Visual Recognition using Mappings that Replicate Margins

Transcription

1 Vsual Recognton usng Mappngs that Replcate Margns Lor Wolf and Nathan Manor The Blavatnk School of Computer Scence Tel-Avv Unversty Abstract We consder the problem of learnng to map between two vector spaces gven pars of matchng vectors, one from each space. Ths problem naturally arses n numerous vson problems, for example, when mappng between the mages of two cameras, or when the annotatons of each mage s multdmensonal. We focus on the common asymmetrc case, where one vector space X s more nformatve than the other Y, and fnd a transformaton from Y to X. We present a new optmzaton problem that ams to replcate n the transformed Y the margns that domnate the structure of X. Ths optmzaton problem s convex, and effcent algorthms are presented. Lnks to varous exstng methods such as CCA and SVM are drawn, and the effectveness of the method s demonstrated n several vsual domans. 1. Introducton A great deal of attenton s devoted to recognton tasks n whch the predcton value s a sngle label. Such problems are straghtforward to benchmark, and they ft well wthn conventonal learnng technques. Some attenton was gven, however, to the type of tasks that requre the predcton of a multdmensonal outcome. In [13], efforts to produce a textual descrptors of a vdeo are presented, and n [3] mages are mapped to tags and vce versa. In [7] one vew of a road scene s compared to another, and n [24] bologcal data s mapped to mages. Typcally, n learnng from vsual data, the mages are converted to vectors for reasons of mathematcal and algorthmc convenence. Smlarly, for example, textural descrptons, collectons of tags, and bologcal data are often transformed nto vectors. Multdmensonal perceptual problems, nvolvng mages on one sde and other forms of structured data on the other, are therefore most readly treated by ether recoverng a mappng from one vector space X to another vector space Y (multdmensonal regresson), or by mappng the two vector spaces X, Y onto one common vector space (as n CCA, see Secton. 2). Often, these mappngs are recovered based on a set of matchng tranng pars {(x, y )} such that x X and y Y. In ths work we consder a margn-based soluton for the asymmetrc mappng problem. Maxmum margn methods have proven to be effectve for classfcaton, regresson, and recently also n metrc learnng [23]. Here, we recover a mappng T : Y X that replcates or mmcs n the space of the transformed ponts T Y the margns that exst n X. Note that we do not maxmze the margns, snce ths mght dstort the space T Y. By defnton, margns requre the dvson of the samples nto two groups. When computng the map, we consder many such dvsons and resultng margns. Many prevous attempts to extend vector-to-vector mappngs to use dscrmnatve technques have assumed that a par of {(x, y j )} such that j (.e., the ponts are not pared n the tranng set), form an example of an ncompatble par. In many applcatons we found ths assumpton to be harmful. The reason mght be that snce the data s dstrbuted unevenly, some of the presumably non-matchng pars are very smlar n nature to the matchng pars. Snce non-matchng pars are typcally not provded, and snce they cannot be deduced from the data, we seek another source of dscrmnatve nformaton. To compute the transformaton T we rely on the exstence of a group of hyperplanes that separate the samples n the space of X. These hyperplanes are obtaned n varous ways n accordance wth the applcaton at hand. If such hyperplanes are not avalable, random hyperplanes are used successfully. 2. Prevous work The problem of learnng statstcal connecton between matchng vectors from two vector spaces s well studed. The vectors are often combned to form two matrces X and Y wth the same order of columns. Once the model lnkng the two matrces s learned, several tasks can be performed. For example, decde, gven two new vectors x new and y new whether they are matchng accordng to the model. A second task s the multdmensonal regresson problem, where the vector x new s to be predcted gven a vector y new. Ths second task s a harder one n the sense that by solvng t, the frst task can be solved, but not the other way around. 1

2 Multdmensonal regresson problems are often solved by extendng one dmensonal problems such as lnear rdge regresson [10], Support Vector Regresson [20, 14], or Kernel Regresson [9]. Sometmes such solutons are appled one coordnate at a tme, and sometmes the entre output s predcted at once [19]. Whle our method s an asymmetrc method, and therefore belongs to the famly of regresson methods, we do not try to fnd a transformaton that maps y to x. Instead, we am to replcate a specfc aspect of the dstrbuton of the ponts n X n the transformed ponts of the space Y. Ths structure s defned by a set of hyperplanes n X and multple labels for each tranng sample. Often, we take the set of hyperplanes to be unformly sampled, and the labels to be the projectons of the datapont n X by these random hyperplanes. In ths case, our method can be shown to be related to symmetrcal methods such as Canoncal Correlaton Analyss (CCA). In CCA [8], two transformatons are found that cast samples from X and Y to a common target vector-space such that the matchng vectors from the two spaces are transformed to smlar vectors n the sense of maxmal correlaton coeffcent. Addtonal constrants enfore the components of the resultng vectors to be parwse uncorrelated and of unt varance. The CCA formulaton s, therefore: Problem 1. non-regularzed CCA max U X,U Y n x U X UY y, subject to =1 n UX x x U X = =1 n UY y y U Y = I =1 The dmenson of the target vector space s typcally the mnmum of the two dmensons d M and d P and shall be denoted by l. Thus U X s a matrx of dmensons d M l and U Y s of dmensons d P l. For the matchng task above, the vectors x new and y new are consdered a match f the dstance (UX x new UY y new j ) s small. Snce oftenly the feature vectors for both vector spaces are of dmensons sgnfcantly hgher than the number of tranng samples, statstcal regularzaton must be used to avod overfttng. A regularzed verson of CCA suggested by [22] can then be used. Two regularzaton parameters need to be determned η X and η Y, to regularze the computatons n each of the vector spaces. Another modfcaton of CCA s kernel CCA (e.g. [2, 27]), whch uses the kernel trck for capturng non-lnear transformatons. CCA mnmzes the dstances between matchng vectors n X and Y, whle dsregardng the dstances between nonmatchng vectors. An alternatve optmzaton goal combnes the mnmzaton of dstances between matchng pars wth the maxmzaton of dstances between non-matchng pars, thus attemptng to acheve maxmal dscrmnatve ablty. Ths approach has been developed n [12]. We have extensvely tested ths approach, and have come to the concluson that the combned energy (ncludng the maxmzaton of the dstances for non-matchng pars) does not outperform CCA when the non-matchng pars are obtaned by mxng the matchng pars. The reason for the lack of mprovement s that by usng such a procedure many samples that are very smlar to matchng samples are obtaned and labeled non-matchng. One soluton we have tred s to delberately sample pars (x, y j ) that are unlke the matchng pars,.e., we exclude from the samplng of non-matchng pars those pars (x, y j ) for whch x s smlar to x j or y s smlar to y j. Ths does not seem to help f the threshold for excluson s too hgh, the obtaned non-matchng pars are not very nformatve. If, however, the threshold s set low, the pars are too smlar to matchng pars. Prevous attempts to formulate a max-margn CCA-lke problem nclude the Maxmal Margn Robot [16]. Rather than maxmzng the sum of correlatons, Maxmal Margn Robot maxmzes the mnmal correlaton. Robustness to outlers s mantaned through the ncluson of slack varables. Ths formalzaton produces the followng quadratc programmng Problem 2 (Maxmal Margn Robot). mn T,ξ 1 2 T 2 F C1 ξ, subject to 1 n y T x 1 ξ ξ 0 Smlarly to our method, a transformaton T s recovered. However, t s assumed that both X and Y are of the same dmenson. The Nearest Neghbor Transfer [24] s a smple alternatve that does not requre optmzaton. Gven a new vector x new, ths smple method chooses out of the tranng vectors of X the closest one arg n mn x x new =1 and predcts the vector y n Y. For the task of selectng a matchng vector for x new out of a set of vectors {y newk Y} k, the methods selects the vector most smlar to the y : arg mn y newk y. k Many of the above vector to vector mappng technques are symmetrc,.e., replacng the roles of the two vector spaces does not change the outcome. Ths s n contrast to the nature of many applcatons, where often one vector space s much more relable than the other. Our method, presented n Secton 3 maps from the less relable vector space Y to the more relable space X such that strong dstnctons n X are presented n the mapped data from Y.

3 3. Problem formulaton We are gven n pars of matchng vectors (x, y ) n =1. For example, x X mght depct the encodng of an mage ndexed n one camera, whle y Y depcts the a matchng frame from another vew of the same scene. As mentoned above, our framework s asymmetrc and transforms data from the less nformatve space to the more relable one. Assume that X s the relable vector space and Y s the less relable one. For example, wde feld of vew may be better suted for dentfyng the scene than a narrow feld of vew. Moreover, assume that we have an addtonal set of k hyperplanes n the space X each known to provde a good separaton of the n samples x 1,..., x n nto two classes, assocated wth labels L j {1, 1, 0} for the th hyperplane and the j th sample (each hyperplane separates the samples dfferently). The source of these hyperplanes and labels s applcaton dependent, and n [26] we provde several examples on how to obtan them. Denote the set of hyperplanes as w 1,.., w k. We look for a transformaton T : Y X, and scalars b 1,.., b k such that for each, the hyperplane w separates the n samples T y 1,.., T y n around b smlarly to the labels L j,.e., we encourage smlarty between L j and the sgn of the projectons w T y j b for all = 1..k and j = 1..n for whch L j 0. Typcally, the hyperplanes and the labels are such that L j = sgn(w x jb 0 ) for some scalars b0 1,.., b 0 k, however, b 0 and b often dffer, and are not part of the nput. Defnng the smlarty between labels and the projectons usng correlatons one would obtan an extenson of CCA, n whch an extra set of hyperplanes s used. Alternatvely, nspred by the success of large-margn methods such as SVM [4], and LMNN [23], our method mnmzes the hnge-loss. In contrast to SVM and LMNN, the margns are not maxmzed n our framework; Instead, they are replcated,.e., and we seek to have the same margns n the space of transformed samples T Y as exst n the space X. For each of the k separators w, = 1..k, we compute or otherwse obtan a goal margn m. Typcally, ths margn s computed n the vector space X on the tranng samples x j as m = mn j:lj>0 w x j max j:lj<0 w x j. Alternatvely, any defnton of soft margns can be employed, e.g., n the non-separable case. Usng the above defntons, we defne the followng optmzaton problem, whch yelds a mappng T : Y X that replcates the margns n X. Problem 3. (MRM: Margn Replcatng Mappng) [m L j (w T y j b )] mn T, b j Ths s an unconstraned pecewse lnear convex optmzaton problem that can be rewrtten as lnear programmng by addng slack hnges: Problem 4. (MRM as lnear programmng) ξ j, subject to mn T, b j ξ j m L j (w T y j b ) ξ j 0 In Sec. 4 we develop a gradent descent algorthm for MRM. Its frst teraton (when startng at zero) s n the lnear regons of the objectve functon, and no hnges are actve. We show that n ths case, there s a deep relaton to CCA (see Secton 5), therefore, n a sense, the frst teraton follows CCA and then evolves. 4. Optmzaton For such convex cost functons, the gradent descend method s guaranteed to converge to the global mnma, however, ths convergence s slow for a large number of separators. To speed up the convergence, we have developed a sutable sample and backtrack method depcted n Fgure 1. The method s modeled after the daggng and backtrackng modules of boostng methods [18, 5]. In the future, to further speed up the computaton, we plan to ncorporate wthn the procedure a column-generaton lnearprogrammng module [6]. The reason that the gradent descent methods would converge slowly on Problem 3 s that for every combnaton of a separator and a sample, there s a term of the form of max(, 0), thus, there are many non-dfferentablty break ponts. Durng the gradent decent procedure (or more precsely sub-gradent), we cannot have a step sze whch s larger than the dstance to the next break pont (to avod the rsk of overshootng ), and the progress s therefore slow. In order to reduce the effectve number of break ponts we sample a random subsets of hnge terms - thereby makng the gradent step both longer and faster to compute. Each teraton samples a dfferent subset ensurng dversty of search drectons and a faster convergence. 5. Connecton to other methods In ths secton we show that both CCA and SVM are specal cases of MRM. Also SVR and MMR are closely related. Frst, note that unlke most learnng algorthms, MRM objectve s bounded, thus does not ntrnscally requre regulaton. Yet, regularzed MRM can be defned: Problem 5. (bounded MRM) mn [m L j (w T y j b )] w.r.t. T,b j (w T y j) 2 1 j Ths s exactly the problem of 3, where the soluton s bounded. Snce the range of projectons T s convex, a

4 functon [T,b]=MRM({ y j }, { w }, {m }, {l j }) Intalze T:=0 and b:=0 Repeat c tmes Select at random for = 1..p: j,k s.t. l j > 0 k j,k s.t. l j < 0 k Set T := T //to allow backtrackng Set dt := [(w k ) (w k )] Set ξ := m k w k Set ψ := m k Set dξ := m k Set dψ := m k y j T y j b k w k T y j b k w k dt y j b k w k dt y j b k Let r be a random number n [0,3] Repeat 10 r tmes //try both long and short loops Mnmze λ := mn [mn( ξ, ψ )] dξ dψ Save argmn: ndex 0. Put s 0 := 1 f ξ, -1 f ψ Set T := T λ dt Set ξ := ξ λ dξ and ψ := ψ λ dψ f s 0 = 1 w := w k 0 y := y j 0 y j f s 0 = 1 w := w k y := y j 0 0 Set dt := dt s 0 wy Set dξ := dξ s 0 (w w k Set dψ := dψ s 0 (w w k )(y y j ) )(y y j End nner loop rebalance b va convex 1D optmzaton. If global objectve functon evaluated on T s worse than that of T then backtrack by: Set T := T End man loop Return T and b Fgure 1. Pseudocode of a method for computng the transformaton T and bases b n the Margn Replcatng Mappng mehtod. global soluton can be found by usng a gradent decent algorthm smlar to the one of Sec. 4. In our experence the bound s superfluous, however, snce the lnkng to other methods s done asymptotcally, such boundng smplfes the arguments. Unlke MRM, the methods we compare to, namely, CCA (Problem 1), Maxmaum Margn Robot (Problem 2) and SVR, do not employ hyperplanes as nput. The lnkng s therefore performed for the case where random hyperplanes are used. The margns are set wth accordance to the desred reducton. Specfcally, for the random hyperplane case, Problem 5 would reduce to CCA f all m approach nfnty and would reduce to an algorthm whch closely resembles MMR f we set m = 1. Lastly, MRM reduces to SVM, for a undmensonal X takng the role of labels. ) 5.1. connecton to SVM and SVR In the 1D case, where the dmenson of X s 1, w becomes a scalar, and the MRM problem reduces to: Problem 6. 1D MRM: [m L j (t y j b)] mn t,b j Problem 6 s equvalent to problem 7 below when γ = 0, as can be seen by dvdng objectve above by the postve (exhaugenc) constant m and substtute u := t m. Problem 7. SVM as unconstraned mnmzaton: mn u,b (γ u 2 [1 L j (u y j b)] ) j Problem 7 s exactly the optmzaton problem of SVM, formulated va the hnge loss. Note that both problems, SVM and MRM, are often bounded even wthout regularzaton. Yet, SVM s typcally regularzed, to make sure the largest margn s selected among all classfers for whch the penalty s mnmzed. Smlarly, MRM can be also be regularzed. However, expermentally, we found out that t results n lttle gan for hgh dmensonal X. In such cases, the penaltes for msclassfed T y j are strong enough even for low norm of T to stablze the soluton. Snce we dsregard the regularzaton term of MRM, we obtan an algorthm whch has no parameters other than the nput data and set of hyperplanes on X. Even n the case of random hyperplanes, T s forced to match the desred structure as occur n Co-Krng based solutons to SVR [21]. In contrast to sem-parametrc methods such as [15], MRM assumes no pror on bas terms b and learn them from nput. Of course, such prors, whenever avalable, mght mprove the algorthm s performance Connecton to CCA Lnks between MRM and the non-regularzed CCA (problem 1) are presented n [26]. Smlarly to CCA, the proposed varant receves zero mean nputs (centralzed data). Then, gven large m s (forcng the hnge terms to become loose), MRM wll behave smlarly to CCA. Note that the frst teraton of MRM gnores all hnges, snce the hnge functons are lnear near the orgn. Thus, the frst teraton of MRM acts smlarly to the case of m =. Further teratons refne the produced transformaton. Ths observaton was verfed on synthetc data [26]. 6. MRM modfcatons The MRM framework s flexble, allowng for several adaptatons. The frst two modfcatons below have been mplemented and tested. The weghted MRM typcally produces results that are smlar to those of the vanlla MRM.

5 Kernel MRM s often used to speed up experments nvolvng hgh dmensonal data. The mplementaton of the thrd, sparse MRM, s left for future work Weghted MRM Ths varant specfes a confdence level for each label L j. One natural choce of confdence s to use, for a hyperplane w and example j the value L j = w x j b 0 as the sgned confdence value (recall that b 0 s the bas for hyperplane w n the space X ). The sgn of ths value sgn(l j ) s the label, and ts absolute value L j a measure of confdence (n practce t s benefcal to trm hgh confdence values). The followng problem s optmzed: Problem 8. Weghted MRM: L j [m sgn(l j )(w T y j b )] mn T,b j 6.2. kernelzed MRM Ths varant uses the kernel trck to allow for non-lnear mappngs, and to reduce computatonal tme for hgh dmensonal vector spaces. Let φ : X H X and ψ : Y H Y be two transformatons that embed the nput samples n dmensonal Hlbert spaces. We mplctly recover a transformaton τ : H Y H X, wthout performng operatons n H X or n H Y. Snce the recovered T s a lnear combnaton of outer products, t can be represented as products of the form w ψ(y j ). Durng the algorthm run, the transformaton T s appled to the dataponts n H Y, and the results s correlated wth w l, thus obtanng scalars of the form K φ (w l, w )K ψ (y j, y ). We used ths kernalzaton to speed up the lnear case when the dmensonalty s much larger than the number of examples. Whle the complexty of each teraton n the Algorthm of Fgure 1 (domnated by re-calculatons of the objectve functon) s O(nkd), the complexty of each teraton n the kernelzed algorthm s O(kn) f W W and Y Y are precomputed (ths precomputaton takes O(dn 2 dk 2 )) sparse MRM To acheve an ntrnsc feature selecton, the columns of T mght be encouraged to become zero, by bndng approprate cost terms (as suggested by [17]). We wsh to elmnate entre columns, and therefore apply these terms to the maxmal absolute value of each column: Problem 9. mnmze over T,β,ξ,ψ of L j ξ j C ψ r w.r.t. r j cr ψ r T cr, cr ψ r T cr j ξj m sgn(l j )[ T rc y rj w c rc r j ξ j 0 β r y rj w r ] Fgure 2. Precson/Recall statstcs for matchng car mages between two cameras. We overlay our results on top of the results of Fgure 5 n [7]. MRM performs smlarly to the best magematchng technque of [7] eventhough a smple representaton s used. Regularzed CCA (best parameter found) yelds poor results wth ths poorly dscrmnatve data and Lnear Rdge-Regresson performed even worse (not shown). 7. Results We present results for the MRM algorthm for varous applcatons. These nclude camera to camera mappng, and mmckng the performance of face recognton technques. Other examples are provded n [26]. Below, 2000 random hyperplanes were used n all examples Matchng between cameras In several applcatons t s useful to compare the the output of multple cameras. We employ two such datasets. The frst was presented n [7] and contans the output of detected cars n two traffc cameras. The other s collected by us. In the frst dataset, the task s to dentfy car based on a sngle vew from the other camera, whle tranng s on such matched pars, but of dfferent cars. The tranng set contans about 120 cars, and the test set contans 50 dfferent car types. Ths yelds a tranng set wth 2922 samples, each s a 1200 dmensonal vector, descrbng the three channels of rescaled 20x20 pxel mages. Tranng tme was 17 mnutes on a standard PC. Results are comparable to [7], whch employs a sophstcated mage matchng technque. See Fg. 2. We also receved 4 mnutes of two synchronzed vdeo sequences wth partly overlappng felds of vew. We used half of the vdeo to apply MRM wth random hyperplanes. To test, we randomly pcked from the second half of the vdeo one frame from one camera, and 10 frames from the other camera, out of whch 9 are random dstractors.

6 (a) (b) Fgure 3. Sample frames from two synchronzed vdeos. Frames from vew (b) are mapped to vew (a). The vews overlap only partly, and are taken at very dfferent angles. The red crosses mark the most promnent locaton n the vew of (a) are pxels n range (.e. columns n mappng matrx T ) that ther norm s > 10% of the strongest. These turn out to be at the expected locatons wthn the regon vewed by both cameras. We run also the opposte drecton and got analog marks n (b). Here algorthm have chosen to focus on the area when vast majorty of moton occurs. Note that the two frames here are not matched n tme. Fgure 3 depcts two sample frames. In ths experment, both feature spaces were smply 1200 dmentonal vectors obtaned by reszng the mages to 40x30 pxels. Tranng MRM wth 290 frames has taken 2 mnutes. On the frame from the camera we used to map from, we mark red crosses to sgn the output coordnates whose weght (norm of relevant column n mappng matrx) f at least 10% the maxmum (most mportant pxel n spatal doman). Note that marks are spatally contnuous, as desrable, although we dd not employ such constrants. Fgure 4(a,b) shows the performance of the varous algorthms: Nearest Neghbor Transfer (Secton 2), regularzed CCA, and our MRM method. MRM consderably outperforms all other methods. We have also tred to analyze the opposte (less plausble) drecton: map from the wde feld of vew nto the narrow one. Analogous graphs appear n Fg. 4(c,d). As expected results are less mpressve, however MRM stll outperforms the two other methods Mmckng algorthm performance One task that s framed naturally n the vector-to-vector learnng framework s the task of mmckng an unknown algorthm. We test ths applcaton n the context of face recognton, usng the LFW database [11]. We wsh to recognze face mages based on the well-proven LBP feature set [1]. LBP represents each face mage as a vector. For the task of comparng two face mages and decdng whether they belong to the same person, we consder the vector v 12 of absolute dfferences v 1 v 2, where v 1 and v 2 are the LBP encodngs of these mages. Tranng lnear SVM on ths vector, usng 9 splts of the LFW benchmark for tranng and one for testng, and repeatng ten tmes, yelds an average performance of 69.5%, (a) (c) (d) Fgure 4. Performance charts for the vdeo to vdeo matchng task. (a) hstogram of the rankng of the true frame wthn the lst of dstractors as gven by each algorthm. As can be seen, MRM typcally ranks the hghest the correct frame. (b) ROC curves for same vs not-same queres n the same settngs,.e., the accuracy n whch each method s able to dstngush between matchng pars and non-matchng pars of frames. (c,d) Performance charts for the vdeo to vdeo matchng task, now mappng n the opposte drecton. whch s slghtly hgher than applyng a smple threshold to the Eucldean dstance (67.2%) between v 1 and v 2. To further mprove performance, we employ the output of more advance algorthms. We employ the raw outputs of the 16 dfferent classfers employed n [25] to obtan a results of 78.5%. Out of these 16 methods, only one can be drectly computed from the vector of absolute LBP dfferences v j (the Eucldean dstance of LBP descrptors). Usng MRM, we map these vectors of absolute dfferences to the 16D output obtaned by the methods of [25]. Thus, the nput to MRM contans 4800 samples (pars of faces, only 8 tranng splts, snce one s used as the background sample set of [25]), each consstng of a vector of 3717 absolute dfference and a vector of 16 classfer outputs. Tranng takes 26 mnutes. We then tran SVM on the values of T v j and obtan an average performance level of 74.6%, whch s not as hgh as the 16 classfers together, but s much easer to compute: snce both T and the learned SVM are lnear, the fnal classfer s lnear as well. CCA and lnear rdge regresson, appled n a smlar settng, both produce results lower than what s obtaned wth SVM tself (69.5%), for a wde range of regularzaton parameters. We have mmcked the performance of the face recognton algorthm based on the raw output of the classfers (b)

7 (real numbers). If the algorthms are gve as a black box, and only the fnal predcton s gven, a confdence level can be estmated emprcally, by addng nose to the nput data. Ths s left for future research. 8. Conclusons The problem of perceptual nference based on pars of matchng vectors s applcable to a wde range of computer vson problems. Despte some effort, the classcal regularzed CCA algorthm seems to performs better than many of the more recent contrbutons. Ths mght stem from the addton of dscrmnatve nformaton of non-matchng pars that s ether much less relevant than the nformaton obtaned by matchng pars, or s obtaned usng assumptons that are often false. Here we employ margns, however, unlke prevous work ths s done by usng dscrmnatve nformaton provded n the feature space tself. Another contrbuton s that we refran from maxmzng the margns n the transformed space, and nstead am to replcate the nput margns. Snce the nput margns are gven n one of the two spaces, and replcated n the other, the problem s asymmetrc by nature. We note that many of the applcatons n vector-to-vector learnng are ndeed asymmetrc. Whle our method maps one space to the other, we do not perform classcal regresson analyss. Our optmzaton framework allows the mapped ponts to consderably dffer from the matchng ponts n the other space. The emphass s put on smlar margns, not smlar coordnates, and the optmzaton computes a new set of bas terms. Acknowledgments Lor Wolf s supported by the Israel Scence Foundaton (grant No. 1214/06), and The Mnstry of Scence and Technology Russa-Israel Scentfc Research Cooperaton. Ths research was carred out n partal fulfllment of the requrements for the Ph.D. degree of Nathan Manor. References [1] T. Ahonen, A. Hadd, and M. Petkanen. Face descrpton wth local bnary patterns: Applcaton to face recognton. PAMI, 28(12): , Dec [2] F. R. Bach and M. I. Jordan. Kernel ndependent component analyss. JMLR, [3] K. Barnard and D. Forsyth. Learnng the semantcs of words and pctures. In CVPR, [4] G. I. Boser B. and V. V. An tranng algorthm for optmal margn classfers. In COLT, [5] T. Bylander and L. Tate. Usng valdaton sets to avod overfttng n adaboost. In Artfcal Intellgence Research Socety Conference, [6] A. Demrz, K. P. Bennett, and J. Shawe-Taylor. Lnear programmng boostng va column generaton. Mach. Learn., 46(1-3): , [7] A. Ferencz, E. G. Learned-Mller, and J. Malk. Learnng to locate nformatve features for vsual dentfcaton. IJCV, , 5 [8] H. H. Relatons between two sets of varates. In Bometrka, Vol. 28, pages , [9] T. Haste, R. Tbshran, and J. Fredman. The elements of statstcal learnng: data mnng, nference and predcton. Sprnger, [10] A. E. Hoerl and R. Kennard. Rdge regresson: based estmaton for nonorthogonal problems. Technometrcs, [11] G. Huang, M. Ramesh, T. Berg, and E. Learned-Mller. Labeled faces n the wld: A database for studyng face recognton n unconstraned envronments. Unversty of Massachusetts, Amherst, TR 07-49, [12] D. Ln and X. Tang. Inter-modalty face recognton. ECCV, pages 13 26, [13] H.-H. Nagel. Steps toward a cogntve vson system. AI Mag., [14] B. Schölkopf, A. Smola, R. C. Wllamson, and P. L. Bartlett. New support vector algorthms. Neural Computaton, 12: , [15] A. J. Smola, T. T. Freß, and B. Schölkopf. Semparametrc support vector and lnear programmng machnes. In NIPS, [16] B. T. Szedmak S. and H. D.R. A metamorphoss of canoncal correlaton analyss nto multvarate maxmum margn learnng. In ESANN, [17] R. Tbshran. Regresson shrnkage and selecton va the lasso. Journal Royal Statststcs, [18] K. M. Tng and I. H. Wtten. Stackng bagged and dagged models. In ICML, [19] I. Tsochantards, T. Hofmann, T. Joachms, and Y. Altun. Support vector learnng for nterdependent and structured output spaces. In ICML, [20] V. Vapnk. Statstcal learnng theory. Wley, [21] E. Vazquez and E. Walter. Mult-output support vector regresson. In 13th IFAC Symposum on System Identfcaton, pages Cteseer, [22] H. D. Vnod. Canoncal rdge and econometrcs of jont producton. Journal of Econometrcs, [23] B. J. Wenberger K. and S. L. Dstance metrc learnng for large margn nearest neghbor classfcaton. In NIPS, , 3 [24] L. Wolf and Y. Donner. An expermental study of employng vsual appearance as a phenotype. CVPR, , 2 [25] L. Wolf, T. Hassner, and Y. Tagman. Descrptor based methods n the wld. In Faces n Real-Lfe Images Workshop n ECCV, [26] L. Wolf and N. Manor. Vsual recognton usng mappngs that replcate margns. Extended Techncal Report, avalable at wolf, , 4, 5 [27] L. Wolf and A. Shashua. Learnng over sets usng kernel prncpal angles. J. Mach. Learn. Res., 4: ,