Adaptive algorithms for low-rank and sparse matrix recovery with truncated nuclear norm

Recent studies have shown that the use of the truncated nuclear norm (TNN) in low-rank and sparse matrix decomposition (LRSD) can realize a better approximation to rank function of matrix, and achieve effectively recovery effects. This paper addresses the algorithms for LRSD with adaptive TNN (LRSD-ATNN), and designs an efficient algorithmic frame inspired by the alternating direction method of multiple (ADMM) and the accelerated proximal gradient approach (APG). To establish the adaptive algorithms, the method of singular value estimate is utilized to find adaptively the number of truncated singular value. Experimental results on synthetic data as well as real visual data show the superiority of the proposed algorithm in effectiveness in comparison with the state-of-the-art methods.

[1]  Rong Li,et al.  Extracting contrast-filled vessels in X-ray angiography by graduated RPCA with motion coherency constraint , 2017, Pattern Recognit..

[2]  Yilun Wang,et al.  Truncated Nuclear Norm Minimization for Image Restoration Based On Iterative Support Detection , 2014, ArXiv.

[3]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[4]  Arvind Ganesh,et al.  Fast Convex Optimization Algorithms for Exact Recovery of a Corrupted Low-Rank Matrix , 2009 .

[5]  Andrzej Cichocki,et al.  Total Variation Regularized Tensor RPCA for Background Subtraction From Compressive Measurements , 2015, IEEE Transactions on Image Processing.

[6]  Xiaochun Cao,et al.  Total Variation Regularized RPCA for Irregularly Moving Object Detection Under Dynamic Background , 2016, IEEE Transactions on Cybernetics.

[7]  Allen Y. Yang,et al.  Fast ℓ1-minimization algorithms and an application in robust face recognition: A review , 2010, 2010 IEEE International Conference on Image Processing.

[8]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[9]  Yadong Wang,et al.  Low‐rank and sparse decomposition based shape model and probabilistic atlas for automatic pathological organ segmentation , 2017, Medical Image Anal..

[10]  Hong Li,et al.  Small Infrared Target Detection Based on Low-Rank and Sparse Matrix Decomposition , 2012 .

[11]  Bastian Cheng,et al.  Beyond cost function masking: RPCA-based non-linear registration in the context of VLSM , 2016, 2016 International Workshop on Pattern Recognition in Neuroimaging (PRNI).

[12]  Xiaodong Li,et al.  Stable Principal Component Pursuit , 2010, 2010 IEEE International Symposium on Information Theory.

[13]  Bing Liu,et al.  Infrared small target detection in heavy sky scene clutter based on sparse representation , 2017 .

[14]  Julien Mairal,et al.  Optimization with Sparsity-Inducing Penalties , 2011, Found. Trends Mach. Learn..

[15]  Zhenghua Zhou,et al.  Recovering low-rank and sparse matrix based on the truncated nuclear norm , 2017, Neural Networks.

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

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

[18]  Rémi Gribonval,et al.  Performance measurement in blind audio source separation , 2006, IEEE Transactions on Audio, Speech, and Language Processing.

[19]  Zhiyao Duan,et al.  Singing Voice Separation by Low-Rank and Sparse Spectrogram Decomposition with Prelearned Dictionaries , 2017 .

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

[21]  Yan Liu,et al.  Weighted Schatten $p$ -Norm Minimization for Image Denoising and Background Subtraction , 2015, IEEE Transactions on Image Processing.

[22]  Soon Ki Jung,et al.  Moving Object Detection on RGB-D Videos Using Graph Regularized Spatiotemporal RPCA , 2017, ICIAP Workshops.

[23]  Quan Zheng,et al.  ε-Uniform Convergence of the Hybrid Difference Scheme on the Shishkin Mesh for Singularly Perturbed Problems , 2013 .

[24]  Yao Hu,et al.  Online robust principal component analysis via truncated nuclear norm regularization , 2016, Neurocomputing.

[25]  Jeffrey A. Fessler,et al.  Low-Rank and Adaptive Sparse Signal (LASSI) Models for Highly Accelerated Dynamic Imaging , 2016, IEEE Transactions on Medical Imaging.

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

[27]  George Atia,et al.  Randomized subspace learning approach for high dimensional low rank plus sparse matrix decomposition , 2015, 2015 49th Asilomar Conference on Signals, Systems and Computers.

[28]  Ming Yang,et al.  RAP: Scalable RPCA for Low-rank Matrix Recovery , 2016, CIKM.

[29]  Qing Liu,et al.  A Truncated Nuclear Norm Regularization Method Based on Weighted Residual Error for Matrix Completion , 2016, IEEE Transactions on Image Processing.

[30]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[31]  S. Yun,et al.  An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems , 2009 .

[32]  Paris Smaragdis,et al.  Singing-voice separation from monaural recordings using robust principal component analysis , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[33]  Victor Vianu,et al.  Invited articles section foreword , 2010, JACM.

[34]  Xiongwei Zhang,et al.  Auditory mask estimation by RPCA for monaural speech enhancement , 2017, 2017 IEEE/ACIS 16th International Conference on Computer and Information Science (ICIS).

[35]  Chinh T. Dang,et al.  RPCA-KFE: Key Frame Extraction for Video Using Robust Principal Component Analysis , 2014, IEEE Transactions on Image Processing.

[36]  Daniel K Sodickson,et al.  Low‐rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components , 2015, Magnetic resonance in medicine.

[37]  Babak Hassibi,et al.  A simplified approach to recovery conditions for low rank matrices , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[38]  G. Sapiro,et al.  A collaborative framework for 3D alignment and classification of heterogeneous subvolumes in cryo-electron tomography. , 2013, Journal of structural biology.

[39]  Zhizhong Fu,et al.  Infrared and visible images fusion based on RPCA and NSCT , 2016 .

[40]  Zhengrong Zuo,et al.  A Dim Small Infrared Moving Target Detection Algorithm Based on Improved Three-Dimensional Directional Filtering , 2013 .

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

[42]  Jianchao Bai,et al.  A New Model for Sparse and Low-Rank Matrix Decomposition , 2017 .

[43]  George Atia,et al.  A Subspace Learning Approach for High Dimensional Matrix Decomposition with Efficient Column/Row Sampling , 2016, ICML.

[44]  Thierry Bouwmans,et al.  Robust PCA via Principal Component Pursuit: A review for a comparative evaluation in video surveillance , 2014, Comput. Vis. Image Underst..

[45]  Yujie He,et al.  Small infrared target detection based on low-rank and sparse representation , 2015 .

[46]  Ivan W. Selesnick,et al.  Sparse Signal Estimation by Maximally Sparse Convex Optimization , 2013, IEEE Transactions on Signal Processing.

[47]  Lei Zhang,et al.  Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision , 2016, International Journal of Computer Vision.

[48]  Arvind Ganesh,et al.  Fast algorithms for recovering a corrupted low-rank matrix , 2009, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[49]  Longxin Lin,et al.  Group object detection and tracking by combining RPCA and fractal analysis , 2018, Soft Comput..

[50]  George Atia,et al.  High Dimensional Low Rank Plus Sparse Matrix Decomposition , 2015, IEEE Transactions on Signal Processing.

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

[52]  Andrew Chi-Sing Leung,et al.  Sparse and Truncated Nuclear Norm Based Tensor Completion , 2017, Neural Processing Letters.

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

[54]  Ebroul Izquierdo,et al.  Foreground Segmentation with Tree-Structured Sparse RPCA , 2018, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[55]  Xuelong Li,et al.  Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[56]  Yin Zhang,et al.  Fixed-Point Continuation for l1-Minimization: Methodology and Convergence , 2008, SIAM J. Optim..

[57]  M. Fukushima,et al.  A generalized proximal point algorithm for certain non-convex minimization problems , 1981 .