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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 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.

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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.

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

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

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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

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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

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

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Saturation and shadow noise Camera noise … Yale B faces:

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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)

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MLE Model Use EM algorithm to solve it!

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E Step: M Step:

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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

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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

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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.

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Face modeling experiments

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Explanation Saturation and shadow noise Camera noise

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Background Subtraction

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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

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Thanks!

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