Multi-view Clustering with Adaptively Learned Graph

Transcription

1 Proceedngs of Machne Learnng Research 77: , 217 ACML 217 Mult-vew Clusterng wth Adaptvely Learned Graph Hong Tao Chenpng Hou Jubo Zhu ju bo Dongyun Y dongyun.y@gmal.com College of Scence, Natonal Unversty of Defense Technology, Changsha, Hunan, 4173, Chna Edtors: Yung-Kyun Noh and Mn-Lng Zhang Abstract Mult-vew clusterng, whch ams to mprove the clusterng performance by explorng the data s multple representatons, has become an mportant research drecton. Graph based methods have been wdely studed and acheve promsng performance for mult-vew clusterng. However, most exstng mult-vew graph based methods perform clusterng on the fxed nput graphs, and the results are dependent on the qualty of nput graphs. In ths paper, nstead of fxng the nput graphs, we propose Mult-vew clusterng wth Adaptvely Learned Graph (MALG), learnng a new common smlarty matrx. In our model, we not only consder the mportance of multple graphs from vew level, but also focus on the performance of smlartes wthn a vew from sample-par level. Sample-par-specfc weghts are ntroduced to explot the connecton across vews n more depth. In addton, the obtaned optmal graph can be parttoned nto specfc clusters drectly, accordng to ts connected components. Expermental results on toy and real-world datasets demonstrate the effcacy of the proposed algorthm. Keywords: Mult-vew Clusterng, Graph-based Learnng, Sample Par Sgnfcance 1. Introducton In many real-world applcatons, such as vdeo survellance and mage retreval, heterogeneous features can be obtaned to represent the same nstance. For example, an mage can be characterzed by varous descrptors, such as SIFT (Lowe (24)), hstograms of orented gradents (HOG) (Dalal and Trggs (25)), GIST (Olva and Torralba (21)) and local bnary pattern (LBP) (Ojala et al. (22)); a webpage can be descrbed by ts content, the text of webpages lnkng to t, and the lnk structure of lnked pages. Snce these heterogeneous features summarze the objects characterstcs from dstnct perspectves, they are regarded as multple vews of the data. Mult-vew learnng, whch explores the nformaton from dfferent vews to mprove the learnng performance, has become an mportant research drecton (Xu et al. (213); Hou et al. (217)). Clusterng s the task to partton objects nto meanngful groups wthout label nformaton. To properly ntegrate the nformaton from multple vews, many mult-vew clusterng approaches have been proposed (Chaudhur et al. (29); Kumar and Daumé (211); Lu et al. (213); Ca et al. (213)). Among these methods, a number of graph based mult-vew clusterng algorthms were presented wth good performance. Kumar et al. (211) extended spectral clusterng for mult-vew learnng by co-regularzng the clusterng hypotheses to c 217 H. Tao, C. Hou, J. Zhu & D. Y.

2 Tao Hou Zhu Y make graphs from dfferent vews agree wth each other. Wth a nonnegatve constrant on the relaxed cluster assgnment matrx, Ca et al. (211) developed the mult-modal spectral clusterng (MMSC) algorthm to learn a commonly shared graph Laplacan matrx. Based on bpartte graph, L et al. (215) utlzed local manfold fuson to ntegrate heterogeneous features and proposed a new large-scale mult-vew spectral approach (MVSC). Ne et al. (217) proposed a mult-vew clusterng method wth adaptve neghbors (MLAN), ths method performs clusterng and local structure learnng smultaneously. Despte the effcacy of graph based mult-vew clusterng methods, there stll exsts some lmts. On one hand, some methods conduct the subsequent procedures based on the gven smlarty matrces wthout modfyng them. Thus, the clusterng results are dependent on the qualty of the nput graphs. To solve ths problem, Ne et al. (216) proposed a constraned laplacan rank (CLR) model to learn a new smlarty matrx based on the gven ntal affnty matrx. By mposng rank constrant on the correspondng Laplacan matrx, the resultant smlarty matrx has c exact connected components (c s the number of clusters) and each component corresponds to a cluster. However, CLR s developed n sngle-vew settng and s napplcable to dealng wth multple graphs smultaneously. On the other hand, those methods combnng dfferent vews only consder the dversty of vew contrbutons, but gnore the qualty of parwse relatonshps wthn the same vew. In practce, the nose s usually non-homogeneously dstrbuted, smlartes of dfferent sample pars wthn the same vew mght also have vared capablty for dstngushng the true cluster structure. Thus, sample par sgnfcance analyss s very mportant for learnng the detaled connecton across vews. Instead of treatng smlartes of dfferent sample pars equally (usng the same weght for a certan vew), a promsng choce s to assume that dfferent smlartes usually have dstnct weghts accordng to ther sample par sgnfcance. Specfcally, we ntroduce sample-par-specfc weghts to measure the sample par sgnfcance. How to adaptvely learn these sample-par-specfc weghts s crucal to fnd the rght clusterng structure. Self-paced learnng (SPL) (Kumar et al. (21); Jang et al. (214)), whch ncorporates samples from easy to complex nto the tranng process by adaptvely assgnng them weghts, offers a feasble strategy for addressng ths problem. There are two weghtng manners n SPL,.e., hard weghtng and soft weghtng. Soft weghtng s demonstrated more effectve n prevous studes (Xu et al. (215); Zhao et al. (215)) and also fts our settng better. Hence, we transplant the soft weghtng manner of SPL nto mult-vew graph based clusterng. In ths paper, based on CLR, we propose Mult-vew clusterng wth Adaptvely Learned Graph (MALG), consderng both l 1 -norm and l 2 -norm dstances. Specfcally, gven ntal affnty matrces from multple vews, we attempt to learn a new common smlarty matrx wth explct clusterng structure by forcng rank constrant on the correspondng Laplacan matrx. Meanwhle, both the qualty dversty of multple graphs and the sample par sgnfcance wthn a certan graph are taken nto consderaton by ntroducng the sample-par-specfc weghts. Based on the above two aspects, the resultng model s able to adaptvely learn a new common smlarty matrx and explot the connecton between dfferent vews n more depth. There are several benefts of our approach: The proposed approach modfyng the smlarty matrx durng each teraton untl reach to the optmal one, and the outputted smlarty matrx explctly gve the clusterng result; The sample-par-specfc weghts are 114

3 Mult-vew Clusterng wth Adaptvely Learned Graph frst employed to measure the qualty of smlartes of dfferent sample pars for mult-vew graph based clusterng; The proposed objectve functons can be solved by a smple yet effectve algorthm; The clusterng performance of the proposed method s exaed on toy and real-world datasets and our approach outperforms other related methods n most cases. 2. Related Works We frst ntroduce the notatons used n ths paper. Matrces and vectors are wrtten as boldface uppercase letters and boldface lowercase letters respectvely. For matrx M, the -th row and the -th element of M are denoted as m and m, respectvely. The trace of matrx M s denoted by T r(m). The transpose of matrx M s denoted as M T. The l 1 - norm and l 2 -norm of vector v s denoted by v 1 and v 2, the Frobenus and the l 1 -norm of matrx M are represented as M F and M 1 respectvely. 1 denotes a column vector wth all ones, and the dentty matrx s denoted by I CLR Revsed Gven an ntal smlarty matrx A R n n, CLR ams to learn a new smlarty matrx S R n n wth exact c connected components, where n s the number of data ponts and c s the number of clusters. The Laplacan matrx related to S s defned as L S = D S (S T + S)/2, where D S s a dagonal matrx wth -th dagonal element as j (s + s j )/2. There s an mportant property of the Lapalcan matrx as follows (Chung (1997)). Theorem 1 The multplcty c of the egenvalue of the Laplacan matrx L S s equal to the number of connected components n the graph assocated wth S. Based on ths observaton, Ne et al. (216) constraned the rank of L S to be n c, and proposed the followng CLR model for graph based clusterng wth both l 1 -norm and l 2 -norm dstances: J CLR l1 = j s =1,s,rank(L S )=n c S A 1, (1) J CLR l2 = S j s =1,s,rank(L S )=n c A 2 F. (2) It has been shown that CLR acheves superor performance for clusterng (Ne et al. (216)). Note that the CLR method only works for sngle-vew data. It cannot deal wth graphs from multple vews smultaneously SPL By stmulatng the learnng process of humans, Bengo et al. (29) proposed currculum learnng to learn a model by gradually ncludng samples nto tranng from easy to complex. Instead of usng the predetered currculum, Kumar et al. (21) derved the SPL framework to learn the currculum and the model smultaneously. The general SPL model ncludes a weghted loss term on all samples and a regularzer term mposed on weghts: Θ,w [,1] n n w l (Θ) + f(w; λ), (3) =1 115

4 Tao Hou Zhu Y where l (Θ) denotes the loss functon of -th sample, Θ represents the model parameter, w = [w 1,, w n ] T denote the weght varables reflectng the samples mportance, f(w; λ) s the self-paced regularzer, and λ s a parameter. There are two knds of regularzaton n SPL: one assgns bnary weghts to samples (hard regularzaton), and the other allocates real-valued weghts (soft regularzaton). In real-world applcatons, the nose contaned n data s usually non-homogeneously dstrbuted. Under ths crcumstance, t has been shown that soft weghtng s more effectve (Zhao et al. (215); Xu et al. (215)). 3. Mult-vew Clusterng wth Adaptvely Learned Graph 3.1. The Proposed Formulaton Let A (v) R n n denote the smlarty matrx from the v-th vews (v = 1,, V ). In multvew clusterng, the clusterng results of dfferent vews should be consstent. We assume there s a common shared smlarty matrx S whch has exact c connected components. Thus, n the mult-vew settng, Eq. (1) and Eq. (2) can be extended to the followng formulatons: S A (v) 1, (4) S j s =1,s,rank(L S )=n c S j s =1,s,rank(L S )=n c S A (v) 2 F. (5) As dfferent vews may have dstnct physcal meanngs, treatng multple graphs equally s often dffcult to fnd the optmal soluton. What s more, the above two formulatons fal to measure the dfferences n ndvdual sample pars wthn and across vews. In order to brdge ths gap, we ntroduce sample-par-specfc weghts w (v) nto the mult-vew graph based clusterng model. Consderng both l 1 -norm and l 2 -norm dstances, the objectve functons of MALG are formulated as follows: S,{W (v) } V s.t. S,{W (v) } V s.t. W (v) (S A (v) ) 1 + f(w (v) ; λ) j s = 1, s, rank(l S ) = n c, W (v) [, 1] n n, 1 v V, W (v) (S A (v) ) 2 + F f(w(v) ; λ) j s = 1, s, rank(l S ) = n c, W (v) [, 1] n n, 1 v V, (6) (7) where W (v) = (w (v) ) 1,j n s composed of the weghts of n n elements n the v-th vew, W (v) calculates element-wse square root of W (v) (.e., the so-called Hadamard s root), s the element-wse product (Hadamard product) operaton of matrces, f(w (v) ; λ) denotes the regularzer mposed on the weghts, and λ s a parameter. The frst term n Eq. (6) or (7) s the weghted loss between the learned smlarty and ther nput values across dfferent 116

5 Mult-vew Clusterng wth Adaptvely Learned Graph vews. Weghts of hgh-qualty smlartes are encouraged to be larger n order to obtan small losses. In order to learnng the weghts w (v) automatcally durng the optmzaton process, followng Xu et al. (215), the regularzaton term f(w (v) ; λ) s set as follows: f(w (v) ; λ) = ln (1 + e λ w (v) w (v) )(1+e λ ) + ln (w (v) )w(v) λw (v). (8) The optmal weght of the ()-th element n the smlarty matrx n the v-th vew can be obtaned by solvng l(v) + f(w (v) ; λ), (9) where l (v) represents s (v) w (v) the dervatve wth respect to w (v) [,1] w (v) a (v) or (s(v) a (v) )2 for the convenence of notaton. Settng to zero, then we have w (v) = 1 + e λ. (1) 1 + e l(v) λ It s worth to note that Eq. (1) can be regarded as an adapted logstc functon, whch nherts the merts of logstc functon and provdes probablstc weghts. By combnng Eqs. (6) - (7) and (8) respectvely, we obtan the resultng objectve functons. The learned common smlarty matrx S can be used for clusterng drectly accordng to Tarjan s strongly connected components algorthm (Tarjan (1972)) Optmzaton Denote σ (L S ) as the -th smallest egenvalue of L S. Snce L S s postve sem-defnte, we c have σ (L S ). So the constrant rank(l S ) = n c wll be ensured f σ (L S ) =. Accordng to Ky Fan s Theorem (Fan (1949)), we have c σ (L S ) = T r(f T L S F) (11) F R n c,f T F=I =1 Hence, wth a large enough γ, problems (6) and (7) are equvalent to the followng problems: =1 S,F,{W (v) } V s.t. W (v) (S A (v) ) 1 + 2γT r(f T L S F) + f(w (v) ; λ) j s = 1, s, F T F = I, F R n c, W (v) [, 1] n n, 1 v V, (12) S,F,{W (v) } V s.t. W (v) (S A (v) ) 2 + 2γT F r(ft L S F) + f(w (v) ; λ) j s = 1, s, F T F = I, F R n c, W (v) [, 1] n n, 1 v V. (13) 117

6 Tao Hou Zhu Y The optmal soluton to these two problems wll make equaton c =1 σ (L S ) = holds. The problems (12) and (13) can be solved n an alternatng fashon,.e., dvdng varables nto dsjont blocks and alternatvely optmzng one of them wth the others fxed. Fx S and F, update {W (v) } V. When S and F s fxed, W(v) s optmzed by solvng the followng problem, { w (v) l(v) + f(w (v) }, ; λ) (14) W (v) 1,j n where l (v) = s (v) a (v) for problem (12) or l(v) = (s (v) a (v) )2 for problem (13) wth the current obtaned S. Ths optmzaton s separable wth respect to each w (v), and thus can be easly solved by Eq. (1). Fx S and {W (v) } V, update F. (13) are transformed nto When S and {W(v) } V are fxed, problems (12) and F R n c,f T F=I T r(f T L S F). (15) The optmal soluton F s formed by the c egenvectors that correspond to the c smallest egenvalues of L S. Fx {W (v) } V and and F, update S. S j s =1,s,j S j s =1,s,j Respectvely, problems (12) and (13) become w (v) s a (v) + γ,j w (v) (s a (v) )2 + γ,j f f j 2 2 s, (16) f f j 2 2 s, (17) where f denotes the -th row ( = 1,, n) of F. As there s no dependence between dfferent s, the above two problems can be solved for each ndvdual : j s =1,s j s =1,s j j w (v) s a (v) + γ,j w (v) (s a (v) )2 + γ,j f f j 2 2 s, (18) f f j 2 2 s. (19) Denote b = f f j 2 2, and denote b as a vector wth the j-th element as b (same for s, a (v) and w (v) ), then problems (18) and (19) can be wrtten n vector form as s T 1=1,s dag(w (v) )(s a (v) + γs T b, (2) 1 ) 118

7 Mult-vew Clusterng wth Adaptvely Learned Graph s T 1=1,s dag( w (v) )(s a (v) ) 2 F + γs T b, (21) where dag(x) returns a dagonal matrx wth the elements of x on the man dagonal. Usng the teratve re-weghtng method (Ne et al. (216)), problem (2) can be addressed by solvng the followng problem teratvely: s T 1=1,s T r(s a (v) ) T U (v) (s a (v) )+γs T b, (22) where U (v) s a dagonal matrx wth the j-th dagonal element as u (v) jj = w (v) 2 s a (v) (23), and s s the current soluton. For problem (21) wth the l 2 -norm dstance, denote U (v) = dag(w (v) ), then t can also be unfed n the form of problem (22). Now, we focus on solvng problem (22). Let p = U = U (v), (24) then problem (22) can be further smplfed as The Lagrangan functon of problem (26) s U (v) a (v) γ 2 b, (25) s T Us s T p. (26) s T 1=1,s L(s, η, β ) = 1 2 st Us s T p η(s T 1 1) β T s, (27) where η and β are the Lagrangan multplers. Accordng to the KKT condton, the optmal soluton of s s s = ( 1 η + 1 p ) +, (28) u u where (x) + = max(, x). Defne the followng functon g (x) = ( 1 u x + 1 u p ) + 1. (29) Then accordng to Eqs. (28) and (29), and the constrant s T 1 = 1, t holds that g (η) =. (3) 119

8 Tao Hou Zhu Y That s to say, the value of η s the root of functon g (x). Note that g (x) s a pecewse lnear and monotoncally ncreasng functon, thus ts root can be easly found wth Newton method. Once η s solved, the optmal soluton to problem (26) can be obtaned by Eq. (28). As shown n Eq. (23), the U (v) defned for problem (2) s dependent on S and thus s also an unknown varable. Wth alternatvely updated U (v) and S, problem (2) can be fnally solved. It s proved that ths teratve method decreases the objectve of problem (2) n each teraton and wll converge to the optmal soluton (Tao et al. (216)). For problem (21), t can be drectly resolved, snce there s no dependence between U (v) and S. The algorthm s descrbed n Algorthm 1. Algorthm 1 Mult-vew Clusterng wth Adaptvely Learned Graph Input: The smlarty matrces for V vews {A (1),, A (V ) }, A (v) R n n, number of clusters c, a large enough γ, the regularzaton parameter λ, ntal smlarty matrx S R n n. Intalzaton: calculate {l (v) } 1,j n, v = 1,, V, t Repeat 1. Update W (v) t = (w (v),t ) accordng to Eq. (1). 2. Update each row of S t by solvng problem (2) or (21). 3. Update F t by solvng problem (15). 4. Compute current {l (v) } 1,j n, v = 1,, V. 5. t t + 1. Untl convergence. Output: The smlarty matrx S R n n wth exact c connected components. 4. Experment In ths secton, we evaluate MALG on both synthetc and real-world datasets. For smplcty, we denote MALG wth l 1 -norm and l 2 -norm dstances as MALG(l 1 ) and MALG(l 2 ), respectvely Toy Example We construct a synthetc dataset, whch contans three graphs sharng the same nodes (vertces) but multple types of nteractons. Ths synthetc dataset has 3 clusters, and each cluster has 5 members. Fg. 1(a) - 1(c) dsplay the ntal smlarty matrces from the three graphs. On each vew, two of three clusters entangle wth each other. The affnty data wthn each block s randomly generated n the range of and 1, whle the nose data s randomly generated n the range of and.8 Moreover, to make ths clusterng task more challengng, we randomly choose m nose data and set ther value to be 1, from Vew1 to Vew3, m s 2, 3, and 4 respectvely. Note that when parttonng data ponts nto groups accordng to the connected components of a graph, we merely care whether a element of the smlarty matrx s zero or not and gnore ts detaled values. Thus, we use the adjacent matrx whch only contans 12

9 Mult-vew Clusterng wth Adaptvely Learned Graph bnary values ( or 1) to represent the parwse relatonshps learned by related methods. We run all methods usng the l 1 -norm dstances. Fg. 1(d) - 1(f ) show the learned parwse relatonshps by perforg CLR on each vew. It can be observed that sngle-vew CLR fals to fnd the true clusterng structure. To valdate the effcency of sample-par-specfc weghts n MALG, we compare our method wth combnng multple graphs wthout weght (denoted as MvCLR-nave) as shown n Eq. (4) as well as vew-weghtng CLR (Vew-CLR for short). The objectve of vew-weghtng CLR wth l 1 -norm dstance s formulated as S,α s.t. α (v) S A (v) 1 j s = 1, s, rank(l S ) = n c α (v) = 1, α (v) >, (31) where α (v) s the weght of the v-th vew. The learned parwse relatonshps of MvCLR-nave, Vew-CLR and MALG are presented n Fg. 1(g), 1(h) and 1(), respectvely. It can be seen that the performances of both MvCLR-nave and Vew-CLR are affected by the nose. The dstorton of parwse relatonshps for MvCLR-nave and Vew-CLR are manly brought by Vew3, snce t has the most nose data. In comparson, our proposed MALG elates the mpact of nose and obtans the deal structure for clusterng Mult-vew Clusterng Comparson on Real-world Datasets In ths subsecton, we evaluate our approach on fve real-world datasets. Snce the proposed method s knd of graph based learnng model, we compare t wth other related graph based mult-vew clusterng algorthms. In partcular, we focus on comparng wth followng algorthms: Best sngle-vew CLR (BestCLR) (Ne et al. (216)). On each vew, the sngle-vew CLR algorthm s performed n both l 1 -norm and l 2 -norm dstances. The best results are reported. Co-regularzed mult-vew spectral clusterng (CoRegSC) (Kumar et al. (211)). Ths method co-regularzes the clusterng hypotheses from multple vews to enforce each vew to have the same cluster membershp. We mplement the centrod-based coregularzaton approach. Co-traned mult-vew spectral clusterng (CoTranSC) (Kumar and Daumé (211)). CoTranSC utlzes the spectral embeddng from one vew to constran the affnty graph used for the other vews. By teratvely applyng ths approach, the clusterng of multple vews tend to be the same. As there s no specfc stop crteron, we run 2 teratons. Mult-modal spectral clusterng (MMSC) (Ca et al. (211)). MMSC learns a commonly shared graph Laplacan matrx by mzng the dfferences between the common Laplacan graph and that of each vew. 121

10 Tao Hou Zhu Y (a) (b) (c) (d ) (g) (h) 1 15 (f ) 15 5 (e) ( ) Fgure 1: Vsualzaton of the toy example. (a) - (c) The generated synthetc nput smlarty matrces of three vews. (d) - (f) The parwse relatonshps learned by CLR on each vew. (g) The parwse relatonshps learned by MvCLR-nave. (h) The parwse relatonshps learned Vew-CLR. () The parwse relatonshps learned by our MALG. Note that we use the adjacent matrx whch contans bnary values ( or 1) to represent the parwse relatonshps of samples. 122

11 Mult-vew Clusterng wth Adaptvely Learned Graph Mult-vew spectral clusterng (MVSC) (L et al. (215)). Local manfold fuson s adopted to ntegrate heterogeneous features based on the bpartte graph. Mult-vew clusterng wth adaptve neghbors (MLAN) (Ne et al. (217)). MLAN performs clusterng and local structure learnng smultaneously, whch s also based on the property of the Laplacan matrx llustrated n Theorem 1. The mplementatons of CLR, CoRegSC, CoTranSC, MMSC and MLAN are downloaded from ther authors homepages Datasets Descrptons The detaled descrptons of the datasets are presented as follows and the statstcs of them are summarzed n Table 1. 3sources s a mult-vew text dataset, collected from three onlne news sources: BBC, Reuters, and The Guardan. We use the 169 stores that were reported n all three sources. Each source corresponds to a vew. Accordng to the prmary secton headngs used across the three news sources, each story was manually annotated wth one of the sx topcal labels: busness, entertanment, health, poltcs, sport, technology. Snce the orgnal dmenson of each vew s hgh (all above 3,), we frst perform PCA on each vew and reduce the dmenson to 1 for each vew. MSRC-v1 dataset contans 24 mages n 8 classes. Followng Ca et al. (211), we select 7 classes composed of tree, buldng, arplane, cow, face, car, bcycle and each class has 3 mages. We extract fve vsual features from each mage: color moment (CM) wth dmenson 48, local bnary pattern (LBP) wth 256 dmenson, HOG wth 1 dmenson, GIST wth 512 dmenson, and CENTRIST (Wu and Rehg (211)) feature wth 132 dmenson. Caltech11 s an object recognton dataset consstng of 11 categores of mages. We follow prevous work (L et al. (215)) and select the wdely used 7 classes,.e. Dolla-Bll, Face, Garfeld, Motorbkes, Snoopy, Stop-Sgn and Wndsor-Char and get 441 mages. The selected dataset s referred to as Caltech11-7. The same fve types of features wth MSRC-v1 are extracted. Trecvd23 s a vdeo dataset, whch contans 178 manually labeled vdeo shots that belong to 5 categores. Each shot s represented as a 1894-dmensonal vector of text features and a 165-dmensonal vector of HSV color hstogram, whch s extracted from the assocated keyframe. GRAZ2 s a database for object categorzaton. It contans mages wth objects of hgh complexty and hgh ntra-class varablty. It conssts of 365 mages wth bkes, 311 mages wth persons, 42 mages wth cars and 38 mages not contanng one of these objects. We extract the followng 3 vsual features: SIFT, GIST, and LBP. 123

12 Tao Hou Zhu Y Table 1: Statstcs of the mult-vew datasets used n our experments. Dataset Instances Vews Clusters 3sources MSRC-v Caltech Trecvd GRAZ Table 2: Clusterng performance on 3sources. Methods NMI ACC F-score BestCLR.59().651().627() CoRegSC.24(.34).433(.39).36(.24) CoTranSC.476(.37).573(.3).493(.39) MMSC.621(.23).596(.35).585(.37) MVSC.314(.27).55(.55).444(.43) MLAN.59().586().556() MALG(l 1 ).733().85().795() MALG(l 2 ).746().811().798() Experment Setup The sngle-vew CLR approach s conducted on each vew wth both l 1 -norm and l 2 -norm dstances and the best results are reported. In CLR, MLAN and our proposed MALG, there s a common parameter γ brought by the Laplacan matrx rank constrant. For smple mplementaton and acceleratng the convergence speed, we ntalze γ = 8 for these three methods, and decrease t (γ = γ/4) f the connected components of S s greater than the number of clusters c or ncrease t (γ = γ 4) f smaller than c n each teraton. There s another parameter n our method,.e., the regularzaton parameter λ. To obtan a better model, t s set vew-by-vew: λ (v) = π (v) + log((π (v) ) 2 + 1)t,where π (v) denotes the medan of the losses l (v) of all smlartes on the v-th vew, and t s the number of teratons. For CoRegSC, CoTranSC and MMSC, ther trade-off parameters are all selected from {.1,.1, 1, 1, 1}, the best results are reported. For MVSC, the number of salent ponts s set as 1% of the number of total samples. On all datasets, each sample s assgned 1 nearest neghbors to construct graph. We use the graph constructon method descrbed n Ne et al. (216) to ntalze the graphs for CLR and our method. Approaches based on spectral clusterng need perform post-processng, such as k-means, to get the fnal clusterng results. We repeat experments for 5 tmes for all methods and report the average results and the standard devaton. The clusterng performance s measured usng three evaluaton metrcs: clusterng accuracy (ACC), normalzed mutual nformaton (NMI) and F-score. For all three metrcs, hgher value means better clusterng qualty. 124

13 Mult-vew Clusterng wth Adaptvely Learned Graph Table 3: Clusterng performance on MSRC-v1. Methods NMI ACC F-score BestCLR.692().79().651() CoRegSC.772(.21).841(.44).745(.32) CoTranSC.797(.29).86(.53).764(.39) MMSC.795(.21).825(.58).742(.4) MVSC.531(.2).613(.25).485(.17) MLAN.778().81().722() MALG(l 1 ).924().962().924() MALG(l 2 ).96().952().94() Table 4: Clusterng performance on Caltech11-7. Methods NMI ACC F-score BestCLR.576().58().428() CoRegSC.623(.32).661(.37).68(.34) CoTranSC.6(.23).657(.49).595(.38) MMSC.698(.27).696(.29).661(.37) MVSC.583(.29).64(.47).568(.46) MLAN.438().488().444() MALG(l 1 ).739().77().78() MALG(l 2 ).728().664().633() Performance Evaluaton The clusterng results are presented n Table 2-6. We have followng observatons. It can be seen that the proposed method acheves the best clusterng results n most cases. As shown n Table 2 and 3, on 3sources and MSRC-v1, our proposed MALG n both l 1 -norm and l 2 -norm dstances outperforms the compared approaches sgnfcantly n terms of all the three metrcs. Compared wth the second best baselne, MALG acheves over 1% mprovements wth respect to NMI, ACC and F-score. Table 4 shows that, on Caltech11-7, despte the performance superorty s not obvous, MALG stll gets the best clusterng results when adoptng the l 1 -norm dstance. In terms of ACC and F-score, MALG(l 1 ) obtans the hghest scores on Trecvd23, as dsplayed n Table 5. When measurng wth NMI, MMSC performs the best, and MALG(l 1 ) ranks second. On GRAZ2, MALG(l 2 ) and MALG(l 1 ) respectvely get the best NMI and F-score, whle MMSC s slghtly better than MALG wth respect to ACC (Table 6). Except for Caltech11-7 and GRAZ2, the mpact of the utlzaton of dfferent dstances (l 1 -norm or l 2 -norm) n MALG on the clusterng results s not markedly on the other three datasets. On Caltech11-7, MALG(l 1 ) gets hgher ACC and F-score values wth a obvous gap compared wth that of MALG(l 2 ), whereas on GRAZ2, MALG(l 2 ) outperforms MALG(l 1 ) n terms of ACC. MLAN s most closely related wth our method, as t s also developed based on the Laplacan rank constrant. However, n comparson wth our method, the clusterng results 125

14 Tao Hou Zhu Y Table 5: Clusterng performance on Trecvd23. Methods NMI ACC F-score BestCLR.219().433().427() CoRegSC.177(.5).43(.12).32(.7) CoTranSC.215(.4).42(.13).343(.11) MMSC.266(.3).431(.32).365(.18) MVSC.189(.36).43(.37).344(.15) MLAN.194().434().412() MALG(l 1 ).252().497().434() MALG(l 2 ).254().497().435() Table 6: Clusterng performance on GRAZ2. Methods NMI ACC F-score BestCLR.14().457().363() CoRegSC.79(.1).48(.6).36(.1) CoTranSC.6(.5).393(.16).292(.4) MMSC.142(.2).477(.2).356(.1) MVSC.85(.1).426(.2).311(.) MLAN.45().268().396() MALG(l 1 ).153().387().419() MALG(l 2 ).169().475().4() obtaned by MLAN are much worse. Especally, on Caltech11-7, the NMI (ACC, F-score) of MALG(l 1 ) mproves about 3% (21%, 26%) over that of MLAN. The reason mght be that MALG consder the sgnfcance of sample pars both wthn vew and across vews, whle MLAN treat sample pars wthn the same vew equally. Note that the standard devaton values of CLR, MLAN and our method are all zeros, snce they need not post-processng wth k-means. 5. Concluson In ths paper, we propose a mult-vew graph based clusterng method to adaptvely learn a common graph wth exact c connected components, whch s an deal structure for clusterng. Instead of treatng sample pars wthn each vew equally, sample-par-specfc weghts are ntroduced to evaluate the mportance of smlartes n a partcular vew. Two objectve functons wth both l 1 -norm and l 2 -norm dstances are formulated, and a smple yet effectve algorthm s derved to solve them. The effectveness of our method s valdated on fve realworld datasets for news artcle clusterng and mage clusterng. Expermental results show that the proposed method acheves robust performance and outperforms several related methods. 126

15 Mult-vew Clusterng wth Adaptvely Learned Graph Acknowledgments The authors would lke to thank the anonymous revewers. Ths work was supported by Chna Natonal Scence Foundaton under Grants No and No References Yoshua Bengo, Jérôme Louradour, Ronan Collobert, and Jason Weston. Currculum learnng. In ICML, pages 41 48, 29. Xao Ca, Fepng Ne, Heng Huang, and Farhad Kamangar. Heterogeneous mage feature ntegraton va mult-modal spectral clusterng. In CVPR, pages IEEE, 211. Xao Ca, Fepng Ne, and Heng Huang. Mult-vew k-means clusterng on bg data. In IJCAI, pages , 213. Kamalka Chaudhur, Sham M Kakade, Karen Lvescu, and Karthk Srdharan. Mult-vew clusterng va canoncal correlaton analyss. In ICML, pages , 29. Fan RK Chung. Spectral graph theory. Number 92. Amercan Mathematcal Socety, Navneet Dalal and Bll Trggs. Hstograms of orented gradents for human detecton. In CVPR, volume 1, pages , 25. K. Fan. On a theorem of weyl concernng egenvalues of lnear transformatons I. Proceedngs of the Natonal Academy of Scences, 35(11): , Chenpng Hou, Fepng Ne, Hong Tao, and Dongyun Y. Mult-vew unsupervsed feature selecton wth adaptve smlarty and vew weght. IEEE Transactons on Knowledge and Data Engneerng, PP(99):1 1, 217. Lu Jang, Deyu Meng, Shoou Yu, Zhenzhong Lan, Shguang Shan, and Alexander Hauptmann. Self-paced learnng wth dversty. In NIPS, pages , 214. Abhshek Kumar and Hal Daumé. A co-tranng approach for mult-vew spectral clusterng. In ICML, pages 393 4, 211. Abhshek Kumar, Pyush Ra, and Hal Daume. Co-regularzed mult-vew spectral clusterng. In NIPS, pages , 211. M Pawan Kumar, Benja Packer, and Daphne Koller. varable models. In NIPS, pages , 21. Self-paced learnng for latent Yeqng L, Fepng Ne, Heng Huang, and Junzhou Huang. Large-scale mult-vew spectral clusterng va bpartte graph. In AAAI, pages , 215. Jalu Lu, Ch Wang, Jng Gao, and Jawe Han. Mult-vew clusterng va jont nonnegatve matrx factorzaton. In ICDM, pages , 213. Davd G Lowe. Dstnctve mage features from scale-nvarant keyponts. Internatonal Journal of Computer Vson, 6(2):91 11,

16 Tao Hou Zhu Y Fepng Ne, Xaoqan Wang, Mchael I. Jordan, and Heng Huang. The constraned laplacan rank algorthm for graph-based clusterng. In AAAI, pages , 216. Fepng Ne, Guohao Ca, and Xuelong L. Mult-vew clusterng and sem-supervsed classfcaton wth adaptve neghbours. In AAAI, pages , 217. Tmo Ojala, Matt Petkänen, and Top Mäenpää. Multresoluton gray-scale and rotaton nvarant texture classfcaton wth local bnary patterns. IEEE Transactons on Pattern Analyss and Machne Intellgence, 24(7): , Jul. 22. Aude Olva and Antono Torralba. Modelng the shape of the scene: A holstc representaton of the spatal envelope. Internatonal Journal of Computer Vson, 42(3): , 21. H. Tao, C. Hou, F. Ne, Y. Jao, and D. Y. Effectve dscratve feature selecton wth nontrval soluton. IEEE Transactons on Neural Networks and Learnng Systems, 27 (4):796 88, 216. Robert Tarjan. Depth-frst search and lnear graph algorthms. SIAM Journal on Computng, 1(2):146 16, J. Wu and J. M. Rehg. CENTRIST: A vsual descrptor for scene categorzaton. IEEE Transactons on Pattern Analyss and Machne Intellgence, 33(8): , 211. Chang Xu, Dacheng Tao, and Chao Xu. A survey on mult-vew learnng. Computer Scence, 213. Chang Xu, Dacheng Tao, and Chao Xu. Mult-vew self-paced learnng for clusterng. In IJCAI, pages , 215. Qan Zhao, Deyu Meng, Lu Jang, Q Xe, Zongben Xu, and Alexander G Hauptmann. Self-paced learnng for matrx factorzaton. In AAAI, pages ,