1 Robust Nonnegative Matrix Factorization Yining Zhang

2 Outline  Review of nonnegative matrix factorization  Robust nonnegative matrix factorization using L21-norm  Robust nonnegative matrix factorization through sparse learning  Further works

3 Outline  Review of nonnegative matrix factorization  Robust nonnegative matrix factorization using L21-norm  Robust nonnegative matrix factorization through sparse learning  Further works

4 Review of nonnegative matrix factorization

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7 Clustering Interpretation

8 Outline  Review of nonnegative matrix factorization  Robust nonnegative matrix factorization using L21-norm  Robust nonnegative matrix factorization through sparse learning  Further works

9 Robust nonnegative matrix factorization using L21-norm

10 Shortcoming of Standard NMF

11 L21-norm

12 From Laplacian noise to L21 NMF

13 Illustration of robust NMF on toy data

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15 Illustration of robust NMF on real data

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Computation algorithm for L21NMF 17

Convergence of the algorithm  Theorem 1. (A) Updating G using the rule of Eq.(17) while fixing F, the objective function monotonically decreases. (B) Updating F using the rule of Eq.(16) while fixing G, the objective function monotonically decreases. 18

Updating G 19

Correctness of the algorithm  Theorem 7. At convergence, the converged solution rule of Eq.(17) satisfies the KKT condition of the optimization theory. 20

A general trick about the NMF 21 KKT condition Updating formula Auxiliary function Prove monotonicity

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Experiments on clustering 23

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25 Outline  Review of nonnegative matrix factorization  Robust nonnegative matrix factorization using L21-norm  Robust nonnegative matrix factorization through sparse learning  Further works

26 Robust nonnegative matrix factorization through sparse learning

27 Motivation Motivated by robust pca

Optimization 28

29 Experimental results-1 A case study

Experimental results 2- Face clustering on Yale 30

Experimental results 3- Face recognition on AR 31

32 Outline  Review of sparse learning  Efficient and robust feature selection via joint l 2,1 -norm minimzation  Exploiting the entire feature space with sparsity for automatic image annotation  Further works

33 Future works-1 (1) Direct robust matrix factorization for anomaly detection ICDM.

Future works-2 34

35 [1]Deguang Kong, Chris Ding, Heng Huang. Robust nonnegative matrix factorization using L21-norm. CIKM [2]Lijun Zhang, Zhengguang Chen, Miao Zheng, Xiaofei He. Robust non- negative matrix factorization. Front. Electr. Eng.China [3]Chris Ding, Tao Li, Michael I.Jordan. Convex and Semi-nonnegative matrix factorization. IEEE T.PAMI, Reference

36 Thanks! Q&A