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# Co clustering by-block_value_decomposition

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### Co clustering by-block_value_decomposition

1. 1. Author / Bo Long, Zhongfei Zhang and Philip S. Yu<br />Source / ACM KDD’05, August 21-24, 2005, pp. 635 – pp. 640<br />Presenter / Allen Wu<br />Co-clustering by Block Value Decomposition<br />1<br />
2. 2. Outline<br />Introduction<br />Block value decomposition<br />Derivation of the Algorithm<br />Empirical Evaluation<br />Conclusion<br />2<br />
3. 3. Introduction<br />Dyadic data refer to a domain with two finite sets of objects in which observations are made for dyads.<br />Co-clustering can effectively deal with the high dimensional and sparse data between rows and columns.<br />In this paper, a new co-clustering framework, Block Value Decomposition(BVD), had been proposed.<br />3<br />
4. 4. Introduction (cont.)<br />This paper develop a specific novel co-clustering algorithm for a special yet very popular case – non-negative dyadic data.<br />The algorithm performs an implicitly adaptive dimensionality reduction, which works well for typical sparse data.<br />The dyadic data matrix is factorized into three components.<br />The row-coefficient matrix – R<br />The block value matrix– B<br />The column-coefficient matrix– C<br />4<br />
5. 5. The definition of dyadic data<br />5<br />The notion dyadic refers to a domain with two sets of objects X={x1, …, xn} and Y={y1, …, ym}<br />The data can be organized as an n by m two-dimensional matrix Z.<br />Each w(x,y) corresponds to one element of Z.<br />
6. 6. 6<br />×<br />×<br />=<br />n×k<br />k ×l<br />l×m<br />n×m<br />k ×l<br />
7. 7. 7<br />y1 y2 y3 y4<br />x1<br />x2<br />x3<br />x4<br />×<br />×<br />C<br />B<br />R<br />y1 y2 y3 y4<br />=<br />y1 y2 y3 y4<br />x1<br />x2<br />x3<br />x4<br />x1<br />x2<br />x3<br />x4<br />RBC<br />Z<br />
8. 8. Block value decomposition definition<br />8<br />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 <br /> f(R, B, C) = ||Z – RBC||2<br /> 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.<br />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 <br /> f(S, B,) = ||Z – SBST||2<br /> ij: Sij  0, and Bij  0, where S n×k, B k×k and k<<n.<br />
9. 9. Derivation of the algorithm<br />9<br />The objective function is convex in R, B and C respectively. However, it is not convex in all of them simultaneously.<br /> Thus, it is unrealistic to expect an algorithm to find the global minimum.<br />Theorem 1. If R, B and C are a local minimizer of the objective function , then the equations<br /> (ZCTBT )。R + (RBCCTBT )。R = 0 <br />(RTZCT )。B +(RTRBCCT )。B = 0<br /> (BTRTZ)。C + (BTRTRBC)。C = 0<br /> are satisified, where 。denotes the Hadamard product of two matrices.<br />
10. 10. Derivation of the algorithm (cont.)<br />10<br />Let λ1, λ2, and λ3 be the Lagrange multipliers for the constraint R, B, and C  0, respectively, where λ1k×n, λ2l×k and λ3m×l. The Lagrange function L(R, B, C, λ1, λ2, λ3 ) becomes:<br /> L = f(R;B;C) -tr(λ1 RT ) -tr(λ2BT ) - tr(λ3 CT )<br />The Kuhn-Tucker conditions are:<br /> L/ R = L/ B = L/ C = 0 <br /> λ1。R = λ2。B = λ3。C = 0<br />Taking the derivatives, we obtain the following three equations, respectively.<br /> 2ZCTBT - 2RBCCTBT + λ1 = 0 <br />2RTZCT - 2RTRBCCT + λ2 = 0 <br /> 2BTRTZ - 2BTRTRBC + λ3 = 0<br />
11. 11. Derivation of the algorithm (cont.)<br />11<br />Based on Theorem 1, we propose following updating rules.<br />If the R=CT, we derive the updating rules for symmetric matrix, that the symmetric NBVD provides only one clustering result.<br />
12. 12. EMPIRICAL EVALUATIONS<br />12<br />The experiment dataset is collected from the 20-Newsgroup data and CLASSIC3 dataset.<br />We measure the clustering performance using the accuracy given by the confusion matrix of the obtained clusters and the "real" classes.<br />
13. 13. EMPIRICAL EVALUATIONS (cont.)<br />13<br />
14. 14. EMPIRICAL EVALUATIONS (cont.)<br />14<br />
15. 15. Conclusion<br />15<br />In this paper, we have proposed a new co-clustering frame work for dyadic data called Block Value Decomposition.<br />Under this framework, we focus on a special but also very popular case, Non-negative Block Value Decomposition.<br />We have shown the correctness of the NBVD algorithm theoretically.<br />According to the empirical evaluations, the effectiveness and the great potential of the BVD framework.<br />