# American Institute of Mathematical Sciences

April  2017, 13(2): 609-622. doi: 10.3934/jimo.2016035

## A new semi-supervised classifier based on maximum vector-angular margin

 College of Science, China Agricultural University, No.17 Tsing Hua East Road, Hai Dian District, Beijing 100083, China

* Corresponding author: Liming Yang

Received  December 2014 Revised  December 2015 Published  May 2016

Semi-supervised learning is an attractive method in classification problems when insufficient training information is available. In this investigation, a new semi-supervised classifier is proposed based on the concept of maximum vector-angular margin, (called S$^3$MAMC), the main goal of which is to find an optimal vector $c$ as close as possible to the center of the dataset consisting of both labeled samples and unlabeled samples. This makes S$^3$MAMC better generalization with smaller VC (Vapnik-Chervonenkis) dimension. However, S$^3$MAMC formulation is a non-convex model and therefore it is difficult to solve. Following that we present two optimization algorithms, mixed integer quadratic program (MIQP) and DC (difference of convex functions) program algorithms, to solve the S$^3$MAMC. Compared with the supervised learning methods, numerical experiments on real and synthetic databases demonstrate that the S$^3$MAMC can improve generalization when the labelled samples are relatively few. In addition, the S$^3$MAMC has competitive experiment results in generalization compared to the traditional semi-supervised classification methods.

Citation: Liming Yang, Yannan Chao. A new semi-supervised classifier based on maximum vector-angular margin. Journal of Industrial and Management Optimization, 2017, 13 (2) : 609-622. doi: 10.3934/jimo.2016035
##### References:
 [1] L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 133 (2005), 23-46.  doi: 10.1007/s10479-004-5022-1. [2] L. T. H. An, H. M. Le, V. V. Nguyen and P. D. Tao, A DC programming approach for feature selection in support vector machines learning, Advances in Data Analysis and Classification, 2 (2008), 259-278.  doi: 10.1007/s11634-008-0030-7. [3] A. Asuncion and D. J. Newman, UCI machine learning repository, School of Information and Computer Sciences, University of California Irvine, 2007, http://www.ics.uci.edu/~mlearn/MLRepository.html. [4] K. Bennett and A. Demiriz, Semi-supervised support vector machines, In Advances in Neural Information Processing Systems, MIT Press, Cambridge, 12 (1998), 368–374. [5] W. Changzhi, L. Chaojie and L. Qiang, A DC programming approach for sensor network localization with uncertainties in anchor positions, Journal of Industrial and Management Optimization, 10 (2014), 817-826.  doi: 10.3934/jimo.2014.10.817. [6] O. Chapelle, V. Sindhwani and S. Keerthi, Optimization Techniques for Semi-Supervised Support Vector Machines, Journal of Machine Learning Research, 9 (2008), 203-233. [7] T. Fawcett, An introduction to ROC analysis, Pattern Recognition Letters, 27 (2006), 861-874. [8] G. Fung and O. Mangasarian, Semi-Supervised Support Vector Machines for Unlabeled Data Classification, Optimization methods & software, 15 (2001), 29-44. [9] W. Guan and A. Gray, Sparse high-dimensional fractional-norm support vector machine via DC programming, Computational Statistics and Data Analysis, 67 (2013), 136-148.  doi: 10.1016/j.csda.2013.01.020. [10] W. J. Hu, F. L. Chung and L. SH. Wang, The Maximum Vector-Angular Margin Classifier and its fast training on large datasets using a core vector machine, Neural Networks, 27 (2012), 60-73. [11] P. D. Tao and L. T. T. An, Convex analysis approaches to DC programming: Theory, algorithms and applications, Acta Mathematica, 22 (1997), 287-367. [12] B. Scholkopf, A. J. Smola, R. C. Williamson and P. L. Bartlett, New support vector algorithms, Neural Computation, 12 (2000), 1207-1245. [13] X. Xiao, J. Gu, L. Zhang and S. Zhang, A sequential convex program method to DC program with joint chance constraints, Journal of Industrial and Management Optimization, 8 (2012), 733-747.  doi: 10.3934/jimo.2012.8.733. [14] L. M. Yang and L. SH. Wang, A class of smooth semi-supervised SVM by difference of convex functions programming and algorithm, Knowledge-Based Systems, 41 (2013), 1-7. [15] YALMIP Toolbox. http://control.ee.ethz.ch/~joloef/wiki/pmwiki.php. [16] Y. B. Yuan, Canonical duality solution for alternating support vector machine, Journal of Industrial and Management Optimization, 8 (2012), 611-621.  doi: 10.3934/jimo.2012.8.611. [17] V. N. Vapnik, Statistical Learning Theory, New York: Wiley. 1998.

show all references

##### References:
 [1] L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 133 (2005), 23-46.  doi: 10.1007/s10479-004-5022-1. [2] L. T. H. An, H. M. Le, V. V. Nguyen and P. D. Tao, A DC programming approach for feature selection in support vector machines learning, Advances in Data Analysis and Classification, 2 (2008), 259-278.  doi: 10.1007/s11634-008-0030-7. [3] A. Asuncion and D. J. Newman, UCI machine learning repository, School of Information and Computer Sciences, University of California Irvine, 2007, http://www.ics.uci.edu/~mlearn/MLRepository.html. [4] K. Bennett and A. Demiriz, Semi-supervised support vector machines, In Advances in Neural Information Processing Systems, MIT Press, Cambridge, 12 (1998), 368–374. [5] W. Changzhi, L. Chaojie and L. Qiang, A DC programming approach for sensor network localization with uncertainties in anchor positions, Journal of Industrial and Management Optimization, 10 (2014), 817-826.  doi: 10.3934/jimo.2014.10.817. [6] O. Chapelle, V. Sindhwani and S. Keerthi, Optimization Techniques for Semi-Supervised Support Vector Machines, Journal of Machine Learning Research, 9 (2008), 203-233. [7] T. Fawcett, An introduction to ROC analysis, Pattern Recognition Letters, 27 (2006), 861-874. [8] G. Fung and O. Mangasarian, Semi-Supervised Support Vector Machines for Unlabeled Data Classification, Optimization methods & software, 15 (2001), 29-44. [9] W. Guan and A. Gray, Sparse high-dimensional fractional-norm support vector machine via DC programming, Computational Statistics and Data Analysis, 67 (2013), 136-148.  doi: 10.1016/j.csda.2013.01.020. [10] W. J. Hu, F. L. Chung and L. SH. Wang, The Maximum Vector-Angular Margin Classifier and its fast training on large datasets using a core vector machine, Neural Networks, 27 (2012), 60-73. [11] P. D. Tao and L. T. T. An, Convex analysis approaches to DC programming: Theory, algorithms and applications, Acta Mathematica, 22 (1997), 287-367. [12] B. Scholkopf, A. J. Smola, R. C. Williamson and P. L. Bartlett, New support vector algorithms, Neural Computation, 12 (2000), 1207-1245. [13] X. Xiao, J. Gu, L. Zhang and S. Zhang, A sequential convex program method to DC program with joint chance constraints, Journal of Industrial and Management Optimization, 8 (2012), 733-747.  doi: 10.3934/jimo.2012.8.733. [14] L. M. Yang and L. SH. Wang, A class of smooth semi-supervised SVM by difference of convex functions programming and algorithm, Knowledge-Based Systems, 41 (2013), 1-7. [15] YALMIP Toolbox. http://control.ee.ethz.ch/~joloef/wiki/pmwiki.php. [16] Y. B. Yuan, Canonical duality solution for alternating support vector machine, Journal of Industrial and Management Optimization, 8 (2012), 611-621.  doi: 10.3934/jimo.2012.8.611. [17] V. N. Vapnik, Statistical Learning Theory, New York: Wiley. 1998.
ACC versus µ=1 for ν=1 and 10 on Thyroid data
Training samples of the synthetic data
Comparison of DCA-S3MAMC and MIQP-S3MAMC in terms of CPU-time
Comparison of DCA-S3MAMC and MIQP-S3MAMC in terms of ACC
Comparison of S$^3$MAMC, MAMC and $\nu$-SVC with the ratio of labelled to unlabelled samples being 2:8 in terms of generalization
 data Classification models G-ACC (%) ACC (%) MCC (%) $F_1$-measure (%) DCA-S$^3$MAMC 100 100 100 100 MIQP-S$^3$MAMC 100 100 100 100 Wine MAMC 94.31 94.39 89.06 94.16 $(107 \times 13)$ $\nu$-SVM 99.30 99.30 98.62 99.31 DCA-S$^3$MAMC 99.81 99.81 99.63 99.81 MIQP-S$^3$MAMC 95.55 95.65 91.64 95.45 Tryroid MAMC 94.87 95.00 90.45 95.24 $(65 \times 5)$ $\nu$-SVM 87.94 88.34 77.76 89.23 DCA-S$^3$MAMC 92.69 93.76 86.73 93.49 MIQP-S$^3$MAMC 96.35 96.35 91.20 96.16 Cancer MAMC 92.26 93.36 86.03 93.02 $(569 \times 30)$ $\nu$-SVM 91.27 91.46 85.30 90.93 DCA-S$^3$MAMC 62.40 62.40 24.81 62.65 MIQP-S$^3$MAMC 62.42 63.02 26.44 65.98 Sonar MAMC 60.11 60.41 20.97 62.67 $(208 \times 60)$ $\nu$-SVM 60.11 60.19 20.40 58.99 DCA-S$^3$MAMC 86.65 87.61 75.00 87.98 MIQP-S$^3$MAMC 85.34 86.69 75.91 88.20 Ionosphere MAMC 80.94 81.25 63.14 82.49 $(350 \times 34)$ $\nu$-SVM 86.34 86.74 74.51 87.76 DCA-S$^3$MAMC 72.82 73.66 48.51 76.28 MIQP-S$^3$MAMC 78.26 78.65 58.00 80.19 Hepatitis MAMC 70.60 71.76 45.04 74.98 $(155 \times 19)$ $\nu$-SVM 70.00 71.21 43.93 74.53 DCA-S$^3$MAMC 86.39 86.40 72.80 86.54 MIQP-S$^3$MAMC 84.72 84.79 69.76 85.31 Heart MAMC 81.82 81.86 63.83 82.35 $(155 \times 19)$ $\nu$-SVM 84.70 84.72 69.46 84.95 DCA-S$^3$MAMC 94.78 94.81 89.69 94.91 MIQP-S$^3$MAMC 95.61 95.62 91.29 95.69 Vote MAMC 90.11 90.29 81.07 90.80 $(432 \times 16)$ $\nu$-SVM 94.51 94.54 89.09 94.60 DCA-S$^3$MAMC 92.50 92.50 85.00 92.54 MIQP-S$^3$MAMC 93.25 93.25 86.50 93.23 Synthesis MAMC 86.45 86.49 73.07 86.82 $(200 \times 2)$ $\nu$-SVM 82.50 82.50 65.00 82.41
 data Classification models G-ACC (%) ACC (%) MCC (%) $F_1$-measure (%) DCA-S$^3$MAMC 100 100 100 100 MIQP-S$^3$MAMC 100 100 100 100 Wine MAMC 94.31 94.39 89.06 94.16 $(107 \times 13)$ $\nu$-SVM 99.30 99.30 98.62 99.31 DCA-S$^3$MAMC 99.81 99.81 99.63 99.81 MIQP-S$^3$MAMC 95.55 95.65 91.64 95.45 Tryroid MAMC 94.87 95.00 90.45 95.24 $(65 \times 5)$ $\nu$-SVM 87.94 88.34 77.76 89.23 DCA-S$^3$MAMC 92.69 93.76 86.73 93.49 MIQP-S$^3$MAMC 96.35 96.35 91.20 96.16 Cancer MAMC 92.26 93.36 86.03 93.02 $(569 \times 30)$ $\nu$-SVM 91.27 91.46 85.30 90.93 DCA-S$^3$MAMC 62.40 62.40 24.81 62.65 MIQP-S$^3$MAMC 62.42 63.02 26.44 65.98 Sonar MAMC 60.11 60.41 20.97 62.67 $(208 \times 60)$ $\nu$-SVM 60.11 60.19 20.40 58.99 DCA-S$^3$MAMC 86.65 87.61 75.00 87.98 MIQP-S$^3$MAMC 85.34 86.69 75.91 88.20 Ionosphere MAMC 80.94 81.25 63.14 82.49 $(350 \times 34)$ $\nu$-SVM 86.34 86.74 74.51 87.76 DCA-S$^3$MAMC 72.82 73.66 48.51 76.28 MIQP-S$^3$MAMC 78.26 78.65 58.00 80.19 Hepatitis MAMC 70.60 71.76 45.04 74.98 $(155 \times 19)$ $\nu$-SVM 70.00 71.21 43.93 74.53 DCA-S$^3$MAMC 86.39 86.40 72.80 86.54 MIQP-S$^3$MAMC 84.72 84.79 69.76 85.31 Heart MAMC 81.82 81.86 63.83 82.35 $(155 \times 19)$ $\nu$-SVM 84.70 84.72 69.46 84.95 DCA-S$^3$MAMC 94.78 94.81 89.69 94.91 MIQP-S$^3$MAMC 95.61 95.62 91.29 95.69 Vote MAMC 90.11 90.29 81.07 90.80 $(432 \times 16)$ $\nu$-SVM 94.51 94.54 89.09 94.60 DCA-S$^3$MAMC 92.50 92.50 85.00 92.54 MIQP-S$^3$MAMC 93.25 93.25 86.50 93.23 Synthesis MAMC 86.45 86.49 73.07 86.82 $(200 \times 2)$ $\nu$-SVM 82.50 82.50 65.00 82.41
Comparison of S$^3$MAMC, MAMC and $\nu$-SVM with the ratio of labelled to unlabelled samples being 1:9 in terms of accuracy (ACC)
 models DCA-S$^3$MAMC $(\%)$ MAMC $(\%)$ $\nu$-SVM $(\%)$ Tryroid 92.59 85.19 86.29 Ionosphere 83.37 71.43 71.74 Sonar 60.45 55.56 53.89 Cancer 93.15 89.88 84.52 Heart 85.19 74.31 75.49 Hepatitis 73.33 64.44 71.11 Vote 93.65 86.95 89.68 Synthesis 91.56 70.39 63.66
 models DCA-S$^3$MAMC $(\%)$ MAMC $(\%)$ $\nu$-SVM $(\%)$ Tryroid 92.59 85.19 86.29 Ionosphere 83.37 71.43 71.74 Sonar 60.45 55.56 53.89 Cancer 93.15 89.88 84.52 Heart 85.19 74.31 75.49 Hepatitis 73.33 64.44 71.11 Vote 93.65 86.95 89.68 Synthesis 91.56 70.39 63.66
Comparisons of the S$^3$MAMC with other semi-supervised learning methods by accuracy (ACC)
 models MIQP-S$^3$MAMC $(\%)$ DCA-S$^3$MAMC $(\%)$ MILP-S$^3$VM $(\%)$ VS$^3$VM $(\%)$ Ionosphere 86.69 87.61 89.40 87.36 Sonar 63.02 62.40 78.10 66.12 Cancer 96.35 93.76 96.60 97.46 Heart 84.79 86.40 84.00 84.70 Hepatitis 78.65 73.66 70.36 65.13 Synthesis 93.25 92.50 81.11 85.67
 models MIQP-S$^3$MAMC $(\%)$ DCA-S$^3$MAMC $(\%)$ MILP-S$^3$VM $(\%)$ VS$^3$VM $(\%)$ Ionosphere 86.69 87.61 89.40 87.36 Sonar 63.02 62.40 78.10 66.12 Cancer 96.35 93.76 96.60 97.46 Heart 84.79 86.40 84.00 84.70 Hepatitis 78.65 73.66 70.36 65.13 Synthesis 93.25 92.50 81.11 85.67
 [1] Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial and Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529 [2] Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial and Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611 [3] Hao Li, Honglin Chen, Matt Haberland, Andrea L. Bertozzi, P. Jeffrey Brantingham. PDEs on graphs for semi-supervised learning applied to first-person activity recognition in body-worn video. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4351-4373. doi: 10.3934/dcds.2021039 [4] Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018 [5] Jian Luo, Shu-Cherng Fang, Yanqin Bai, Zhibin Deng. Fuzzy quadratic surface support vector machine based on fisher discriminant analysis. Journal of Industrial and Management Optimization, 2016, 12 (1) : 357-373. doi: 10.3934/jimo.2016.12.357 [6] Xin Li, Ziguan Cui, Linhui Sun, Guanming Lu, Debnath Narayan. Research on iterative repair algorithm of Hyperchaotic image based on support vector machine. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1199-1218. doi: 10.3934/dcdss.2019083 [7] Fatemeh Bazikar, Saeed Ketabchi, Hossein Moosaei. Smooth augmented Lagrangian method for twin bounded support vector machine. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021027 [8] Xin Yan, Hongmiao Zhu. A kernel-free fuzzy support vector machine with Universum. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021184 [9] Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1835-1861. doi: 10.3934/jimo.2021046 [10] Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027 [11] Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial and Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557 [12] Robert L. Peach, Alexis Arnaudon, Mauricio Barahona. Semi-supervised classification on graphs using explicit diffusion dynamics. Foundations of Data Science, 2020, 2 (1) : 19-33. doi: 10.3934/fods.2020002 [13] Radu Ioan Boţ, Anca Grad, Gert Wanka. Sequential characterization of solutions in convex composite programming and applications to vector optimization. Journal of Industrial and Management Optimization, 2008, 4 (4) : 767-782. doi: 10.3934/jimo.2008.4.767 [14] Ning Lu, Ying Liu. Application of support vector machine model in wind power prediction based on particle swarm optimization. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1267-1276. doi: 10.3934/dcdss.2015.8.1267 [15] Huiqin Zhang, JinChun Wang, Meng Wang, Xudong Chen. Integration of cuckoo search and fuzzy support vector machine for intelligent diagnosis of production process quality. Journal of Industrial and Management Optimization, 2022, 18 (1) : 195-217. doi: 10.3934/jimo.2020150 [16] Qianru Zhai, Ye Tian, Jingyue Zhou. A SMOTE-based quadratic surface support vector machine for imbalanced classification with mislabeled information. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2021230 [17] René Henrion, Christian Küchler, Werner Römisch. Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 363-384. doi: 10.3934/jimo.2008.4.363 [18] Louis Caccetta, Syarifah Z. Nordin. Mixed integer programming model for scheduling in unrelated parallel processor system with priority consideration. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 115-132. doi: 10.3934/naco.2014.4.115 [19] Elham Mardaneh, Ryan Loxton, Qun Lin, Phil Schmidli. A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1601-1623. doi: 10.3934/jimo.2017009 [20] Edward S. Canepa, Alexandre M. Bayen, Christian G. Claudel. Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming. Networks and Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783

2020 Impact Factor: 1.801