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Author / Bo Long, Zhongfei Zhang and Philip S. Yu Source / ACM KDD’05, August 21-24, 2005, pp. 635 – pp. 640 Presenter / Allen Wu Co-clustering by Block Value Decomposition 1
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Outline Introduction Block value decomposition Derivation of the Algorithm Empirical Evaluation Conclusion 2
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Introduction Dyadic data refer to a domain with two finite sets of objects in which observations are made for dyads. Co-clustering can effectively deal with the high dimensional and sparse data between rows and columns. In this paper, a new co-clustering framework, Block Value Decomposition(BVD), had been proposed. 3
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Introduction (cont.) This paper develop a specific novel co-clustering algorithm for a special yet very popular case – non-negative dyadic data. The algorithm performs an implicitly adaptive dimensionality reduction, which works well for typical sparse data. The dyadic data matrix is factorized into three components. The row-coefficient matrix – R The block value matrix– B The column-coefficient matrix– C 4
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The definition of dyadic data 5 The notion dyadic refers to a domain with two sets of objects X={x1, …, xn} and Y={y1, …, ym} The data can be organized as an n by m two-dimensional matrix Z. Each w(x,y) corresponds to one element of Z.
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7 y1 y2 y3 y4 x1 x2 x3 x4 × × C B R y1 y2 y3 y4 = y1 y2 y3 y4 x1 x2 x3 x4 x1 x2 x3 x4 RBC Z
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Block value decomposition definition 8 Non-negative block value decomposition of a non-negative data matrix Z n×m(i.e. ij: Zij 0) is given by the minimization of f(R, B, C) = ||Z – RBC||2 subject to the constraints ij: Rij 0, Bij 0 and Cij 0, where R n×k, B k×l, C l×m, k<<n, and l<<m. If R=CT, symmetric non-negative block value decomposition of a symmetric non-negative data matrix Z n×n(i.e. ij: Zij 0) is given by the minimization of f(S, B,) = ||Z – SBST||2 ij: Sij 0, and Bij 0, where S n×k, B k×k and k<<n.
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Derivation of the algorithm 9 The objective function is convex in R, B and C respectively. However, it is not convex in all of them simultaneously. Thus, it is unrealistic to expect an algorithm to find the global minimum. Theorem 1. If R, B and C are a local minimizer of the objective function , then the equations (ZCTBT )。R + (RBCCTBT )。R = 0 (RTZCT )。B +(RTRBCCT )。B = 0 (BTRTZ)。C + (BTRTRBC)。C = 0 are satisified, where 。denotes the Hadamard product of two matrices.
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Derivation of the algorithm (cont.) 10 Let λ1, λ2, and λ3 be the Lagrange multipliers for the constraint R, B, and C 0, respectively, where λ1k×n, λ2l×k and λ3m×l. The Lagrange function L(R, B, C, λ1, λ2, λ3 ) becomes: L = f(R;B;C) -tr(λ1 RT ) -tr(λ2BT ) - tr(λ3 CT ) The Kuhn-Tucker conditions are: L/ R = L/ B = L/ C = 0 λ1。R = λ2。B = λ3。C = 0 Taking the derivatives, we obtain the following three equations, respectively. 2ZCTBT - 2RBCCTBT + λ1 = 0 2RTZCT - 2RTRBCCT + λ2 = 0 2BTRTZ - 2BTRTRBC + λ3 = 0
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Derivation of the algorithm (cont.) 11 Based on Theorem 1, we propose following updating rules. If the R=CT, we derive the updating rules for symmetric matrix, that the symmetric NBVD provides only one clustering result.
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EMPIRICAL EVALUATIONS 12 The experiment dataset is collected from the 20-Newsgroup data and CLASSIC3 dataset. We measure the clustering performance using the accuracy given by the confusion matrix of the obtained clusters and the "real" classes.
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Conclusion 15 In this paper, we have proposed a new co-clustering frame work for dyadic data called Block Value Decomposition. Under this framework, we focus on a special but also very popular case, Non-negative Block Value Decomposition. We have shown the correctness of the NBVD algorithm theoretically. According to the empirical evaluations, the effectiveness and the great potential of the BVD framework.
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