A New Model for Sparse and Low-Rank Matrix Decomposition

The robust principal component analysis (RPCA) model is a popular method for solving problems with the nuclear norm and `1 norm. However, it is time-consuming since in general one has to use the singular value decomposition in each iteration. In this paper, we introduce a novel model to reformulate the existed model by making use of low-rank matrix factorization to surrogate the nuclear norm for the sparse and low-rank decomposition problem. In such case we apply the Penalty Function Method (PFM) and Augmented Lagrangian Multipliers Method (ALMM) to solve this new nonconvex optimization problem. Theoretically, corresponding to our methods, the convergence analysis is given respectively. Compared with classical RPCA, some practical numerical examples are simulated to show that our methods are much better than RPCA.

[1]  Massimiliano Pontil,et al.  Multi-Task Feature Learning , 2006, NIPS.

[2]  Dale Schuurmans,et al.  Real-Time Discriminative Background Subtraction , 2011, IEEE Transactions on Image Processing.

[3]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[4]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[5]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[6]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[7]  Ernie Esser,et al.  Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman , 2009 .

[8]  David Suter,et al.  Recovering the missing components in a large noisy low-rank matrix: application to SFM , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Francis R. Bach,et al.  Low-rank matrix factorization with attributes , 2006, ArXiv.

[10]  Marc Teboulle,et al.  A proximal-based decomposition method for convex minimization problems , 1994, Math. Program..

[11]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[12]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[13]  Santosh S. Vempala,et al.  Latent Semantic Indexing , 2000, PODS 2000.

[14]  Ian T. Jolliffe,et al.  Principal Component Analysis , 2002, International Encyclopedia of Statistical Science.

[15]  Santosh S. Vempala,et al.  Latent semantic indexing: a probabilistic analysis , 1998, PODS '98.

[16]  Yi Ma,et al.  The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices , 2010, Journal of structural biology.

[17]  G. Watson.  Characterization of the subdifferential of some matrix norms , 1992 .

[18]  Xiaoming Yuan,et al.  Sparse and low-rank matrix decomposition via alternating direction method , 2013 .

[19]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[20]  Takeo Kanade,et al.  Shape and motion from image streams under orthography: a factorization method , 1992, International Journal of Computer Vision.

[21]  M. Mesbahi,et al.  On the rank minimization problem , 2004, Proceedings of the 2004 American Control Conference.

[22]  T. Landauer,et al.  Indexing by Latent Semantic Analysis , 1990 .

[23]  Shiqian Ma,et al.  Fixed point and Bregman iterative methods for matrix rank minimization , 2009, Math. Program..

[24]  Shimon Ullman,et al.  Uncovering shared structures in multiclass classification , 2007, ICML '07.

[25]  R. Glowinski.  Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

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

[27]  Pablo A. Parrilo,et al.  Rank-Sparsity Incoherence for Matrix Decomposition , 2009, SIAM J. Optim..