Principal Component Pursuit with reduced linear measurements

In this paper, we study the problem of decomposing a superposition of a low-rank matrix and a sparse matrix when a relatively few linear measurements are available. This problem arises in many data processing tasks such as aligning multiple images or rectifying regular texture, where the goal is to recover a low-rank matrix with a large fraction of corrupted entries in the presence of nonlinear domain transformation. We consider a natural convex heuristic to this problem which is a variant to the recently proposed Principal Component Pursuit. We prove that under suitable conditions, this convex program guarantees to recover the correct low-rank and sparse components despite reduced measurements. Our analysis covers both random and deterministic measurement models.

[1]  Qun Wan,et al.  Restricted Isometry Property of Principal Component Pursuit with Reduced Linear Measurements , 2013, J. Appl. Math..

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

[3]  Emmanuel J. Candès,et al.  The Power of Convex Relaxation: Near-Optimal Matrix Completion , 2009, IEEE Transactions on Information Theory.

[4]  Sham M. Kakade,et al.  Robust Matrix Decomposition With Sparse Corruptions , 2011, IEEE Transactions on Information Theory.

[5]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

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

[7]  Xiaodong Li,et al.  Compressed Sensing and Matrix Completion with Constant Proportion of Corruptions , 2011, Constructive Approximation.

[8]  M. Rudelson,et al.  Non-asymptotic theory of random matrices: extreme singular values , 2010, 1003.2990.

[9]  Yi Ma,et al.  TILT: Transform Invariant Low-Rank Textures , 2010, ACCV 2010.

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

[11]  Xiaodong Li,et al.  Dense error correction for low-rank matrices via Principal Component Pursuit , 2010, 2010 IEEE International Symposium on Information Theory.

[12]  I. Jolliffe.  Principal Component Analysis , 2002 .

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

[14]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[15]  David Gross,et al.  Recovering Low-Rank Matrices From Few Coefficients in Any Basis , 2009, IEEE Transactions on Information Theory.

[16]  Allen Y. Yang,et al.  Robust Face Recognition via Sparse Representation , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  Yongtian Wang,et al.  Robust Photometric Stereo via Low-Rank Matrix Completion and Recovery , 2010, ACCV.

[18]  John Wright,et al.  RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

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

[20]  E. Candès,et al.  Sparsity and incoherence in compressive sampling , 2006, math/0611957.

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

[22]  Stephen P. Boyd,et al.  Rank minimization and applications in system theory , 2004, Proceedings of the 2004 American Control Conference.

[23]  Joel A. Tropp,et al.  User-Friendly Tail Bounds for Sums of Random Matrices , 2010, Found. Comput. Math..

[24]  John Wright,et al.  Compressive principal component pursuit , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[25]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[26]  Aswin C. Sankaranarayanan,et al.  SpaRCS: Recovering low-rank and sparse matrices from compressive measurements , 2011, NIPS.