
-
Previous Article
Quantitative robustness of localized support vector machines
- CPAA Home
- This Issue
-
Next Article
Learning rates for partially linear functional models with high dimensional scalar covariates
Sparse generalized canonical correlation analysis via linearized Bregman method
1. | School of Statistics and Mathematics, Collaborative Innovation Development Center of, Pearl River Delta Science & Technology Finance Industry, Guangdong University of Finance & Economics, Guangzhou, Guangdong, 510320, China |
2. | School of Electronics and Computer Science, University of Southampton, University Road, Southampton, SO17 1BJ, United Kingdom |
Canonical correlation analysis (CCA) is a powerful statistical tool for detecting mutual information between two sets of multi-dimensional random variables. Unlike CCA, Generalized CCA (GCCA), a natural extension of CCA, could detect the relations of multiple datasets (more than two). To interpret canonical variates more efficiently, this paper addresses a novel sparse GCCA algorithm via linearized Bregman method, which is a generalization of traditional sparse CCA methods. Experimental results on both synthetic dataset and real datasets demonstrate the effectiveness and efficiency of the proposed algorithm when compared with several state-of-the-art sparse CCA and deep CCA algorithms.
References:
[1] |
G. Andrew, R. Arora, J. Bilmes and K. Livescu, Deep canonical correlation analysis, in International Conference on Machine Learning, (2013), 1247–1255. |
[2] |
A. Benton, R. Arora and and M. Dredze,
Learning multiview embeddings of twitter users, Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, 2 (2016), 14-19.
|
[3] |
A. Benton, H. Khayrallah, B. Gujral and et al., Deep Generalized Canonical Correlation Analysis, in Proceedings of the 4th Workshop on Representation Learning for NLP (RepL4NLP-2019), (2019), 1–6. |
[4] |
L. M. Br $\grave{e}$ gman,
A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming, Zh. Vychisl. Mat. Mat. Fiz., 7 (1967), 620-631.
|
[5] |
J. F. Cai, S. Osher and Z. Shen,
Convergence of the linearized bregman iteration for $\ell_1$-norm minimization, Math. Comp., 78 (2009), 2127-2136.
doi: 10.1090/S0025-5718-09-02242-X. |
[6] |
J. F. Cai, S. Osher and Z. Shen,
Linearized bregman iterations for compressed sensing, Math. Comp., 78 (2009), 1515-1536.
doi: 10.1090/S0025-5718-08-02189-3. |
[7] |
J. Carroll,
Equations and tables for a generalization of canonical correlation analysis to three or more sets of variables, Proceedings of Annual Convention of The American Psychological Association, 3 (1968), 227-228.
|
[8] |
M. Chen, C. Gao, Z. Ren and et al., Sparse cca via precision adjusted iterative thresholding, in Proceedings of International Congress of Chinese Mathematicians, (2016). |
[9] |
D. Chu, L. Z. Liao, M. K. Ng and X. W. Zhang,
Sparse canonical correlation analysis: new formulation and algorithm, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 3050-3065.
doi: 10.1109/TPAMI.2013.104. |
[10] |
M. Dettling,
Bagboosting for tumor classification with gene expression data, Bioinformatics, 20 (2004), 3583-3593.
doi: 10.1093/bioinformatics/bth447. |
[11] |
O. Friman, J. Cedefamn, P. Lundberg, H. Borga and H. Knutsson,
Detection of neural activity in functional mri using canonical correlation analysis, Magn. Reson. Med., 45 (2001), 323-330.
doi: 10.1002/1522-2594(200102)45:2<323::aid-mrm1041>3.0.co;2-#. |
[12] |
C. Gao, Z. Ma and H. H. Zhou,
Sparse cca: Adaptive estimation and computational barriers, Ann. Statist., 45 (2017), 2074-2101.
doi: 10.1214/16-AOS1519. |
[13] |
D. R. Hardoon and J. Shawe-Taylor,
Sparse canonical correlation analysis, Mach. Learn., 83 (2011), 331-353.
doi: 10.1007/s10994-010-5222-7. |
[14] |
D. R. Hardoon, S. Szedmak and and J. Shawe-Taylor,
Canonical correlation analysis: An overview with application to learning methods, Neural Comput., 16 (2004), 2639-2664.
doi: 10.1162/0899766042321814. |
[15] |
P. Horst,
Generalized canonical correlations and their applications to experimental data, J. Clin. Psychol., 17 (1961), 331-347.
|
[16] |
H. Hotelling,
Relations between two sets of variates, Biometrika, 2 (1936), 321-377.
|
[17] |
M. Kang, B. Zhang, X. Wu, C. Y. Liu and J. Gao, Sparse generalized canonical correlation analysis for biological model integration: a genetic study of psychiatric disorders, in 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), (2013), 1490–1493. |
[18] |
J. R. Kettenring,
Canonical analysis of several sets of variables, Biometrika, 58 (1971), 433-451.
doi: 10.1093/biomet/58.3.433. |
[19] |
Y. Luo, D. Tao, K. Ramamohanarao, C. Xu and Y. G. Wen,
Tensor canonical correlation analysis for multi-view dimension reduction, IEEE Trans. Knowl. Data Eng., 27 (2015), 3111-3124.
doi: 10.1109/TKDE.2015.2445757. |
[20] |
S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Ying,
An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.
doi: 10.1137/040605412. |
[21] |
R. Steinberger, B. Pouliquen, A. Widiger and et al., The jrc-acquis: A multilingual aligned parallel corpus with 20+ languages, in Proceedings of the 5th International Conference on Language Resources and Evaluation (LREC$\prime$ 2006), (2006), 2142–2147. |
[22] |
J. V$\acute{I}$a, I. Santamar$\acute{I}$a and and J. P$\acute{e}$rez,
A learning algorithm for adaptive canonical correlation analysis of several data sets, Neural Netw., 20 (2007), 139-152.
doi: 10.1016/j.neunet.2006.09.011. |
[23] |
A. Vinokourov, N. Cristianini and J. Shawe-Taylor, Inferring a semantic representation of text via cross-language correlation analysis, in Advances in Neural Information Processing Systems, (2003), 1497–1504. |
[24] |
S. Waaijenborg, P. C. V. de Witt Hamer and A. H. Zwinderman, Quantifying the association between gene expressions and dna-markers by penalized canonical correlation analysis, Stat. Appl. Genet. Mol. Biol., 7 (2008), Art. 3.
doi: 10.2202/1544-6115.1329. |
[25] |
D. M. Witten, R. Tibshirani and T. Hastie,
A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis, Biostatistics, 10 (2009), 515-534.
doi: 10.1093/biostatistics/kxp008. |
[26] |
Y. Yamanishi, J. P. Vert, A. Nakaya and M. Kanehisa, Extraction of correlated gene clusters from multiple genomic data by generalized kernel canonical correlation analysis, Bioinformatics, 19 (2003), i323–i330.
doi: 10.1093/bioinformatics/btg1045. |
[27] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon,
Bregman iterative algorithms for 1-minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1 (2008), 143-168.
doi: 10.1137/070703983. |
show all references
References:
[1] |
G. Andrew, R. Arora, J. Bilmes and K. Livescu, Deep canonical correlation analysis, in International Conference on Machine Learning, (2013), 1247–1255. |
[2] |
A. Benton, R. Arora and and M. Dredze,
Learning multiview embeddings of twitter users, Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, 2 (2016), 14-19.
|
[3] |
A. Benton, H. Khayrallah, B. Gujral and et al., Deep Generalized Canonical Correlation Analysis, in Proceedings of the 4th Workshop on Representation Learning for NLP (RepL4NLP-2019), (2019), 1–6. |
[4] |
L. M. Br $\grave{e}$ gman,
A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming, Zh. Vychisl. Mat. Mat. Fiz., 7 (1967), 620-631.
|
[5] |
J. F. Cai, S. Osher and Z. Shen,
Convergence of the linearized bregman iteration for $\ell_1$-norm minimization, Math. Comp., 78 (2009), 2127-2136.
doi: 10.1090/S0025-5718-09-02242-X. |
[6] |
J. F. Cai, S. Osher and Z. Shen,
Linearized bregman iterations for compressed sensing, Math. Comp., 78 (2009), 1515-1536.
doi: 10.1090/S0025-5718-08-02189-3. |
[7] |
J. Carroll,
Equations and tables for a generalization of canonical correlation analysis to three or more sets of variables, Proceedings of Annual Convention of The American Psychological Association, 3 (1968), 227-228.
|
[8] |
M. Chen, C. Gao, Z. Ren and et al., Sparse cca via precision adjusted iterative thresholding, in Proceedings of International Congress of Chinese Mathematicians, (2016). |
[9] |
D. Chu, L. Z. Liao, M. K. Ng and X. W. Zhang,
Sparse canonical correlation analysis: new formulation and algorithm, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 3050-3065.
doi: 10.1109/TPAMI.2013.104. |
[10] |
M. Dettling,
Bagboosting for tumor classification with gene expression data, Bioinformatics, 20 (2004), 3583-3593.
doi: 10.1093/bioinformatics/bth447. |
[11] |
O. Friman, J. Cedefamn, P. Lundberg, H. Borga and H. Knutsson,
Detection of neural activity in functional mri using canonical correlation analysis, Magn. Reson. Med., 45 (2001), 323-330.
doi: 10.1002/1522-2594(200102)45:2<323::aid-mrm1041>3.0.co;2-#. |
[12] |
C. Gao, Z. Ma and H. H. Zhou,
Sparse cca: Adaptive estimation and computational barriers, Ann. Statist., 45 (2017), 2074-2101.
doi: 10.1214/16-AOS1519. |
[13] |
D. R. Hardoon and J. Shawe-Taylor,
Sparse canonical correlation analysis, Mach. Learn., 83 (2011), 331-353.
doi: 10.1007/s10994-010-5222-7. |
[14] |
D. R. Hardoon, S. Szedmak and and J. Shawe-Taylor,
Canonical correlation analysis: An overview with application to learning methods, Neural Comput., 16 (2004), 2639-2664.
doi: 10.1162/0899766042321814. |
[15] |
P. Horst,
Generalized canonical correlations and their applications to experimental data, J. Clin. Psychol., 17 (1961), 331-347.
|
[16] |
H. Hotelling,
Relations between two sets of variates, Biometrika, 2 (1936), 321-377.
|
[17] |
M. Kang, B. Zhang, X. Wu, C. Y. Liu and J. Gao, Sparse generalized canonical correlation analysis for biological model integration: a genetic study of psychiatric disorders, in 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), (2013), 1490–1493. |
[18] |
J. R. Kettenring,
Canonical analysis of several sets of variables, Biometrika, 58 (1971), 433-451.
doi: 10.1093/biomet/58.3.433. |
[19] |
Y. Luo, D. Tao, K. Ramamohanarao, C. Xu and Y. G. Wen,
Tensor canonical correlation analysis for multi-view dimension reduction, IEEE Trans. Knowl. Data Eng., 27 (2015), 3111-3124.
doi: 10.1109/TKDE.2015.2445757. |
[20] |
S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Ying,
An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.
doi: 10.1137/040605412. |
[21] |
R. Steinberger, B. Pouliquen, A. Widiger and et al., The jrc-acquis: A multilingual aligned parallel corpus with 20+ languages, in Proceedings of the 5th International Conference on Language Resources and Evaluation (LREC$\prime$ 2006), (2006), 2142–2147. |
[22] |
J. V$\acute{I}$a, I. Santamar$\acute{I}$a and and J. P$\acute{e}$rez,
A learning algorithm for adaptive canonical correlation analysis of several data sets, Neural Netw., 20 (2007), 139-152.
doi: 10.1016/j.neunet.2006.09.011. |
[23] |
A. Vinokourov, N. Cristianini and J. Shawe-Taylor, Inferring a semantic representation of text via cross-language correlation analysis, in Advances in Neural Information Processing Systems, (2003), 1497–1504. |
[24] |
S. Waaijenborg, P. C. V. de Witt Hamer and A. H. Zwinderman, Quantifying the association between gene expressions and dna-markers by penalized canonical correlation analysis, Stat. Appl. Genet. Mol. Biol., 7 (2008), Art. 3.
doi: 10.2202/1544-6115.1329. |
[25] |
D. M. Witten, R. Tibshirani and T. Hastie,
A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis, Biostatistics, 10 (2009), 515-534.
doi: 10.1093/biostatistics/kxp008. |
[26] |
Y. Yamanishi, J. P. Vert, A. Nakaya and M. Kanehisa, Extraction of correlated gene clusters from multiple genomic data by generalized kernel canonical correlation analysis, Bioinformatics, 19 (2003), i323–i330.
doi: 10.1093/bioinformatics/btg1045. |
[27] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon,
Bregman iterative algorithms for 1-minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1 (2008), 143-168.
doi: 10.1137/070703983. |


Algorithm 1: Sparse Generalized CCA algorithm. |
Input: Training data Output: Sparse canonical variates 1: Compute reduced SVD for each 2: Compute top 3: Construct 4: Let 5: while error 6: Compute 7: error= 8:end while 9:return |
Algorithm 1: Sparse Generalized CCA algorithm. |
Input: Training data Output: Sparse canonical variates 1: Compute reduced SVD for each 2: Compute top 3: Construct 4: Let 5: while error 6: Compute 7: error= 8:end while 9:return |
GCCA | Deep GCCA | WGCCA | SGCCA | ||
Trerror | 3.7277e-30 | 0.5043 | 1.1118e-07 | 1.6354e-10 | |
Tserror | 0.4756 | 0.2955 | 0.6860 | 0.4554 | |
Spar1 (%) | 2.73 | 0.01 | 70.01 | 99.47 | |
Spar2 (%) | 5.15 | 0.03 | 78.19 | 99.67 | |
Spar3 (%) | 7.54 | 0.09 | 83.02 | 99.72 |
GCCA | Deep GCCA | WGCCA | SGCCA | ||
Trerror | 3.7277e-30 | 0.5043 | 1.1118e-07 | 1.6354e-10 | |
Tserror | 0.4756 | 0.2955 | 0.6860 | 0.4554 | |
Spar1 (%) | 2.73 | 0.01 | 70.01 | 99.47 | |
Spar2 (%) | 5.15 | 0.03 | 78.19 | 99.67 | |
Spar3 (%) | 7.54 | 0.09 | 83.02 | 99.72 |
Type | Data | ||||
Gene Data | Leukemia | 3571 | 72 | 2 | 1 |
Lymphomia | 4026 | 62 | 3 | 2 | |
Prostate | 6033 | 102 | 2 | 1 | |
Brain | 5597 | 42 | 5 | 4 |
Type | Data | ||||
Gene Data | Leukemia | 3571 | 72 | 2 | 1 |
Lymphomia | 4026 | 62 | 3 | 2 | |
Prostate | 6033 | 102 | 2 | 1 | |
Brain | 5597 | 42 | 5 | 4 |
Methods | Spar1( |
Spar2( |
Accuracy ( |
SCORR | ||
Leukemia | CCA | 0.03 | 0 | 97.20 | 0.8945 | |
PMD CCA | 98.48 | 0 | 72.22 | 0.8944 | ||
GCCA | 5.80 | 50 | 97.20 | 0.9227 | ||
Deep CCA | 0 | 0 | 94.40 | 0.8662 | ||
SGCCA | 98.99 | 50 | 100 | 0.8916 | ||
Lyphomia | CCA | 0.04 | 0 | 83.87 | 1.8206 | |
PMD CCA | 85.38 | 0 | 77.97 | 1.7911 | ||
GCCA | 5.77 | 50 | 90.32 | 1.7423 | ||
Deep CCA | 0 | 0 | 80.65 | 0.9754 | ||
SGCCA | 99.23 | 50 | 96.77 | 1.6933 | ||
Prostate | CCA | 0.1 | 0 | 88.24 | 0.7646 | |
PMD CCA | 99.19 | 0 | 60.78 | 0.2978 | ||
GCCA | 4.59 | 50 | 88.24 | 0.7645 | ||
Deep CCA | 0 | 0 | 79.41 | 0.7610 | ||
SGCCA | 99.14 | 50 | 85.27 | 0.7758 | ||
Brain | CCA | 0.07 | 0 | 71.43 | 2.9749 | |
PMD CCA | 76.40 | 74.65 | 47.62 | 3.0669 | ||
GCCA | 4.52 | 34.9 | 61.90 | 2.8307 | ||
Deep CCA | 0 | 0 | 61.90 | 2.9674 | ||
SGCCA | 99.62 | 34.9 | 66.67 | 2.5729 |
Methods | Spar1( |
Spar2( |
Accuracy ( |
SCORR | ||
Leukemia | CCA | 0.03 | 0 | 97.20 | 0.8945 | |
PMD CCA | 98.48 | 0 | 72.22 | 0.8944 | ||
GCCA | 5.80 | 50 | 97.20 | 0.9227 | ||
Deep CCA | 0 | 0 | 94.40 | 0.8662 | ||
SGCCA | 98.99 | 50 | 100 | 0.8916 | ||
Lyphomia | CCA | 0.04 | 0 | 83.87 | 1.8206 | |
PMD CCA | 85.38 | 0 | 77.97 | 1.7911 | ||
GCCA | 5.77 | 50 | 90.32 | 1.7423 | ||
Deep CCA | 0 | 0 | 80.65 | 0.9754 | ||
SGCCA | 99.23 | 50 | 96.77 | 1.6933 | ||
Prostate | CCA | 0.1 | 0 | 88.24 | 0.7646 | |
PMD CCA | 99.19 | 0 | 60.78 | 0.2978 | ||
GCCA | 4.59 | 50 | 88.24 | 0.7645 | ||
Deep CCA | 0 | 0 | 79.41 | 0.7610 | ||
SGCCA | 99.14 | 50 | 85.27 | 0.7758 | ||
Brain | CCA | 0.07 | 0 | 71.43 | 2.9749 | |
PMD CCA | 76.40 | 74.65 | 47.62 | 3.0669 | ||
GCCA | 4.52 | 34.9 | 61.90 | 2.8307 | ||
Deep CCA | 0 | 0 | 61.90 | 2.9674 | ||
SGCCA | 99.62 | 34.9 | 66.67 | 2.5729 |
GCCA | Deep GCCA | WGCCA | SGCCA | ||
Trerror | 2.2078e-30 | 3.0859e-02 | 8.4776e-12 | 5.5140e-11 | |
Tserror | 4.1496e-02 | 1.7648e-02 | 0.7138 | 3.8799e-02 | |
Spar1 (%) | 7.00 | 0.1140 | 10.27 | 96.56 | |
Spar2 (%) | 7.24 | 0.6850 | 9.96 | 96.85 | |
Spar3 (%) | 6.09 | 0.1100 | 8.49 | 95.64 |
GCCA | Deep GCCA | WGCCA | SGCCA | ||
Trerror | 2.2078e-30 | 3.0859e-02 | 8.4776e-12 | 5.5140e-11 | |
Tserror | 4.1496e-02 | 1.7648e-02 | 0.7138 | 3.8799e-02 | |
Spar1 (%) | 7.00 | 0.1140 | 10.27 | 96.56 | |
Spar2 (%) | 7.24 | 0.6850 | 9.96 | 96.85 | |
Spar3 (%) | 6.09 | 0.1100 | 8.49 | 95.64 |
[1] |
Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889 |
[2] |
Xiaojuan Deng, Xing Zhao, Mengfei Li, Hongwei Li. Limited-angle CT reconstruction with generalized shrinkage operators as regularizers. Inverse Problems and Imaging, 2021, 15 (6) : 1287-1306. doi: 10.3934/ipi.2021019 |
[3] |
Chengxiang Wang, Li Zeng. Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections. Inverse Problems and Imaging, 2016, 10 (3) : 829-853. doi: 10.3934/ipi.2016023 |
[4] |
Yunmei Chen, Xianqi Li, Yuyuan Ouyang, Eduardo Pasiliao. Accelerated bregman operator splitting with backtracking. Inverse Problems and Imaging, 2017, 11 (6) : 1047-1070. doi: 10.3934/ipi.2017048 |
[5] |
Mridul Nandi, Tapas Pandit. Efficient fully CCA-secure predicate encryptions from pair encodings. Advances in Mathematics of Communications, 2022, 16 (1) : 37-72. doi: 10.3934/amc.2020098 |
[6] |
Wenye Ma, Stanley Osher. A TV Bregman iterative model of Retinex theory. Inverse Problems and Imaging, 2012, 6 (4) : 697-708. doi: 10.3934/ipi.2012.6.697 |
[7] |
Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043 |
[8] |
Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043 |
[9] |
Jin-Zan Liu, Xin-Wei Liu. A dual Bregman proximal gradient method for relatively-strongly convex optimization. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021028 |
[10] |
Michael Renardy. Backward uniqueness for linearized compressible flow. Evolution Equations and Control Theory, 2015, 4 (1) : 107-113. doi: 10.3934/eect.2015.4.107 |
[11] |
Guillaume Bal, Chenxi Guo, Francçois Monard. Linearized internal functionals for anisotropic conductivities. Inverse Problems and Imaging, 2014, 8 (1) : 1-22. doi: 10.3934/ipi.2014.8.1 |
[12] |
Vladimir Sharafutdinov. The linearized problem of magneto-photoelasticity. Inverse Problems and Imaging, 2014, 8 (1) : 247-257. doi: 10.3934/ipi.2014.8.247 |
[13] |
Orlando Lopes. A linearized instability result for solitary waves. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 115-119. doi: 10.3934/dcds.2002.8.115 |
[14] |
Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems and Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173 |
[15] |
Jiaqing Yang, Bo Zhang, Ruming Zhang. Reconstruction of penetrable grating profiles. Inverse Problems and Imaging, 2013, 7 (4) : 1393-1407. doi: 10.3934/ipi.2013.7.1393 |
[16] |
Jorge Tejero. Reconstruction of rough potentials in the plane. Inverse Problems and Imaging, 2019, 13 (4) : 863-878. doi: 10.3934/ipi.2019039 |
[17] |
Felipe Ponce-Vanegas. Reconstruction of the derivative of the conductivity at the boundary. Inverse Problems and Imaging, 2020, 14 (4) : 701-718. doi: 10.3934/ipi.2020032 |
[18] |
Horst Heck, Gunther Uhlmann, Jenn-Nan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems and Imaging, 2007, 1 (1) : 63-76. doi: 10.3934/ipi.2007.1.63 |
[19] |
Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems and Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399 |
[20] |
Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]