• Previous Article
    A smooth QP-free algorithm without a penalty function or a filter for mathematical programs with complementarity constraints
  • NACO Home
  • This Issue
  • Next Article
    A perturbation-based approach for continuous network design problem with emissions
2015, 5(2): 127-134. doi: 10.3934/naco.2015.5.127

Rank-one and sparse matrix decomposition for dynamic MRI

1. 

Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, China, China

Received  November 2014 Revised  April 2015 Published  June 2015

We introduce a rank-one and sparse matrix decomposition model for dynamic magnetic resonance imaging (MRI). Since $l_p$-norm $(0 < p < 1)$ is generally nonconvex, nonsmooth, non-Lipschitz, we propose reweighted $l_1$-norm to surrogate $l_p$-norm. Based on this, we put forward a modified alternative direction method. Numerical experiments are also given to illustrate the efficiency of our algorithm.
Citation: Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127
References:
[1]

K. Amin, W. Xu, A. Avestimehr and B. Hassibi, Weighted $l_1$ minimization for sparse recovery with prior information,, IEEE International Symposium on Information Theory, 2 (2009), 483.

[2]

O. Banerjee, L. Ghaouiand and A. D'Aspremont, Model selection through sparse maximum likelihood estimation for multivariate Gaussian or Binary data,, The Journal of Machine Learning Research, 9 (2008), 485.

[3]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via alternating direction method of multipliers,, Foundations and Trends in Machine Learning, 3 (2011), 1.

[4]

E. Candès, M. Wakin and S. Boyd, Enhancing sparsity by reweighted $l_1$ minimization,, Journal of Fourier Analysis and Applications, 14 (2008), 877. doi: 10.1007/s00041-008-9045-x.

[5]

V. Chandrasekaran, S. Sanghavi, P. Parrilo and A. Willsky, Rank-sparsity incoherence for matrix decomposition,, SIAM Journal on Optimization, 21 (2011), 572. doi: 10.1137/090761793.

[6]

X. Chen, D. Ge, Z. Wang and Y. Ye, Complexity of unconstrained $L_2-L_p$ minimization,, Mathematical Programming, 143 (2014), 371. doi: 10.1007/s10107-012-0613-0.

[7]

I. Daubechies, R. DeVore, M. Fornasier and C. Güntürk, Iteratively reweighted least squares minimization for sparse recovery,, Communications on Pure and Applied Mathematics, 63 (2010), 1. doi: 10.1002/cpa.20303.

[8]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_p$-minimization for 0 < q < 1,, Applied and Computational Harmonic Analysis, 26 (2009), 395. doi: 10.1016/j.acha.2008.09.001.

[9]

D. Ge, X. Jiang and Y. Ye, A note on the complexity of $l_p$ minimization,, Mathematical Programming, 129 (2011), 285. doi: 10.1007/s10107-011-0470-2.

[10]

X. Li, M. Ng and X. Yuan, Nuclear-norm-free variational models for background extraction from surveillance video,, submitted to IEEE Transactions on Image Processing, (2013).

[11]

Z. Lin, M. Chen and Y. Ma, The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, Preprint, (2010).

[12]

R. Otazo, E. Candès and D. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,, Magnetic Resonance in Medicine, 73 (2015), 1125.

[13]

J. Wright, A. Ganesh, S. Rao, Y. Peng and Y. Ma, Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization,, Advances in Neural Information Processing Systems, (2009), 2080.

[14]

S. Wright, R. Nowak and M. Figueiredo, Sparse reconstruction by separable approximation,, IEEE Transactions on Signal Processing, 57 (2009), 2479. doi: 10.1109/TSP.2009.2016892.

[15]

X. Xiu, L. Kong and S. Zhou, Modified iterative reweighted $l_1$ algorithm for surveillance video,, Preprint, (2014).

[16]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pacific Journal of Optimization, 9 (2013), 167.

[17]

Y. Zhao and D. Li, Reweighted $l_1$-minimization for sparse solutions to underdetermined linear systems,, SIAM Journal on Optimization, 22 (2012), 1065. doi: 10.1137/110847445.

[18]

S. Zhou, N. Xiu, Y. Wang and L. Kong, Exact recovery for sparse signal via weighted $l_1$ minimization,, Preprint, (2014).

show all references

References:
[1]

K. Amin, W. Xu, A. Avestimehr and B. Hassibi, Weighted $l_1$ minimization for sparse recovery with prior information,, IEEE International Symposium on Information Theory, 2 (2009), 483.

[2]

O. Banerjee, L. Ghaouiand and A. D'Aspremont, Model selection through sparse maximum likelihood estimation for multivariate Gaussian or Binary data,, The Journal of Machine Learning Research, 9 (2008), 485.

[3]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via alternating direction method of multipliers,, Foundations and Trends in Machine Learning, 3 (2011), 1.

[4]

E. Candès, M. Wakin and S. Boyd, Enhancing sparsity by reweighted $l_1$ minimization,, Journal of Fourier Analysis and Applications, 14 (2008), 877. doi: 10.1007/s00041-008-9045-x.

[5]

V. Chandrasekaran, S. Sanghavi, P. Parrilo and A. Willsky, Rank-sparsity incoherence for matrix decomposition,, SIAM Journal on Optimization, 21 (2011), 572. doi: 10.1137/090761793.

[6]

X. Chen, D. Ge, Z. Wang and Y. Ye, Complexity of unconstrained $L_2-L_p$ minimization,, Mathematical Programming, 143 (2014), 371. doi: 10.1007/s10107-012-0613-0.

[7]

I. Daubechies, R. DeVore, M. Fornasier and C. Güntürk, Iteratively reweighted least squares minimization for sparse recovery,, Communications on Pure and Applied Mathematics, 63 (2010), 1. doi: 10.1002/cpa.20303.

[8]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_p$-minimization for 0 < q < 1,, Applied and Computational Harmonic Analysis, 26 (2009), 395. doi: 10.1016/j.acha.2008.09.001.

[9]

D. Ge, X. Jiang and Y. Ye, A note on the complexity of $l_p$ minimization,, Mathematical Programming, 129 (2011), 285. doi: 10.1007/s10107-011-0470-2.

[10]

X. Li, M. Ng and X. Yuan, Nuclear-norm-free variational models for background extraction from surveillance video,, submitted to IEEE Transactions on Image Processing, (2013).

[11]

Z. Lin, M. Chen and Y. Ma, The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, Preprint, (2010).

[12]

R. Otazo, E. Candès and D. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,, Magnetic Resonance in Medicine, 73 (2015), 1125.

[13]

J. Wright, A. Ganesh, S. Rao, Y. Peng and Y. Ma, Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization,, Advances in Neural Information Processing Systems, (2009), 2080.

[14]

S. Wright, R. Nowak and M. Figueiredo, Sparse reconstruction by separable approximation,, IEEE Transactions on Signal Processing, 57 (2009), 2479. doi: 10.1109/TSP.2009.2016892.

[15]

X. Xiu, L. Kong and S. Zhou, Modified iterative reweighted $l_1$ algorithm for surveillance video,, Preprint, (2014).

[16]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pacific Journal of Optimization, 9 (2013), 167.

[17]

Y. Zhao and D. Li, Reweighted $l_1$-minimization for sparse solutions to underdetermined linear systems,, SIAM Journal on Optimization, 22 (2012), 1065. doi: 10.1137/110847445.

[18]

S. Zhou, N. Xiu, Y. Wang and L. Kong, Exact recovery for sparse signal via weighted $l_1$ minimization,, Preprint, (2014).

[1]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[2]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[3]

Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015

[4]

Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191

[5]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-53. doi: 10.3934/dcdsb.2019155

[6]

Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems & Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025

[7]

Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391

[8]

Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002

[9]

Robert D. Sidman, Marie Erie, Henry Chu. A method, with applications, for analyzing co-registered EEG and MRI data. Conference Publications, 2001, 2001 (Special) : 349-356. doi: 10.3934/proc.2001.2001.349

[10]

Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems & Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917

[11]

Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773

[12]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[13]

Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601

[14]

Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289

[15]

Giuseppe Geymonat, Françoise Krasucki. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Communications on Pure & Applied Analysis, 2009, 8 (1) : 295-309. doi: 10.3934/cpaa.2009.8.295

[16]

Wanyou Cheng, Zixin Chen, Donghui Li. Nomonotone spectral gradient method for sparse recovery. Inverse Problems & Imaging, 2015, 9 (3) : 815-833. doi: 10.3934/ipi.2015.9.815

[17]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[18]

Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067

[19]

Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019029

[20]

Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006

 Impact Factor: 

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]