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Matrix Factorization with Unknown Noise Deyu Meng 参考文献: ① Deyu Meng, Fernando De la Torre. Robust Matrix Factorization with Unknown Noise. International.

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Presentation on theme: "Matrix Factorization with Unknown Noise Deyu Meng 参考文献: ① Deyu Meng, Fernando De la Torre. Robust Matrix Factorization with Unknown Noise. International."— Presentation transcript:

1 Matrix Factorization with Unknown Noise Deyu Meng 参考文献: ① Deyu Meng, Fernando De la Torre. Robust Matrix Factorization with Unknown Noise. International Conference of Computer Vision (ICCV), ② Qian Zhao, Deyu Meng, Zongben Xu, Wangmeng Zuo, Lei Zhang. Robust principal component analysis with complex noise, International Conference of Machine Learning (ICML), 2014.

2 Structure from Motion (E.g.,Eriksson and Hengel,2010) Photometric Stereo (E.g., Zheng et al.,2012) Face Modeling (E.g., Candes et al.,2012)(E.g. Candes et al.,2012) Background Subtraction  Low-rank matrix factorization are widely used in computer vision.

3  Complete, clean data (or with Gaussian noise) SVD: Global solution

4  Complete, clean data (or with Gaussian noise) SVD: Global solution  There are always missing data  There are always heavy and complex noise

5 L2 norm model  Young diagram (CVPR, 2008)  L2 Wiberg (IJCV, 2007)  LM_S/LM_M (IJCV, 2008)  SALS (CVIU, 2010)  LRSDP (NIPS, 2010)  Damped Wiberg (ICCV, 2011)  Weighted SVD (Technometrics, 1979)  WLRA (ICML, 2003)  Damped Newton (CVPR, 2005)  CWM (AAAI, 2013)  Reg-ALM-L1 (CVPR, 2013) Pros: smooth model, faster algorithm, have global optimum for non- missing data Cons: not robust to heavy outliers

6 L2 norm model L1 norm model  Young diagram (CVPR, 2008)  L2 Wiberg (IJCV, 2007)  LM_S/LM_M (IJCV, 2008)  SALS (CVIU, 2010)  LRSDP (NIPS, 2010)  Damped Wiberg (ICCV, 2011)  Weighted SVD (Technometrics, 1979)  WLRA (ICML, 2003)  Damped Newton (CVPR, 2005)  CWM (AAAI, 2013)  Reg-ALM-L1 (CVPR, 2013)  Torre&Black (ICCV, 2001)  R1PCA (ICML, 2006)  PCAL1 (PAMI, 2008)  ALP/AQP (CVPR, 2005)  L1Wiberg (CVPR, 2010, best paper award)  RegL1ALM (CVPR, 2012) Pros: smooth model, faster algorithm, have global optimum for non- missing data Cons: not robust to heavy outliers Pros: robust to extreme outliers Cons: non-smooth model, slow algorithm, perform badly in Gaussian noise data

7  L2 model is optimal to Gaussian noise  L1 model is optimal to Laplacian noise  But real noise is generally neither Gaussian nor Laplacian

8 Saturation and shadow noise Camera noise … Yale B faces:

9 We propose Mixture of Gaussian (MoG) Universal approximation property of MoG Any continuous distributions MoG  E.g., a Laplace distribution can be equivalently expressed as a scaled MoG (Maz’ya and Schmidt, 1996) (Andrews and Mallows, 1974)

10 MLE Model  Use EM algorithm to solve it!

11  E Step:  M Step:

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13 Good measures to estimate groundtruth subspace What L2 and L1 methods optimize Synthetic experiments  Six error measurements  Three noise cases  Gaussian noise  Sparse noise  Mixture noise

14 Gaussian noise experiments Sparse noise experiments Mixture noise experiments L2 methods L1 methods Our method  MoG performs similar with L2 methods, better than L1 methods.  MoG performs as good as the best L1 method, better than L2 methods.  MoG performs better than all L2 and L1 competing methods

15 Why MoG is robust to outliers?  L1 methods perform well in outlier or heavy noise cases since it is a heavy-tail distribution.  Through fitting the noise as two Gaussians, the obtained MoG distribution is also heavy tailed.

16 Face modeling experiments

17 Explanation Saturation and shadow noise Camera noise

18 Background Subtraction

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26 Summary  We propose a LRMF model with a Mixture of Gaussians (MoG) noise  The new method can well handle outliers like L1- norm methods but using a more efficient way.  The extracted noises are with certain physical meanings

27 Thanks!


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