fnlly o h (noy ounn op). Th oln, hh opond o h onl hh vlu, llo fo n dny olon bn h d pon. Aloh, h vulzon uppo h opolol nh, bu h f l ovlppn n 2-D. 6.2 Ipovd Volu dbuon Th ond pobl onn h pnon of ppoxd onou volu,.., h z of lu n ou. A dbd n [24], -bd doon n b ppld o h lndp n od o b fl h l z of hll (onou), hh oh ould only dpnd on h lndp onuon h lf. Thfo, h nl of hll zd odn o h hll opondn lu volu. Bu h volu dbud qully o ll h nl of hll, h nd hll of nd hh hvly dod. Th ply doy h vul xpvn of h hll opondn lu n h lndp. To olv h, hn h nl volu nn. Ind of dvdn bnh volu by h nub of h opondn nl, n h volu bov h f ddl o h h nl of h nd hll nd h volu bnh h f ddl nd qully o ll h nn nl. Alhouh h ould b don o uly, by ondn h volu bn h p of ddl, blv h h dbuon

7 Fu 5: Th flnd nd volu dod opolol lndp of h REUTERS d. Th hh ln nd h olon fl h onl hh vlu (.., h bolu dny vlu) fo lo (blu) o hh (h). Fuho, h dod lnd b fl h ul lu volu. uffnly pon ou h volu of h opondn lu. Fu 5 ho h flnd REUTERS lndp h h z of h hll (lnd) opondn o h lu z (ppoxd by h nub of pon). 6.3 Lbln W ddonlly nhn h vulzon h lbln of boh h hll nd h ll d ph. Sn dln h doun, o lkly n o ho h l f. Bu hon h l of ll doun ould ul n hu ovplon of h lbl. Thfo, hv plnd h b xn lbln ppoh, popod by Fk l. [5]. In h ppoh h u ld ll fou ov h d, lbln only ho d pon hh l nd h ll fou ndo. Th lbl poond ound h fou nd hy onnd h h opondn d n by ln. In h nd, h lbl olod odn o h l of h d pon. If l oo lon, u h o u vn ho von of h l. Follon h d of f-df nd h vo p odl, h lu houh o onu op. Doun vo blonn o h op o lkly h l od h l nfn. Thfo, h doun vo uul n h ubp pnnd by h dnon (od) hy h. Th ol o povd hll (lu) h lbl opondn o h op. Ind of un h doun l-oon, popo quk n nly. Condn h doun on nl hll, h fo h o fqun od() hy h. Fnlly, u h o o fqun od hll op. 7 C LASSIFICATION Aun n unlfd ny ould h on of h vn l, pobl o u ou popod o- po fo lfon pupo: Th LDA, ln dnon duon, poj l hh dnonl vo no h on n h lo dnonl p. Thfo, f u h ldy hv lnd h LDA pojon fo vn l (bd on h lfd npu d), h nu of LDA nu h ddonl l unlfd vo pojd no h lo dnonl. Whl h, on h on hnd, olly od o h pojon lf, n, hov, ly on h f h l vo (lfd o no) opd by h onou of h dny funon. Thfo, lhouh h f pojon hh nu lun n h lo dnonl p, h ond opolol pojon hh vnully v u ool fo dn h lun nd o pfo lfon, bd on lu oon. Fo ou loh pon of v, n unlfd d n h l of h nhbod lfd n follo: W dn h LDA pojon x bd on h lfd d nd u h x o poj boh h lfd nd unlfd d (uln n uulon n h lo dnonl p f h vo l). W no hv o ho: F, obn boh pon loud o v h npu fo h opolol nly, o lnvly pply h opolol nly olly on h lfd pon loud nd ppox h opoloy of h obnd pon loud follo: W xnd h hh ph of h npu pon loud by n d bn h unlfd d pon nd n lfd nhbo (h qul h fndn of h n onou). Afd, opu h dn h d d-pon nd h unlfd pon (n boh, bd on h lfd n). On h xndd hh ph, onnu bfo. Fo boh no, fnd l lfd nd unlfd doun on h bnh n h bnh dopoon, hu nn n unlfd ny h lbl of h o fqun l on h ny bnh (.., lnd n h lndp). 8 D ISCUSSION AND E XAMPLES Anlyzn h l-dnonl nd p, of ou, o xpnv hn nlyzn (loy) 2D p. Fo dld un oplx f o [4], [17] nd [24] nd du o oon un u nd: F, h po LDA pojon ly l h opolol nly of h pon loud dny dbuon n h nd vo p. Th lon du o h f h h dud dnonly ld o l full Gbl ph, uln n l ny dny vluon on h ph d. Sn ph onnvy oonly n xponnlly h h ddonl dnon, h ppoh ld o nfn dffn fo hh dnonl d. Bd h, n lo u h lfon d fo h l on nd pn of h npu d unlfd. By ndoly un,.., only 50% of h npu vo o ln h LDA pojon x, h nn 50% of h vo houh o b pojd oly, du o h ly o h nn d. Of ou, un only p of h npu d ly l h un of h LDA. W ll don h quly of boh lfon nd un only ub of h d l n h on. Fo donon pupo, pply ou hod on N Yok T doun ollon nd pn ollon. A nond, f o h lu fo dld nfoon dn h un of h ndvdul p. To povd ouh udln, h opolol nly of h pvou REUTERS xpl nd h upon xpl n h hp ook ound fou ond h. Ou hn h 8GB oy nd u o 2,6GHz-QudCo poo o bnf fo plll. 8.1 NYT - Doun Collon Th N Yok T Annod Copu4 onn ov 1.8 llon l n nd publhd by h N Yok T bn Jnuy 1, 1987 nd Jun 19, 2007 h l -d povdd by h N Yok T Noo, h N Yok T Indxn Sv nd h onln poduon ff ny.o. A p of h N Yok T ndxn podu, o l nully uzd nd d by ff of lby n. Fo ou n pupo, ond h y 2001 nd x 50 doun p dy. A dbd n Son 3, u h f-df doun- hn nd h volly o dn h doun vo. Thfo, ondn nl doun, ll h op od pund nd h nol f-df hn h ppld o h nn od. Subqunly, 30-dy ldn ndo ud o dn volly fo h of h 365 dy. Thn, u h vn of h volly o hv n odd of ov-y pon of 4 hp://.ld.upnn.du

8 () xul onn, hfo kn nul lnu pon (NLP) ndoy p of oluon fo pn nly. Th h, oplxy, hh dnonly, nd hony of pn d, k lbl vul nly ppoh fo pn nly [15] hd k. On pully lvn yp of d h vlbl fo pn pplon nully nd lfon nfoon. Th lfon nfoon onz h v nub of pn no pdfnd l pnn n hnl o funonl p. Svl dffn h fo pn lfon, uh h Innonl Pn Clfon (IPC), Jpn F-, nd h US lfon, x. Pn off nd n uo lfon of n pn pplon odn o h xn lfon h. In od o vlu ou ppoh, d n h IPC opn o hn 70, 000 l, hhlly onzd no on, l, ubl, n oup, nd ub oup. In h nd, ou on of 1, 552 ndoly ld pn fo dffn IPC hh (up o 200 h): A61K..38/17, C12N...1/21, H04Q...7/22, B41C...1/10, C09D..11/00, C09J...7/02, G01N..33/53, H04Q..11/04 () Fu 6: () 2-D vulzon of h NYT d. Ilnd opond o op nd h z f h lu nub of doun. Th lndp fo hh fl du nd hou volu doon. Th ll ph bn h lnd ud o b oul o no (.., doun h oo oon vobuly). () doun nd h fou lbld h h l. ll voll. Fnlly, lp h f-df vo by h od hh no ud o b voll nouh (bd on h poon n h odd l). A lfon, ndoly hoo doun opondn o 10 dffn, up o 250 p oup. Aloh, h on of 1, 896 doun (pon), dbd by 46, 393 od (dnon). Fu 6 ho o pu of h fnl vulzon. A n b n, h LDA nd h ubqun opoloy-bd pojon don o 2-D pv h 10 opl lu, ufully dbd by h vo p odl. Th popod lbln of h hll ppoply fl h undlyn op, ud by h doun l nd pld by h doun onn. Lookn h vulzon, pon o undnd h lon of lnd do no ply h h opondn op ld n h onl p. Th nfoon lo dun h pojon nd, hfo, pl lonhp only nodd n h hhy of hll. (.., only ub-hll xp pl lon o h pn hll). 8.2 Pn Collon A o pn nfoon of pon fo vy of n oup ody. Bd ny oh pop, h joy of nfoon dbn h nu of pn ll onvyd houh W ud pn d5 fo h Euopn Pn Off (EPO). A ppon p, h d h bn nlyzd nd h x onn od n vozd fo hn h ndx. Fo h ndx h f-df vlu fo ll dnon of h vo hv bn opud. F, xn hh ou lndp fl h nn uu of h hon pn hhy. Fu 7() ho h vulzon fo ou. Mnly fou oup n b dnfd: Th pupl nd bon pon n h upp-h on, blonn o H04Q pn, lly dd nokn, h lbl (nd h doun l) l o, dd,, p nd ll. In f, h H04Q IPC-hhy oz pn blonn o ly (H), l ounon hnqu (H04) nd ln (h, ly, ) (H04Q). Alhouh h oup of pnk, blu nd blk pon on h h-hnd d blon o oplly dffn IPC-on (A61K, C12N, G01N), h opondn pn ll onn dl u n h jo fld: A Hun N, C - Chy, G - Phy. Bu hy h h dl vobuly, hy ll onu on op n h vo p odl. Fnlly, h nd hll blon o h B41C lu nd h n nd oldn pon op lu ld o pplon of l n hy nd lluy (C09D, C09J). W n o u h pn xpl o fuho don h lfon d fo on 7. W, hfo, pl h pn no 50% lfd nn d nd 50% unlfd d, hh n h hnfoh no h l lbl. Fo lluon pupo, hov, b h d l oon fo olon n h lndp. Af dnn h nn LDA pojon x, u o poj boh pn nd u h obnon fo ou opolol nly. Fu 7 ho h (no volu dod) lndp. A n b n, h nn d (pnd by ph) nd h d (pnd by on) blonn o pul l (pnd by h olo) ll hod on nly h on lnd. Th nly onf, h du o h hd vobuly,.., h hd dnon, pn of pf l qully hndld by h LDA, nd h uulon n h lo dnonl p llo u o opololly fnd h dn on obnd lu. Whl h llo fo f LDA opuon, h opolol nodn by n of h jon bnh dopoon lo off y o povd h d h (n) l of bnh / hll nn d l. Rdn ou xpl, h un of h LDA un ll h d ook 28, h un only 50% ook only 12. Rdn h lfon quly, dnd fo h bnh (lnd) ho 5 fo Tx of EP-A doun nd Tx of EP-B doun

9 ACKNOWLEDGEMENTS W hnk Chn Hn fo vlubl k nd npn duon. W lo hnk h nonyou v fo h uful on. Th ok uppod by n fo h Gn Sn Foundon (DFG), hn h h nv on Slbl Vul Anly (SPP 1335). R EFERENCES () Fu 7: () 2-D opolol lndp of h pn d. Th nn uu of h lnd fl h IPC hhy of h d. Th lndp hou volu doon. Evn by lnn h LDA h only 50% of h npu d, h nn pn (h on) pld on h o lnd (lu). ny of h bnh d n h h bnh nn d l (un h d knon l n h ). On v, 89.4% of h d on bnh h h l of h nn d, o o ply 76.7% n h noy on of h dn hplo, nd 99.6% on h nn bnh hh opond o lu bn b pd. 9 C ONCLUSIONS AND F UTURE W ORK In od o lu doun pon loud, fd o h ny of fln ly by dn nd pond o unfo dn, ud by h u of dnonly. W d o llv h f pobl by un volly, hnk h h ul n o opl ld. Connn h dnonly, hod h upvd ppoh ny n vy hh dnonl p. Thfo, popod o fok onn of upvd LDA pojon don o (k 1)-D, follod by d opolol nly of h nd vo p. By don o, bl o pov opbl ppoh hh u loy ond pojon. W lo xndd h vulzon n [17] o fl o p nd l ovlppn nly po n 2-D. Fo lfon pupo, hod ho h LDA nd h u of h bnh dopoon n b ud fo uo doun lfon bd on n xn lfon. Fuho, h quly of h lfon lf n b vfd by xnn h dbuon of olod pon n h lndp. If nl olo ou on vl hll, h lun (nd hfo h lfon) h b nppop. Sn h pud lfon o knd of dbk opd o unupvd ppoh, ou fuu ok ll onn h nvon nd uppo of lfon hod (yb lo opoloy-bd). W lo ond oh d uu o dnfy o oplx opolol fu. [1] R. A. Bk nd W. S. Clvlnd. Buhn plo. Thno, 29(2): , [2] G. Clon. Topoloy nd d. Bulln of h AMS, 46, [3] H. C, J. Snoynk, nd U. Axn. Copun onou n ll dnon. Copuonl Goy, [4] J. Choo, S. Bohn, nd H. Pk. To- Fok fo Vulzon of Clud Hh Dnonl D. In IEEE VAST, [5] J.-D. Fk nd C. Pln. Exn lbln: dyn nhbohood lbln fo d vulzon. In CHI 99: Podn of h SIGCHI onfn on Hun fo n opun y, [6] K. Fukun. Inoduon o l pn onon (2nd d.). Ad P Pofonl, In., Sn Do, CA, USA, [7] R. K. Gbl nd R. R. Sokl. A n l ppoh o oph von nly. Sy Zooloy, 18(3): , [8] A. Hnnbu, C. Al, nd D. A. K. Wh h n nhbo n hh dnonl p? [9] F. Holz nd S. Tnk. Tod uo don nd kn of op hn. In A. Glbukh, do, Po. CICLn 2010, I, LNCS Spn LNCS, pd fo ol pnon. [10] A. Inlb nd B. Ddl. Plll oodn: ool fo vulzn ul-dnonl oy. In VIS 90: Podn of h 1 onfn on Vulzon 90, p , [11] M. Jon, H. Pk, nd J. B. Ron. Dnonl duon bd on nod nd l qu fo ffn pon of x d. In Podn fo h F SIAM Inl. Wokhop on Tx Mnn, [12] T. Joh. Tx ozon h uppo vo hn: Lnn h ny lvn fu, [13] I. T. Jollff. Pnpl oponn nly. Spn, Bln, [14] K. B. Jonhn, J. Goldn, R. Rkhnn, nd U. Shf. Whn n nhbo nnful? In In In. Conf. on Db Thoy, p , [15] S. Koh, H. Boh, M. Gh, nd T. El. Iv non of vul nh dun pn h nd nly. In Po. IEEE Syp. Vul Anly Sn nd Thnoloy, p , [16] J. B. Kukl nd M. Wh. Muldnonl Sln (Qunv Applon n h Sol Sn). SAGE Publon, [17] P. Oln, C. Hn, H. Jnk, nd G. Shunn. Vul nly of hh dnonl pon loud un opolol lndp. In IEEE Pf Vulzon 2010 Podn, [18] G. Slon nd C. Bukly. T hn ppoh n uo x vl. Thnl po, Ih, NY, USA, [19] R. E. Shp nd Y. Sn. Boox: A boon-bd y fo x ozon. In Mhn Lnn, p , [20] F. Sbn nd C. N. D. Rh. Mhn lnn n uod x ozon. ACM Copun Suvy, 34:1 47, [21] M. Snbh, L. Eo z, nd V. Ku. Th hlln of lun hh-dnonl d. In In N V n Sl Phy: Applon n Eonophy, Bonfo, nd Pn Ron., [22] S. Tkhh, I. Fujho, nd M. Okd. Applyn nfold lnn o plon ppox onou. IEEE Tnon on Vulzon nd Copu Gph, 15(6): , [23] S. Tnk, G. Hy, G. Shunn, nd F. Holz. Vulun von Bduunvhbunn n oßn dhonn Dokunkollkonn. Dnbnk-Spku, 31:33 39, Db [24] G. Wb, P.-T. B, nd V. Pu. Topolol lndp: A n pho fo nf d. IEEE Tnon on Vulzon nd Copu Gph, 13(6): , [25] J. W, J. Tho, K. Pnnok, D. Lnp, M. Po, A. Shu, nd V. Co. Vulzn h non-vul: pl nly nd non h nfoon fo x doun. Infoon Vulzon, IEEE Sypou on, 0:51, 1995.