# 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 & Management Optimization, 2017, 13 (2) : 609-622. doi: 10.3934/jimo.2016035
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##### References:
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
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