American Institute of Mathematical Sciences

Advanced
June  2017, 11(3): 539-551. doi: 10.3934/ipi.2017025

Subspace clustering by (k,k)-sparse matrix factorization

Haixia Liu , , Jian-Feng Cai and  Yang Wang

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

* Corresponding author

Received  April 2016 Revised  February 2017 Published  April 2017

High-dimensional data often lie in low-dimensional subspaces instead of the whole space. Subspace clustering is a problem to analyze data that are from multiple low-dimensional subspaces and cluster them into the corresponding subspaces. In this work, we propose a $(k,k)$-sparse matrix factorization method for subspace clustering. In this method, data itself is considered as the "dictionary", and each data point is represented as a linear combination of the basis of its cluster in the dictionary. Thus, the coefficient matrix is low-rank and sparse. With an appropriate permutation, it is also blockwise with each block corresponding to a cluster. With an assumption that each block is no more than $k$-by-$k$ in matrix recovery, we seek a low-rank and $(k,k)$-sparse coefficient matrix, which will be used for the construction of affinity matrix in spectral clustering. The advantage of our proposed method is that we recover a coefficient matrix with $(k,k)$-sparse and low-rank simultaneously, which is better fit for subspace clustering. Numerical results illustrate the effectiveness that it is better than SSC and LRR in real-world classification problems such as face clustering and motion segmentation.

Keywords: Subspace clustering, matrix factorization, (k, k)-sparse, low-rank, ksupport norm.
Mathematics Subject Classification: Primary: 68T04; Secondary: 65F04.
Citation: 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
References:
[1]

P. K. Agarwal and N. H. Mustafa, k-means projective clustering, in Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, ACM, 2004, 155–165. doi: 10.1145/1055558.1055581.  Google Scholar

[2]

A. Argyriou, R. Foygel and N. Srebro, Sparse prediction with the k-support norm, in Advances in Neural Information Processing Systems, 2012, 1457–1465. Google Scholar

[3]

J. Bolte, S. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, 146 (2014), 459-494.  doi: 10.1007/s10107-013-0701-9.  Google Scholar

[4]

V. Chandrasekaran, B. Recht, P. A. Parrilo and A. S. Willsky, The convex geometry of linear inverse problems, Foundations of Computational mathematics,, 12 (2012), 805-849.   Google Scholar

[5]

G. Chen and G. Lerman, Spectral curvature clustering (scc), International Journal of Computer Vision, 81 (2009), 317-330.  doi: 10.1007/s11263-008-0178-9.  Google Scholar

[6]

J. P. Costeira and T. Kanade, A multibody factorization method for independently moving objects, International Journal of Computer Vision, 29 (1998), 159-179.   Google Scholar

[7]

H. Derksen, Y. Ma, W. Hong and J. Wright, Segmentation of multivariate mixed data via lossy coding and compression in Electronic Imaging 2007, International Society for Optics and Photonics (2007), 65080H. doi: 10.1117/12.714912.  Google Scholar

[8]

E. Elhamifar and R. Vidal, Sparse subspace clustering: Algorithm, theory, and applications, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2765-2781.  doi: 10.1109/TPAMI.2013.57.  Google Scholar

[9]

P. Favaro, R. Vidal and A. Ravichandran, A closed form solution to robust subspace estimation and clustering, in 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, 2011, 1801–1807. doi: 10.1109/CVPR.2011.5995365.  Google Scholar

[10]

J. Feng, Z. Lin, H. Xu and S. Yan, Robust subspace segmentation with block-diagonal prior, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2014, 3818–3825. doi: 10.1109/CVPR.2014.482.  Google Scholar

[11]

A. Goh and R. Vidal, Segmenting motions of different types by unsupervised manifold clustering, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2007, 1–6. doi: 10.1109/CVPR.2007.383235.  Google Scholar

[12]

L. N. Hutchins, S. M. Murphy, P. Singh and J. H. Graber, Position-dependent motif characterization using non-negative matrix factorization, Bioinformatics, 24 (2008), 2684-2690.  doi: 10.1093/bioinformatics/btn526.  Google Scholar

[13]

K. Lee, J. Ho and D. Kriegman, Acquiring linear subspaces for face recognition under variable lighting, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27 (2005), 684-698.   Google Scholar

[14]

G. Liu, Z. Lin and Y. Yu, Robust subspace segmentation by low-rank representation, in Proceedings of the 27th international conference on machine learning, 2010, 663–670. Google Scholar

[15]

L. Lu and R. Vidal, Combined central and subspace clustering for computer vision applications, in Proceedings of the 23rd international conference on Machine learning, ACM, 2006, 593–600. doi: 10.1145/1143844.1143919.  Google Scholar

[16]

B. Nasihatkon and R. Hartley, Graph connectivity in sparse subspace clustering, in Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, IEEE, 2011, 2137–2144. doi: 10.1109/CVPR.2011.5995679.  Google Scholar

[17]

A. Y. Ng, M. I. Jordan and Y. Weiss, On spectral clustering: Analysis and an algorithm, Advances in Neural Information Processing Systems, 2 (2002), 849-856.   Google Scholar

[18]

S. Oymak, A. Jalali, M. Fazel, Y. C. Eldar and B. Hassibi, Simultaneously structured models with application to sparse and low-rank matrices, Information Theory, IEEE Transactions on, 61 (2015), 2886-2908.  doi: 10.1109/TIT.2015.2401574.  Google Scholar

[19]

N. Parikh and S. Boyd, Proximal algorithms, Foundations and Trends in optimization, 1 (2014), 127-239.  doi: 10.1561/2400000003.  Google Scholar

[20]

M. J. D. Powell, On search directions for minimization algorithms, Mathematical Programming, 4 (1973), 193-201.  doi: 10.1007/BF01584660.  Google Scholar

[21]

S. R. Rao, R. Tron, R. Vidal and Y. Ma, Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2008, 1–8. doi: 10.1109/CVPR.2008.4587437.  Google Scholar

[22]

E. Richard, G. R. Obozinski and J. -P. Vert, Tight convex relaxations for sparse matrix factorization, in Advances in Neural Information Processing Systems, 2014, 3284–3292. Google Scholar

[23]

Y. Sugaya and K. Kanatani, Geometric structure of degeneracy for multi-body motion segmentation, in Statistical Methods in Video Processing, Springer, 2004, 13–25. doi: 10.1007/978-3-540-30212-4_2.  Google Scholar

[24]

M. E. Tipping and C. M. Bishop, Mixtures of probabilistic principal component analyzers, Neural Computation, 11 (1999), 443-482.  doi: 10.1162/089976699300016728.  Google Scholar

[25]

R. Vidal, A tutorial on subspace clustering, IEEE Signal Processing Magazine, 28 (2010), 52-68.   Google Scholar

[26]

R. Vidal, Y. Ma and S. Sastry, Generalized Principal Component Analysis (GPCA), Interdisciplinary Applied Mathematics, 40. Springer, New York, 2016. doi: 10.1007/978-0-387-87811-9.  Google Scholar

[27]

Y. -X. Wang, H. Xu and C. Leng, Provable subspace clustering: When LRR meets SSC, in Advances in Neural Information Processing Systems, 2013, 64–72. Google Scholar

[28]

J. Yan and M. Pollefeys, A general framework for motion segmentation: Independent, articulated, rigid, non-rigid, degenerate and non-degenerate, in Computer Vision–ECCV 2006, Springer, 2006, 94–106. doi: 10.1007/11744085_8.  Google Scholar

[29]

W. I. Zangwill, Nonlinear Programming: A Unified Approach, vol. 196, Prentice-Hall Englewood Cliffs, NJ, 1969.  Google Scholar

[30]

T. Zhang, A. Szlam, Y. Wang and G. Lerman, Hybrid linear modeling via local best-fit flats, International Journal of Computer Vision, 100 (2012), 217-240.  doi: 10.1007/s11263-012-0535-6.  Google Scholar

show all references

References:
[1]

P. K. Agarwal and N. H. Mustafa, k-means projective clustering, in Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, ACM, 2004, 155–165. doi: 10.1145/1055558.1055581.  Google Scholar

[2]

A. Argyriou, R. Foygel and N. Srebro, Sparse prediction with the k-support norm, in Advances in Neural Information Processing Systems, 2012, 1457–1465. Google Scholar

[3]

J. Bolte, S. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, 146 (2014), 459-494.  doi: 10.1007/s10107-013-0701-9.  Google Scholar

[4]

V. Chandrasekaran, B. Recht, P. A. Parrilo and A. S. Willsky, The convex geometry of linear inverse problems, Foundations of Computational mathematics,, 12 (2012), 805-849.   Google Scholar

[5]

G. Chen and G. Lerman, Spectral curvature clustering (scc), International Journal of Computer Vision, 81 (2009), 317-330.  doi: 10.1007/s11263-008-0178-9.  Google Scholar

[6]

J. P. Costeira and T. Kanade, A multibody factorization method for independently moving objects, International Journal of Computer Vision, 29 (1998), 159-179.   Google Scholar

[7]

H. Derksen, Y. Ma, W. Hong and J. Wright, Segmentation of multivariate mixed data via lossy coding and compression in Electronic Imaging 2007, International Society for Optics and Photonics (2007), 65080H. doi: 10.1117/12.714912.  Google Scholar

[8]

E. Elhamifar and R. Vidal, Sparse subspace clustering: Algorithm, theory, and applications, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2765-2781.  doi: 10.1109/TPAMI.2013.57.  Google Scholar

[9]

P. Favaro, R. Vidal and A. Ravichandran, A closed form solution to robust subspace estimation and clustering, in 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, 2011, 1801–1807. doi: 10.1109/CVPR.2011.5995365.  Google Scholar

[10]

J. Feng, Z. Lin, H. Xu and S. Yan, Robust subspace segmentation with block-diagonal prior, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2014, 3818–3825. doi: 10.1109/CVPR.2014.482.  Google Scholar

[11]

A. Goh and R. Vidal, Segmenting motions of different types by unsupervised manifold clustering, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2007, 1–6. doi: 10.1109/CVPR.2007.383235.  Google Scholar

[12]

L. N. Hutchins, S. M. Murphy, P. Singh and J. H. Graber, Position-dependent motif characterization using non-negative matrix factorization, Bioinformatics, 24 (2008), 2684-2690.  doi: 10.1093/bioinformatics/btn526.  Google Scholar

[13]

K. Lee, J. Ho and D. Kriegman, Acquiring linear subspaces for face recognition under variable lighting, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27 (2005), 684-698.   Google Scholar

[14]

G. Liu, Z. Lin and Y. Yu, Robust subspace segmentation by low-rank representation, in Proceedings of the 27th international conference on machine learning, 2010, 663–670. Google Scholar

[15]

L. Lu and R. Vidal, Combined central and subspace clustering for computer vision applications, in Proceedings of the 23rd international conference on Machine learning, ACM, 2006, 593–600. doi: 10.1145/1143844.1143919.  Google Scholar

[16]

B. Nasihatkon and R. Hartley, Graph connectivity in sparse subspace clustering, in Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, IEEE, 2011, 2137–2144. doi: 10.1109/CVPR.2011.5995679.  Google Scholar

[17]

A. Y. Ng, M. I. Jordan and Y. Weiss, On spectral clustering: Analysis and an algorithm, Advances in Neural Information Processing Systems, 2 (2002), 849-856.   Google Scholar

[18]

S. Oymak, A. Jalali, M. Fazel, Y. C. Eldar and B. Hassibi, Simultaneously structured models with application to sparse and low-rank matrices, Information Theory, IEEE Transactions on, 61 (2015), 2886-2908.  doi: 10.1109/TIT.2015.2401574.  Google Scholar

[19]

N. Parikh and S. Boyd, Proximal algorithms, Foundations and Trends in optimization, 1 (2014), 127-239.  doi: 10.1561/2400000003.  Google Scholar

[20]

M. J. D. Powell, On search directions for minimization algorithms, Mathematical Programming, 4 (1973), 193-201.  doi: 10.1007/BF01584660.  Google Scholar

[21]

S. R. Rao, R. Tron, R. Vidal and Y. Ma, Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2008, 1–8. doi: 10.1109/CVPR.2008.4587437.  Google Scholar

[22]

E. Richard, G. R. Obozinski and J. -P. Vert, Tight convex relaxations for sparse matrix factorization, in Advances in Neural Information Processing Systems, 2014, 3284–3292. Google Scholar

[23]

Y. Sugaya and K. Kanatani, Geometric structure of degeneracy for multi-body motion segmentation, in Statistical Methods in Video Processing, Springer, 2004, 13–25. doi: 10.1007/978-3-540-30212-4_2.  Google Scholar

[24]

M. E. Tipping and C. M. Bishop, Mixtures of probabilistic principal component analyzers, Neural Computation, 11 (1999), 443-482.  doi: 10.1162/089976699300016728.  Google Scholar

[25]

R. Vidal, A tutorial on subspace clustering, IEEE Signal Processing Magazine, 28 (2010), 52-68.   Google Scholar

[26]

R. Vidal, Y. Ma and S. Sastry, Generalized Principal Component Analysis (GPCA), Interdisciplinary Applied Mathematics, 40. Springer, New York, 2016. doi: 10.1007/978-0-387-87811-9.  Google Scholar

[27]

Y. -X. Wang, H. Xu and C. Leng, Provable subspace clustering: When LRR meets SSC, in Advances in Neural Information Processing Systems, 2013, 64–72. Google Scholar

[28]

J. Yan and M. Pollefeys, A general framework for motion segmentation: Independent, articulated, rigid, non-rigid, degenerate and non-degenerate, in Computer Vision–ECCV 2006, Springer, 2006, 94–106. doi: 10.1007/11744085_8.  Google Scholar

[29]

W. I. Zangwill, Nonlinear Programming: A Unified Approach, vol. 196, Prentice-Hall Englewood Cliffs, NJ, 1969.  Google Scholar

[30]

T. Zhang, A. Szlam, Y. Wang and G. Lerman, Hybrid linear modeling via local best-fit flats, International Journal of Computer Vision, 100 (2012), 217-240.  doi: 10.1007/s11263-012-0535-6.  Google Scholar

Figure 1.  The plot of RSS vs. the reduced dimension $s$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The ExtendedYale data B
Figure Options

Download full-size image

Download as PowerPoint slide

Algorithm 1.  ADMM Algorithm to solve Model (11)
    Data: Initialize $E=0$, $U=0$, $V=0$, $\lambda_1=0$, $\lambda_2=0$.
1 while not convergence do
2 │ Update $Z$ with Equation (14);
3 │ Update $E$ with Equation (15) and $U$, $V$ with Equation (16), respectively;
4 │ Update $\lambda_1$ with Equation (17) and $\lambda_2$ with Equation (18);
5 end
Table Options

Download as excel

    Data: Initialize $E=0$, $U=0$, $V=0$, $\lambda_1=0$, $\lambda_2=0$.
1 while not convergence do
2 │ Update $Z$ with Equation (14);
3 │ Update $E$ with Equation (15) and $U$, $V$ with Equation (16), respectively;
4 │ Update $\lambda_1$ with Equation (17) and $\lambda_2$ with Equation (18);
5 end
Algorithm 2.  The algorithm for proximity operator of $\frac{1}{2\alpha}(\|\cdot\|_{sp}^{k})^2$ with input $\mathbf{h}$
    Data: $\mathbf{h}\in\mathbb{R}^n$ and the parameter $\alpha$.
    Result: $\mathbf{p}=\underset{\mathbf{g}}{\arg\min}\left\{\frac{1}{2\alpha}(\|\mathbf{g}\|^{sp}_{k})^2+\frac{1}{2}\|\mathbf{g}-\mathbf{h}\|^2_2\right\}$
1 Let $\tilde{\mathbf{h}}=[\tilde{h}_1,\cdots,\tilde{h}_n]^T$, i.e., $\tilde{h}_i$ is the $i$-th largest element of $|\mathbf{h}|$. Let $\Pi$ be the permutation matrix such that $\tilde{\mathbf{h}}=\Pi|\mathbf{h}|$. For simplicity, define $\tilde{h}_0:=+\infty$, $\tilde{h}_{n+1}:=-\infty$ and $\gamma_{r,l}:=\sum^l_{i=k-r}\tilde{h}_i$
2 Find $r\in\{0,\cdots,k-1\}$ and $l\in\{k,\cdots,n\}$ such that
            $\frac{\tilde{h}_{k-r-1}}{\alpha+1}>\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\ge\frac{\tilde{h}_{k-r}}{\alpha+1}, $
            $\tilde{u}_l>\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\ge \tilde{g}_{l+1}.$
3 Define
            $ q_i = \left\{ \begin{array}{ll} \frac{\alpha}{\alpha+1}\tilde{h}_i & \textrm{if}\; i=1, \cdots, k-r-1\\ \tilde{h}_i-\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\\ & \textrm{if}\; i=k-r, \cdots, l\\ 0 & \textrm{if} \; i=l+1, \cdots, n \end{array} \right.$
4 Set $\mathbf{p}=[p_1,\ldots,p_n]^T$, where $p_i=\mathrm{sign}(h_i)(\Pi^{-1}\mathbf{q})_i$.
Table Options

Download as excel

    Data: $\mathbf{h}\in\mathbb{R}^n$ and the parameter $\alpha$.
    Result: $\mathbf{p}=\underset{\mathbf{g}}{\arg\min}\left\{\frac{1}{2\alpha}(\|\mathbf{g}\|^{sp}_{k})^2+\frac{1}{2}\|\mathbf{g}-\mathbf{h}\|^2_2\right\}$
1 Let $\tilde{\mathbf{h}}=[\tilde{h}_1,\cdots,\tilde{h}_n]^T$, i.e., $\tilde{h}_i$ is the $i$-th largest element of $|\mathbf{h}|$. Let $\Pi$ be the permutation matrix such that $\tilde{\mathbf{h}}=\Pi|\mathbf{h}|$. For simplicity, define $\tilde{h}_0:=+\infty$, $\tilde{h}_{n+1}:=-\infty$ and $\gamma_{r,l}:=\sum^l_{i=k-r}\tilde{h}_i$
2 Find $r\in\{0,\cdots,k-1\}$ and $l\in\{k,\cdots,n\}$ such that
            $\frac{\tilde{h}_{k-r-1}}{\alpha+1}>\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\ge\frac{\tilde{h}_{k-r}}{\alpha+1}, $
            $\tilde{u}_l>\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\ge \tilde{g}_{l+1}.$
3 Define
            $ q_i = \left\{ \begin{array}{ll} \frac{\alpha}{\alpha+1}\tilde{h}_i & \textrm{if}\; i=1, \cdots, k-r-1\\ \tilde{h}_i-\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\\ & \textrm{if}\; i=k-r, \cdots, l\\ 0 & \textrm{if} \; i=l+1, \cdots, n \end{array} \right.$
4 Set $\mathbf{p}=[p_1,\ldots,p_n]^T$, where $p_i=\mathrm{sign}(h_i)(\Pi^{-1}\mathbf{q})_i$.
Table 1.  The error rate (mean % and median %) for face clustering on Extended Yale dataset B
# Classes mean/median SSC LRR (3, 3)-SMF (4, 4)-SMF
error s error s
2 mean 15.83 6.37 3.38 18 3.53 18
median 15.63 6.25 2.34 2.34
3 mean 28.13 9.57 6.19 25 6.06 25
median 28.65 8.85 5.73 5.73
5 mean 37.90 14.86 11.06 35 10.04 35
median 38.44 14.38 9.38 9.06
8 mean 44.25 23.27 23.08 50 22.51 50
median 44.82 21.29 27.54 26.06
10 mean 50.78 29.38 25.36 65 23.91 65
median 49.06 32.97 27.19 27.34
Table Options

Download as excel

# Classes mean/median SSC LRR (3, 3)-SMF (4, 4)-SMF
error s error s
2 mean 15.83 6.37 3.38 18 3.53 18
median 15.63 6.25 2.34 2.34
3 mean 28.13 9.57 6.19 25 6.06 25
median 28.65 8.85 5.73 5.73
5 mean 37.90 14.86 11.06 35 10.04 35
median 38.44 14.38 9.38 9.06
8 mean 44.25 23.27 23.08 50 22.51 50
median 44.82 21.29 27.54 26.06
10 mean 50.78 29.38 25.36 65 23.91 65
median 49.06 32.97 27.19 27.34
Table 2.  The error rate (mean %/median %) for motion segmentation on Hopkins155 dataset
SSC LRR (3, 3)-SMF (4, 4)-SMF
Mean 9.28 8.43 6.61 7.16
Median 0.24 1.54 1.20 1.32
Table Options

Download as excel

SSC LRR (3, 3)-SMF (4, 4)-SMF
Mean 9.28 8.43 6.61 7.16
Median 0.24 1.54 1.20 1.32
[1]

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
[2]

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
[3]

Yun Cai, Song Li. Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery. Inverse Problems & Imaging, 2017, 11 (4) : 643-661. doi: 10.3934/ipi.2017030
[4]

Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001
[5]

Simon Foucart, Richard G. Lynch. Recovering low-rank matrices from binary measurements. Inverse Problems & Imaging, 2019, 13 (4) : 703-720. doi: 10.3934/ipi.2019032
[6]

Hua Huang, Weiwei Wang, Chengwu Lu, Xiangchu Feng, Ruiqiang He. Side-information-induced reweighted sparse subspace clustering. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020019
[7]

Baolan Yuan, Wanjun Zhang, Yubo Yuan. A Max-Min clustering method for $k$-means algorithm of data clustering. Journal of Industrial & Management Optimization, 2012, 8 (3) : 565-575. doi: 10.3934/jimo.2012.8.565
[8]

Ruiqi Yang, Dachuan Xu, Yicheng Xu, Dongmei Zhang. An adaptive probabilistic algorithm for online k-center clustering. Journal of Industrial & Management Optimization, 2019, 15 (2) : 565-576. doi: 10.3934/jimo.2018057
[9]

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
[10]

Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems & Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022
[11]

Changguang Dong. Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4231-4238. doi: 10.3934/dcds.2017180
[12]

Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691
[13]

Guojun Gan, Kun Chen. A soft subspace clustering algorithm with log-transformed distances. Big Data & Information Analytics, 2016, 1 (1) : 93-109. doi: 10.3934/bdia.2016.1.93
[14]

Narciso Román-Roy, Ángel M. Rey, Modesto Salgado, Silvia Vilariño. On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories. Journal of Geometric Mechanics, 2011, 3 (1) : 113-137. doi: 10.3934/jgm.2011.3.113
[15]

Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2291-2303. doi: 10.3934/cpaa.2012.11.2291
[16]

Sangye Lungten, Wil H. A. Schilders, Joseph M. L. Maubach. Sparse inverse incidence matrices for Schilders' factorization applied to resistor network modeling. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 227-239. doi: 10.3934/naco.2014.4.227
[17]

Lejun Shi, Shaocui Guo, Xu Yang. Encryption service protocol based on matrix norm algorithm. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020255
[18]

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
[19]

Sung Ha Kang, Berta Sandberg, Andy M. Yip. A regularized k-means and multiphase scale segmentation. Inverse Problems & Imaging, 2011, 5 (2) : 407-429. doi: 10.3934/ipi.2011.5.407
[20]

Michael Baur, Marco Gaertler, Robert Görke, Marcus Krug, Dorothea Wagner. Augmenting $k$-core generation with preferential attachment. Networks & Heterogeneous Media, 2008, 3 (2) : 277-294. doi: 10.3934/nhm.2008.3.277

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (29)
  • HTML views (18)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences