Matrix Factorization with Unknown Noise Deyu Meng Deyu
- Slides: 27
Matrix Factorization with Unknown Noise Deyu Meng 参考文献: ①Deyu Meng, Fernando De la Torre. Robust Matrix Factorization with Unknown Noise. International Conference of Computer Vision (ICCV), 2013. ②Qian Zhao, Deyu Meng, Zongben Xu, Wangmeng Zuo, Lei Zhang. Robust principal component analysis with complex noise, International Conference of Machine Learning (ICML), 2014.
Ø Low-rank matrix factorization are widely used in computer vision. Structure from Motion (E. g. , Eriksson and Hengel , 2010) Face Modeling (E. g. , Candes et al. , 2012) Photometric Stereo (E. g. , Zheng et al. , 2012) Background Subtraction (E. g. Candes et al. , 2012)
Ø Complete, clean data (or with Gaussian noise) n SVD: Global solution
Ø Complete, clean data (or with Gaussian noise) n SVD: Global solution Ø There always missing data Ø There always heavy and complex noise
L 2 norm model ØYoung diagram (CVPR, 2008) Ø L 2 Wiberg (IJCV, 2007) Ø LM_S/LM_M (IJCV, 2008) Ø SALS (CVIU, 2010) Ø LRSDP (NIPS, 2010) Ø Damped Wiberg (ICCV, 2011) Ø Weighted SVD (Technometrics, 1979) Ø WLRA (ICML, 2003) Ø Damped Newton (CVPR, 2005) Ø CWM (AAAI, 2013) Ø Reg-ALM-L 1 (CVPR, 2013) Pros: smooth model, faster algorithm, have global optimum for nonmissing data Cons: not robust to heavy outliers
L 1 norm model L 2 norm model ØYoung diagram (CVPR, 2008) Ø L 2 Wiberg (IJCV, 2007) Ø LM_S/LM_M (IJCV, 2008) Ø SALS (CVIU, 2010) Ø LRSDP (NIPS, 2010) Ø Damped Wiberg (ICCV, 2011) Ø Weighted SVD (Technometrics, 1979) Ø WLRA (ICML, 2003) Ø Damped Newton (CVPR, 2005) Ø CWM (AAAI, 2013) Ø Reg-ALM-L 1 (CVPR, 2013) Pros: smooth model, faster algorithm, have global optimum for nonmissing data Cons: not robust to heavy outliers Ø Torre&Black (ICCV, 2001) Ø R 1 PCA (ICML, 2006) Ø PCAL 1 (PAMI, 2008) Ø ALP/AQP (CVPR, 2005) Ø L 1 Wiberg (CVPR, 2010, best paper award) Ø Reg. L 1 ALM (CVPR, 2012) Pros: robust to extreme outliers Cons: non-smooth model, slow algorithm, perform badly in Gaussian noise data
Ø L 2 model is optimal to Gaussian noise Ø L 1 model is optimal to Laplacian noise Ø But real noise is generally neither Gaussian nor Laplacian
Yale B faces: … Saturation and shadow noise Camera noise
We propose Mixture of Gaussian (Mo. G) Universal approximation property of Mo. G Any continuous distributions Mo. G (Maz’ya and Schmidt, 1996) Ø E. g. , a Laplace distribution can be equivalently expressed as a scaled Mo. G (Andrews and Mallows, 1974)
MLE Model Ø Use EM algorithm to solve it!
Ø E Step: Ø M Step:
Synthetic experiments Ø Three noise cases Ø Gaussian noise Ø Sparse noise Ø Mixture noise Ø Six error measurements What L 2 and L 1 methods optimize Good measures to estimate groundtruth subspace
Our method L 2 methods L 1 methods Gaussian noise experiments Ø Mo. G performs similar with L 2 methods, better than L 1 methods. Sparse noise experiments Ø Mo. G performs as good as the best L 1 method, better than L 2 methods. Mixture noise experiments Ø Mo. G performs better than all L 2 and L 1 competing methods
Why Mo. G is robust to outliers? Ø L 1 methods perform well in outlier or heavy noise cases since it is a heavy-tail distribution. Ø Through fitting the noise as two Gaussians, the obtained Mo. G distribution is also heavy tailed.
Face modeling experiments
Explanation Saturation and shadow noise Camera noise
Background Subtraction
Background Subtraction
Summary Ø We propose a LRMF model with a Mixture of Gaussians (Mo. G) noise Ø The new method can well handle outliers like L 1 -norm methods but using a more efficient way. Ø The extracted noises are with certain physical meanings
Thanks!
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