Multiple Medoids based Multi-view Relational Fuzzy Clustering with Minimax Optimization

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1 Mutipe Medoids based Muti-view Reationa Fuzzy Custering with Minimax Optimization Yangtao Wang, Lihui Chen 2, Xiaoi Li Institude for Infocomm Research(I2R), A*STAR, Singapore 2 Schoo of Eectrica and Eectronic Engineering, Nanyang Technoogica University, Singapore {wangyt, xi}@i2r.a-star.edu.sg, ELHCHEN@ntu.edu.sg Abstract Muti-view data becomes prevaent nowadays because more and more data can be coected from various sources. Each data set may be described by different set of features, hence forms a muti-view data set or muti-view data in short. To find the underying pattern embedded in an unabeed mutiview data, many muti-view custering approaches have been proposed. Fuzzy custering in which a data object can beong to severa custers with different memberships is widey used in many appications. However, in most of the fuzzy custering approaches, a singe center or medoid is considered as the representative of each custer in the end of custering process. This may not be sufficient to ensure accurate data anaysis. In this paper, a new muti-view fuzzy custering approach based on mutipe medoids and minimax optimization caed M4-FC for reationa data is proposed. In M4-FC, every object is considered as a medoid candidate with a weight. The higher the weight is, the more ikey the object is chosen as the fina medoid. In the end of custering process, there may be more than one medoid in each custer. Moreover, minimax optimization is appied to find consensus custering resuts of different views with its set of features. Extensive experimenta studies on severa muti-view data sets incuding rea word image and document data sets demonstrate that M4-FC not ony outperforms singe medoid based mutiview fuzzy custering approach, but aso performs better than existing muti-view reationa custering approaches. Introduction Muti-view data becomes prevaent nowadays because arge amount of data can be easiy coected from different sources or represented by different features. Mining vauabe information in the prevaent muti-view data is very important for different parties nowadays. Custering as an exceent too of data anaysis to find the structure of pattern and information underining the unabeed data has been widey studied in different areas incuding artificia inteigence, data mining and machine earning. Based on various theories and appications, different custering agorithms have been deveoped to hande different types of data [Jain, 200; Fiippone et a., 2008; Xu and Wunsch, 2005]. As rea word data sets may not be we separated, fuzzy custering agorithms in which each object can be assigned to different custers with different memberships are widey used in many appications. Two main types of data processed by custering agorithms are feature data and reationa data which are represented as feature vectors and reationa (distance) matrix. Reationa data is prevaent as it is a more generic data representation in many rea word appications. Many custering approaches have been proposed to hande these two kinds of data. For exampe, two popuar fuzzy custering approaches Fuzzy c medoids (FCMD)[Krishnapuram et a., 200] and Fuzzy c means (FCM)[Bezdek, 98], hande reationa data and feature data respectivey. Note that FCMD can aso be appied on feature data because it can be transferred to reationa data easiy by cacuating the distance of each pair of feature vectors, whie feature vector-oriented FCM can ony hande feature vector data and can not hande reationa data directy. Different muti-view custering approaches based on FCM and FCMD are aso proposed in the iterature. However, singe center and singe medoid is considered for each custer in FCM and FCMD respectivey, which may not be sufficient for various data sets. In this paper, a new muti-view reationa fuzzy custering approach caed M4-FC is proposed which is abe to achieve the fuzzy partition of objects based on mutipe distance matrices. Two mechanisms are introduced into M4-FC to better use the information of different views and improve the custering performance. First, each object is considered as a medoid candidate by introducing a representative weight to represent the degree of representativeness of each object to its custer. The higher the weight is, the more probabe this object can be considered as a medoid. Second, minimax optimization in which the maximum summation of weighted cost of different views are minimized is appied to find more consensus custering resuts of different views. Severa muti-view fuzzy custering approaches have been proposed in the iterature. For exampe, two FCM based muti-view fuzzy custering CoFKM [Ceuziou et a., 2009] and WV-Co-FCM [Jiang et a., 205] are deveoped. They 297

2 use the same two-step strategy. In the first step, the membership of each view is achieved by minimizing the FCM reated objective function and the disagreements between different views. Then the extra step is needed to get the fina consensus fuzzy membership based on the membership of a views. From an optimization point of view, both CoFKM and WV-Co-FCM formuated the muti-view custering to a minimization probem. Another optimization method used to formuate muti-view custering is minimax optimization. Minimax optimization is used in muti-objective optimization to hep making decision rues. For muti-view custering, custering each view can be considered as one objective and minimax optimization is subsequenty used to find consensus custering resuts of different views. Recenty a muti-view fuzzy custering based on minimax optimization caed MinimaxFCM has been proposed in [Wang and Chen, 207]. The superior experimenta resuts have shown the effectiveness of minimax optimization on muti-view custering. Note that CoFKM, WV-Co-FCM and MinimaxFCM can ony hande feature data. In order to hande reationa data, a muti-view reationa fuzzy custering approach caed MVFCMddV based on FCMD is proposed in [de Carvaho et a., 205]. However, singe medoid is identified for each custer in MVFCMddV which may not be sufficient to capture the pattern structure of various data sets. In this paper, we propose a new mutipe medoids based muti-view fuzzy custering (M4-FC) with minimax optimization to hande muti-view reationa data. Different from MVFCMddV in which a singe medoid is identified for each custer of each view, M4-FC is abe to identify mutipe medoids to better capture the underying custer structure by introducing a representative weight for each object. Inspired by [Wang and Chen, 207], minimax optimization is adopted in M4-FC to integrate different views to hep to generate better consensus custering resuts. The experimenta resuts on different rea word data sets incuding image and document data sets show that M4-FC not ony outperforms singe medoid based muti-view fuzzy custering approach, but aso performs better than other muti-view custering approaches on a the benchmark data sets. The rest of the paper is organized as foows: in the next section, some reated fuzzy custering approaches are reviewed. In section 3, the detais of the proposed muti-view custering approach M4-FC are presented. Experiments on five rea word muti-view data sets are conducted and the resuts are anayzed in section 4. Finay, concusions are drawn in section 5. 2 Reated Work Due to the imited space, we ony review some cosey reated fuzzy custering based approaches for reationa data. 2. Fuzzy C Medoids (FCMD) FCMD [Krishnapuram et a., 200] is one of the most popuar fuzzy custering approaches for reationa data. The foowing objective function is minimized in FCMD agorithm: K J F CMD = u m cidis(x i, δ c ) () c= i= Where K is the custer number, u ci is the membership degree of object i to custer c, m > is the fuzzification constant caed fuzzifier, and δ c is the medoid (a data object) for custer c. Dis(x i, v c ) = x i δ c is the distance between object x i and medoid δ c. FCMD can generate a fuzzy partition of the data set based on its distance matrix and identify singe medoid for each custer. 2.2 MVFCMddV MVFCMddV [de Carvaho et a., 205] is the muti-view extension of the singe view FCMD for muti-view reationa data custering. The reevance weights are introduced for each custer of each view in the objective function as foows: J MV F CMddV = K c= i= u m ci P w cp Dis p (x i, δc p ) (2) Where u ci is consensus membership of mutipe views for object i in fuzzy custer c, w cp is reevance weight of distance matrix Dis p of p-th view on custer c, and Dis p (x i, δc p ) is the distance between object x i and medoid δc p of custer c under p-th view. By minimizing the weighted summation of the distance as shown in the objective function, MVFCMddV is abe to achieve the consensus fuzzy partition and a medoid for each custer under each view. As discussed above, FCMD and MVFCMddV identified ony one medoid for each custer. In our method, a objects are considered as medoid candidates by introducing a weight for each object. The higher the weight is, the more probabe this object can be considered as a medoid. Moreover, minimax optimization is appied to find consensus custering resuts of different views. 3 The Proposed Approach In this section, we formuate the objective function of the proposed M4-FC by integrating two types of mechanisms: introduction of the representative weight for each object in a custer and adoption of minimax optimization to find consensus custering resuts. In M4-FC, the consensus custering resuts are generated based on minimax optimization in which the maximum summation of weighted cost of different views are minimized. Moreover, the weight of each view can be earned automaticay in the custering process. 3. Formuation First the foowing objective function is formuated for M4- FC J M4 F C : J M4 F C = where Q (p) = K min U,{V (p) } P c= i= j= + T v 2 K (v (p) cj )2 c= j= max {α (p) } P u civ (p) cj r(p) ij + T u 2 P (α (p) ) γ Q (p) (3) K c= i= (u ci) 2 (4) 2972

3 subject to K u ci =, for i =, 2,..., N (5) c= u ci 0, for c =, 2,..., K, i =, 2,...N (6) j= v (p) cj =, for c =, 2,..., K (7) v (p) cj 0, for c =, 2,..., K, j =, 2,...N (8) P α (p) = (9) α (p) 0, for p =, 2,..., P (0) In J M4 F C, r (p) ij is the distance between object i and object j in p-th view which is the input. U is the K N membership matrix whose eement in row c and coumn i is the consensus membership u ci for object i of custer c. V (p) is the K N representative weight matrix whose eement in row c and coumn j is the representative weight v (p) cj. v(p) cj represents how we an object j can be considered as a medoid for custer c of p-th view. When weight is higher, this object is more probabe to be considered as a medoid. Q (p) is the cost of p- th view and (α (p) ) γ is the weight of p-th view. The parameter γ [0, ) contros the distribution of weights (α (p) ) γ for different views. The cost of p-th view Q (p) is composed of three parts. The first part is the distance summation of a custers weighted by membership and representative weight. The other two parts are appied as reguarization terms which are penaties on u ci and v (p) cj. T u > 0 and T v > 0 contro the weight of the two parts in the objective function. As shown in (3), the summation of weighted cost is maximized with respect to the weight (α (p) ) γ at first. In this step, the view with higher cost wi be given higher weight. Then the maximum of the summation of weighted cost is minimized with respect to consensus membership U and V (p). In this way, the consensus membership is generated by minimizing the cost under the worst case (maximum summation of weighted cost) to achieve harmonic consensus custering resuts. 3.2 Optimization The custering goa is to conduct a minimax optimization on the objective function P (α (p) ) γ Q (p), and subject to the constraints in (5), (6), (7), (8), (9) and (0). It is difficut to sove the variabes u ci, v(p) cj and α (p) in (3) directy because (3) is nonconvex. Therefore, the aternative optimizaiton (AO) is used to sove the optimization probem by soving one variabe with others fixed. Minimization: Fixing v (p) cj, α(p) and Updating u ci The updating rue of u ci can be derived based on the Lagrangian Mutipier method and KKT conditions as foows: 0 for c q q + P ( (α P (p) ) γ v (p) cj r(p) ij u ci = T u (α (p) ) γ j= q + P f q + j= (α (p) ) γ v (p) q = {c : u ci = 0} q + = {c : u ci > 0} fj r(p) ij )for c q+ () (2) Here, q is custer set in which the membership of object i is zero to the custers. q + is custer set in which the membership of object i is arger than zero. And q +, q are the number of custers in the set. The sets q + and q which are updated in each iteration are determined by a method simiar to the one used in [Mei and Chen, 200] which is outined as foows. Procedure for determining q + and q Initiaize q 0 + =, q 0 =, 2,..., K, s = 0; 2 s s +, q s + = q s + + {m}, q s = qs {m}; where m = arg min c q { P (α (p) ) γ v (p) s j= 3 Check whether u fi > 0 computed by (), where f = arg max c q + s { P (α (p) ) γ v (p) j= cj r(p) ij }; cj r(p) ij }. If yes, go to step 2, ese set q + = q + s, q = q s and terminate. Minimization: Fixing u ci, α(p) and Updating v (p) cj The v (p) cj is updated as foows: 0 for j m v (p) cj = where m + T v ( i= u cir (p) ij m + m = {j : v (p) cj = 0} m + = {j : v (p) cj > 0} f m + i= u cir (p) if ) for j m + (3) (4) Here, m + and m are object sets in which the representative weight for custer c of p-th view is positive and zero respectivey. m and m + represent the size of set m and m + respectivey. Simiar to q + and q, the process for determining m and m + is presented as foows. 2973

4 Procedure for determining m + and m Initiaize m + 0 =, m 0 =, 2,..., N, s = 0; 2 s s +, m + s = m + s + {m}, m s = m s {m}; where m = arg min j m { N u ci r(p) ij }; s i= 3 Check whether v cg (p) > 0 computed by Eq. 3, where g = arg max j m + s { N u ci r(p) ij }. i= If yes, go to step 2, ese set m + = m + s, m = m s and terminate. Maximization: Fixing u ci, v(p) cj and Updating α(p) Based on the Lagrangian Mutipier method, the updating rue α (p) is given as foow: α (p) = (Q(p) ) γ P (Q (j) ) γ j= (5) Here the cost term Q (p) is the weighted distance summation of a pairs of the data objects under p-th view. The arger the vaue of Q (p) is, the arger cost this view wi contribute to the objective function. From (5), we can see that the arger cost of p-th view is, the higher vaue wi be assigned to α (p) which eads to the maximum of the weighted cost. The maximum is then minimized with respect to the memberships u ci and v (p) cj in order to suppress the high cost views and achieve harmonic consensus custering resuts. The whoe agorithm of the proposed approach is isted in Agorithm. First, the consensus membership matrix U0 and α (p) 0 is initiaized. α (p) 0 is initiaized to be /P to consider each view equay. The initiaization method of U0 is introduced in Section 4.3. Then in second step, the representative weight matrix V (p) and α (p) for each view are updated using (3) and (5). After updating a the views, the membership matrix U is updated based on (). This step wi be continued unti the difference between two consecutive membership matrix is ess than ɛ. Agorithm : Input: Distance matrix of P views Rn n,..., Rn n, P number of custers K, parameter T u, T v,γ stopping criterion ɛ Output: Consensus Fuzzy membership matrix U, representative weight matrix of each view V (p) Method: Initiaize U0 and α (p) 0, set iteration number 0 2 Repeat + for p = to P Update V (p) using (3) based on U and α(p) Update α (p) using (5) based on U (p) and V end for Update U using () based on {V and {α ),..., α(p } Unti( U U < ɛ),..., V (P ) } The time compexity of M4-FC is O(P (N 2 )) considering the number views P and the data size N. The sma eements in distance matrix can be set to zero or appying samping approaches to acceerate the computation as reported in [Bezdek et a., 2006]. 4 Experiments In this section, we conduct experimenta studies of the proposed approach on five benchmark data sets. Tabe shows the summarized characteristics of these data sets. Feature Type in Tabe highights the number of data objects, and type of the features used in each feature set (or View) of the corresponding data set. The code is impemented in MAT- LAB and runs on a computer with eight cores and eight gigabytes of memory. 4. Data Sets Mutipe features (MF) : This data set is composed of 2000 handwritten digit images (0-9) which is from a coection of Dutch utiity maps. It has 0 casses and each cass has 200 images. Each image is represented by six different views incuding Fourier coefficients (FOU), profie correations (FAC), Karhunen-Love coefficients (KAR), pixe averages (PIX), Zernike moments (ZER), and morphoogica features (MOR). Image segmentation (IS) 2 : This data set is composed of 230 outdoor images which have 7 casses. Each image is represented by 9 features. The features can be considered as two views which are shape view and RGB view. The shape view consists of 9 features which describe the shape information of each image. The RGB view consists of 0 features which describe the RGB vaues of each image. Oxford Fowers 3 : This data set consists of 360 fower images. It has 7 casses and each cass has 80 images. Each image is represented by 4 views incuding Coour, Histogram of Gradients (HOG), SIFT on the foreground region (SIFTreg) and SIFT on the foreground boundary (SIFTbdy). 3-Sources document (3-S) 4 : This data set is composed of 948 news artices containing 46 distinct news stories. They are coected from three onine news sources: BBC, Guardian and Reuters. We seected 69 news artices which are reported in a three sources (views). It has 6 topic casses which are entertainment, heath, business, poitics, sport and technique. Reuters mutiingua: This data set consists of documents written in five different anguages (Engish, French, German, Spanish and Itaian) and their corresponding transations [Amini et a., 2009]. It has six casses. The origina documents in Engish are used as one view and their four transations are considered as the other four views. 500 documents are randomy samped with each cass having 250 documents. sets/mutipe+features. 2 sets/image+segmentation. 3 vgg/data/fowers/7/index.htm. 4 sets/3sources.htm. 2974

5 Tabe : The characteristics of the muti-view data sets Feature Type MF IS Fowers 3-Sources Reuters PIX(240) Shape(9) Coour(000) BBC(3560) Engish(253) 2 FOU(76) RGB(9) HOG(500) Guardian(363) French(24892) 3 FAC(26) - SIFTreg(8000) Reuters(3068) German(3425) 4 ZER(47) - SIFTbdy(3000) - Spanish(547) 5 KAR(64) Itaian(5506) 6 MOR(6) Tota data # Casses # View # Experimenta Settings In the experiments, we aim to answer the foowing four questions. First, does the proposed M4-FC perform better than singe view Fuzzy c medoids(fcmd)? Second, does M4-FC perform better than FCMD on concatenated features of a the views? Third, does mutipe medoids based M4-FC perform better than singe medoid based MVFCMddV? Last, how effective is M4-FC compared to other muti-view custering approaches for reationa data? In order to answer the four questions, the foowing approaches are seected for comparisons. Singe view Fuzzy c medoids (FCMD): Fuzzy c medoids is run on each singe view. Feature Concatenation Fuzzy c medoids (ConFCMD): The features of a the views are concatenated first and then FCMD is run on the resuted features. MVFCMddV: The singe medoid based muti-view fuzzy c medoids is proposed in [de Carvaho et a., 205]. Muti-view Spectra Custering (MVSC): One of the state-of-the-art muti-view spectra custering methods for reationa data is proposed in [Wang et a., 204]. The time compexity of FCMD, MVFCddV and MVSC are O(N 2 ), O(K N 2 P ) and O(N 3 P ), respectivey. Here N is the number of objects, K is the number of custers, P is the number of views. For fair comparison, the parameter m in FCMD, ConFCMD and MVFCMddV is searched from the range of [. 2] with step 0.. For MVSC, we foow the authors experimenta setting and parameter seecting methods in their paper. For M4-FC, the parameter γ is searched from [0. 0.9] with the step 0.. T u is searched from [0.00, 0.003, 0.0, 0.03, 0., 0.3]. T v is set by a guideine to make T u /K and T v /N cose to each other or be in the same order which aways produces reasonabe resuts in our experimenta study. 4.3 Initiaization Method For FCMD and ConFCMD, the medoids are initiaized by using the same method used in [Krishnapuram et a., 200]. The first medoid is seected as the object which has the minimum distance to a the other objects. We then seect the remaining medoids by choosing the objects which maximize their minima distance to the existing medoids. Seecting meoids by this method heps the agorithm converge to a better position because the meoids are distributed more eveny in the data space than random initiaization. The same method is aso appied in K-means which is used as the fina step of MVSC. For MVFCMddV and M4-FC, the consensus fuzzy membership U is initiaized based on this simiar method. First, we seect medoids for each view by using the same method mentioned above. Then, if object i in p-th view is cosest to the seected medoid of custer m, we set u p mi =, and set the membership of this object to other custers as 0 which is u p ci = 0, c =, 2...K, c m. Last, U is generated based on the membership of a views U p by setting U = P U p /P. Note that using this initiaization method the initiaization of each run is the same, therefore the custering resuts are the same for each run. 4.4 Experimenta Resuts For the experimenta resuts, we report three standard custering evauation metrics : Accuracy (ACC) [Cai et a., 2005], Normaized Mutua Information (NMI) [Streh and Ghosh, 2003] and F-measure [Larsen and Aone, 999]. For MF and IS data sets, the Eucidean distance is used to cacuate the distances. For Fowers data set, the distance matrices are downoaded directy from website. Therefore, there are no resuts for ConFCMD on Fowers data. For 3-Sources data set, cosine distance is used for cacuating the distance. For Reuters data set, the same experimenta setting as in [Kumar et a., 20], Probabiistic Latent Semantic Anaysis (PLSA) [Hofmann, 999] is appied to project the data to a 00-dimensiona space and the custering approaches are conducted on the ow dimensiona data. Tabe 2, Tabe 3 and Tabe 4 show the custering ACC, NMI and F-measure respectivey. As we can see from the tabes, a muti-view approaches perform better than singe view approaches in genera. Our proposed M4-FC constanty performs much better than the singe view FCMD and FCMD with concatenated features. In addition, mutipe medoids based M4-FC constanty outperforms singe medoid based MVFCMddV. Moreover, comparing with muti-view spectra custering, our proposed approach aso performs better on a the data sets. 4.5 Parameter Anaysis In our objective function as shown in (3) and (4), we have three parameters which are γ to contro the view weight, T u and T v to contro the weight of the reguarization term in the objective function. To show the impact of the parameters on 2975

6 the performance of M4-FC, we pot the NMI performance curve w.r.t. γ and T u for each data set in Fig. and Fig. 2, Tabe 2: ACC(%) of different methods on the five data sets Method MF IS Fowers 3-S Reuters FCMD() FCMD(2) FCMD(3) FCMD(4) FCMD(5) FCMD(6) ConFCMD MVFCMddV MVSC M4-FC NMI MF IS Fowers 0. 3 Sources Reuters gamma Figure : Parameter anaysis: NMI vaues with different γ. Tabe 3: NMI(%) of different methods on the five data sets Method MF IS Fowers 3-S Reuters FCMD() FCMD(2) FCMD(3) FCMD(4) FCMD(5) FCMD(6) ConFCMD MVFCMddV MVSC M4-FC NMI MF 0.2 IS Fowers 0. 3 Sources Reuters Tu Figure 2: Parameter anaysis: NMI vaues with different T u. Tabe 4: F-measure(%) of different methods on the five data sets Method MF IS Fowers 3-S Reuters FCMD() FCMD(2) FCMD(3) FCMD(4) FCMD(5) FCMD(6) ConFCMD MVFCMddV MVSC M4-FC respectivey. As mentioned above, T v can be set to make T u /K and T u /N cose to each other or be in the same order. Here we ony show the NMI resuts, the resuts of accuracy and F-measure have a simiar pattern. Fig. and Fig. 2 are generated as foows. First, for Fig. the vaue of the T u and T v which produces the resuts in Tabe 2, Tabe 3 and Tabe 4 are fixed. Then, the NMI resuts are potted w.r.t the parameter γ with vaues from [0. 0.9] with a step of 0.. Respectivey, Fig. 2 shows the NMI resuts with γ and T v fixed and T u taking vaues from [0.00, 0.003, 0.0, 0.03, 0., 0.3]. As shown in Fig. and Fig. 2, the NMI resuts are robust with respect to the parameter γ in the range of [0. 0.7] and T u in the range of [ ] for most of the data sets. In genera, higher γ vaue near 0.9 may achieve worse performance. Moreover, it shows that choosing γ from [0. 0.7] may achieve better and robust performance. For different T u, the performance on IS and Fowers data sets are very stabe. Therefore, we recommend to set γ to a vaue from [0. 0.7] and T u from [ ] to have robust resuts in practice. 5 Concusion We have proposed M4-FC, a new mutipe medoids based muti-view fuzzy custering with minimax optimization for muti-view data anaysis, and appied M4-FC on severa rea word data sets to demonstrate its effectiveness and potentia. In M4-FC, a representative weight for each object is introduced to generate custers with better quaity. In addition, minimax optimization is appied to ensure more harmonic consensus custering resuts. Experimenta resuts show that mutipe medoids based M4-FC not ony outperforms singe medoid based muti-view fuzzy custering, but aso achieves better custering resuts than the reated muti-view custering approaches on the benchmark data sets used in the study. 2976

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