GAUSS-SEIDEL BASED NON-NEGATIVE MATRIX FACTORIZATION FOR GENE EXPRESSION CLUSTERING

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1 GAUSS-SEIDEL BASED NON-NEGAIVE MARIX FACORIZAION FOR GENE EXPRESSION CLUSERING Qing Liao, Naiyang Guan, Qian Zhang Dept. of Computer Science & echnology, he Hong Kong University of Science and echnology el:

2 INRODUCION Gene Expression Clustering g e n e s samples Goal Unearths similar bio-process, gene function, gene regulation, and subtypes of cells. Imbalance he number of probed genes is rather greater than the number of samples. NMF clustering (Brunet et al. 2004) Jean-Philippe Brunet, Pablo amayo, odd R Golub, and Jill P Mesirov, Metagenes and molecular pattern discovery using matrix factorization, Proceedings of the national academy of sciences, vol. 101, no. 12, pp , Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 2

3 INRODUCION Nonnegative Matrix Factorization (NMF) samples Model: min W 0, H 0 X WH 2 F genes V W H Optimization: - Multiplicative Update Rule (MUR) (Lee et al. 2001) - NeNMF (Guan et al. 2012) Daniel D Lee and H Sebastian Seung, Algorithms for nonnegative matrix factorization, in Advances in Neural Information Processing Systems, 2001, pp Naiyang Guan, Dacheng ao, Zhigang Luo, and Bo Yuan, NeNMF: an optimal gradient method for nonnegative matrix factorization, IEEE ransactions on Signal Processing, vol. 60, no. 6, pp , Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 3

4 PROBLEM FORMULAION Gauss-Seidel Based Nonnegative Matrix Factorization (GSNMF) Gauss-Seidel Method - An Example Problem: Given V = and A = , find H = h 11 h 12 h 21 h 22 h 31 h 32 such that V = AH. Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 4

5 PROBLEM FORMULAION Gauss-seidel Based Nonnegative Matrix Factorization (GSNMF) Gauss-Seidel Method - An Example 1. Initialize: H 1 = Decompose: A = U + D + U = 3. Solve the linear system: V = AH = UH + DH + U H. H k V UH k = DH k+1 + U H k h 11 h 12 h 21 h 31 h 22 + h 32 = Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 5 = h 11 h 12 h 21 h 22 H 2 = h 31 h

6 PROBLEM FORMULAION Gauss-seidel Based Nonnegative Matrix Factorization (GSNMF) Gauss-Seidel Method Metric 1. Avoid the inverse operation 2. Fast convergence rate Gauss-Seidel Method Problem GS method: X AH F2, where A = A NMF: X WH F2, where W W How to adopt the GS method to solve NMF? Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 6

7 PROBLEM FORMULAION Gauss-seidel Based Nonnegative Matrix Factorization (GSNMF) Gauss-Seidel Method An approximation model min H 0 W X W WH 2 F Is the approximation reasonable? min W 0, H 0 W X W WH 2 F W X W WH W X WH F F F Constraint: W 2 F W X W WH X WH F F Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 7

8 OPIMIZAION Gauss-seidel Based Nonnegative Matrix Factorization (GSNMF) Divide: ransform: W W U D U W V Update H row by row: W WH W V U H DH UH W V UH U H DH 1 H UH W W H W W H k 1 k k k k k k k 1 i k k j j ( W W ) i ij ji ii j i j i H W Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 8

9 OPIMIZAION Gauss-seidel Based Nonnegative Matrix Factorization (GSNMF) Algorithms ime complexities of one iteration round NMF O(mnr + mr 2 + nr 2 ) NeNMF O(mnr + mr 2 + nr 2 ) + K O(mr 2 + nr 2 ) GSNMF O(mnr + mr 2 + nr 2 ) Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 9

10 EXPERIMENS Datasets able 1. Summarization of six cancer gene expression datasets. Datasets Samples Genes Classes gastricgse2685 2razreda gastricgse LL GSE LL GSE1577 2razreda AE GSE5060 GPL HSS GSE he orange datasets: 2 he GSE data : Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 10

11 EXPERIMENS Results Fig. 1. he box plots of the clustering accuracies for the three NMF algorithms over 100 runs on all the six gene expression datasets. Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 11

12 EXPERIMENS Results Fig. 2. he box plots of the clustering mutual information for the three NMF algorithms over 100 runs on all the six gene expression datasets. Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 12

13 EXPERIMENS Results Fig. 3. he average time consumption plots for the three NMF algorithms over 100 runs on all six gene expression datasets. Gauss-Seidel Based Non-negative Matrix Factorization for Gene Expression Clustering 13

14 hanks! Questions? AVAAR 平民化 的多功能交通大数据平台