Robust nonnegative matrix factorization via L1 norm regularization by multiplicative updating rules

Nonnegative Matrix Factorization (NMF) is a widely used technique in many applications such as face recognition, motion segmentation, etc. It approximates the nonnegative data in an original high dimensional space with a linear representation in a low dimensional space by using the product of two nonnegative matrices. In many applications data are often partially corrupted with large additive noise. When the positions of noise are known, some existing variants of N-MF can be applied by treating these corrupted entries as missing values. However, the positions are often unknown in many real world applications, which prevents the usage of traditional NMF or other existing variants of NMF. This paper proposes a Robust Nonnegative Matrix Factorization (RobustNMF) algorithm that explicitly models the partial corruption as large additive noise without requiring the information of positions of noise. In particular, the proposed method jointly approximates the clean data matrix with the product of two nonnegative matrices and estimates the positions and values of outliers/noise. An efficient iterative optimization algorithm with a solid theoretical justification has been proposed to learn the desired matrix factorization. Experimental results demonstrate the advantages of the proposed algorithm.

[1]  Shuicheng Yan,et al.  Robust Object Tracking with Online Multi-lifespan Dictionary Learning , 2013, 2013 IEEE International Conference on Computer Vision.

[2]  Anil M. Cheriyadat,et al.  Non-negative matrix factorization of partial track data for motion segmentation , 2010, 2009 IEEE 12th International Conference on Computer Vision.

[3]  Anders P. Eriksson,et al.  Efficient computation of robust low-rank matrix approximations in the presence of missing data using the L1 norm , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[4]  DaugmanJohn Face and Gesture Recognition , 1997 .

[5]  Hujun Bao,et al.  Understanding the Power of Clause Learning , 2009, IJCAI.

[6]  Qi Tian,et al.  Task-Dependent Visual-Codebook Compression , 2012, IEEE Transactions on Image Processing.

[7]  Bin Shen,et al.  Visual Tracking via Online Nonnegative Matrix Factorization , 2014, IEEE Transactions on Circuits and Systems for Video Technology.

[8]  Hyunsoo Kim,et al.  Nonnegative Matrix Factorization Based on Alternating Nonnegativity Constrained Least Squares and Active Set Method , 2008, SIAM J. Matrix Anal. Appl..

[9]  A. Ben Hamza,et al.  Reconstruction of reflectance spectra using robust nonnegative matrix factorization , 2006, IEEE Transactions on Signal Processing.

[10]  Wen Gao,et al.  Learning to Distribute Vocabulary Indexing for Scalable Visual Search , 2013, IEEE Transactions on Multimedia.

[11]  Luo Si,et al.  Non-Negative Matrix Factorization Clustering on Multiple Manifolds , 2010, AAAI.

[12]  Wen Gao,et al.  Location Discriminative Vocabulary Coding for Mobile Landmark Search , 2011, International Journal of Computer Vision.

[13]  Michael J. Black,et al.  Robust Principal Component Analysis for Computer Vision , 2001, ICCV.

[14]  Jordi Vitrià,et al.  Non-negative Matrix Factorization for Face Recognition , 2002, CCIA.

[15]  Rama Chellappa,et al.  Large-Scale Matrix Factorization with Missing Data under Additional Constraints , 2010, NIPS.

[16]  H. Sebastian Seung,et al.  Algorithms for Non-negative Matrix Factorization , 2000, NIPS.

[17]  Patrik O. Hoyer,et al.  Non-negative Matrix Factorization with Sparseness Constraints , 2004, J. Mach. Learn. Res..

[18]  Hyunsoo Kim,et al.  Sparse Non-negative Matrix Factorizations via Alternating Non-negativity-constrained Least Squares , 2006 .

[19]  Douglas M. Hawkins,et al.  Inferential, robust non-negative matrix factorization analysis of microarray data , 2007, Bioinform..

[20]  Yun Fu,et al.  Discriminative dictionary learning with low-rank regularization for face recognition , 2013, 2013 10th IEEE International Conference and Workshops on Automatic Face and Gesture Recognition (FG).

[21]  D. Donoho For most large underdetermined systems of equations, the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution , 2006 .

[22]  Wei Hu,et al.  Image inpainting via sparse representation , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[23]  John Wright,et al.  Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization , 2009, NIPS.

[24]  Bin Shen,et al.  Learning dictionary on manifolds for image classification , 2013, Pattern Recognit..

[25]  Fillia Makedon,et al.  Learning from Incomplete Ratings Using Non-negative Matrix Factorization , 2006, SDM.