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doi: 10.3934/jimo.2021046
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Quadratic surface support vector machine with L1 norm regularization

Ahmad Mousavi 1,, , Zheming Gao 2, , Lanshan Han 3, and  Alvin Lim 3,

1. 

Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA
2. 

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110819, China
3. 

Precima, a NielsenIQ Company, Chicago, IL 60606, USA

* Corresponding author: Ahmad Mousavi

Received  May 2020 Revised  November 2020 Early access March 2021

Fund Project: This research was done in the Research and Development Department of Precima, which fully supported the work. The first two authors contributed equally

We propose $ \ell_1 $ norm regularized quadratic surface support vector machine models for binary classification in supervised learning. We establish some desired theoretical properties, including the existence and uniqueness of the optimal solution, reduction to the standard SVMs over (almost) linearly separable data sets, and detection of true sparsity pattern over (almost) quadratically separable data sets if the penalty parameter on the $ \ell_1 $ norm is large enough. We also demonstrate their promising practical efficiency by conducting various numerical experiments on both synthetic and publicly available benchmark data sets.

Keywords: Binary classification, support vector machines, quadratic support vector machines, L1 norm regularization.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021046
References:
[1]

Arthur Asuncion and David Newman, UCI Machine Learning Repository, 2007., Google Scholar

[2]

Y. Bai, X. Han, T. Chen and H. Yu, Quadratic kernel-free least squares support vector machine for target diseases classification, Journal of Combinatorial Optimization, 30 (2015), 850-870.  doi: 10.1007/s10878-015-9848-z.  Google Scholar

[3]

D. P. Bertsekas, Nonlinear programming, Journal of the Operational Research Society, 48 (1997), 334-334.   Google Scholar

[4]

J. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer Science & Business Media, 2010. doi: 10.1007/978-0-387-31256-9.  Google Scholar

[5]

C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, 20 (1995), 273-297.  doi: 10.1007/BF00994018.  Google Scholar

[6] N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods, Cambridge University Press, 2000.  doi: 10.1017/CBO9780511801389.  Google Scholar
[7]

I. Dagher, Quadratic kernel-free non-linear support vector machine, Journal of Global Optimization, 41 (2008), 15-30.  doi: 10.1007/s10898-007-9162-0.  Google Scholar

[8]

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, Journal of Industrial and Management Optimization, 14 (2018), 1651-1666.  doi: 10.3934/jimo.2018025.  Google Scholar

[9]

N. Deng, Y. Tian and C. Zhang, Support Vector Machines: Optimization Based Theory, Algorithms, and Extensions, Chapman and Hall/CRC, 2012.  Google Scholar

[10]

M. Di and E. M. Joo, A survey of machine learning in wireless sensor networks from networking and application perspectives, 2007 6th International Conference on Information, Communications & Signal Processing, IEEE, (2007), 1-5. Google Scholar

[11]

J. Gallier, Schur complements and applications, Geometric Methods and Applications, Springer, (2011), 431-437. doi: 10.1007/978-1-4419-9961-0.  Google Scholar

[12]

Z. Gao, S.-C. Fang, J. Luo and and N. Medhin, A Kernel-Free Double Well Potential Support Vector Machine with Applications, European Journal of Operational Research, 290 (2021), 248-262.  doi: 10.1016/j.ejor.2020.10.040.  Google Scholar

[13]

Z. Gao and G. Petrova, Rescaled pure greedy algorithm for convex optimization, Calcolo, 56 (2019), 15. doi: 10.1007/s10092-019-0311-x.  Google Scholar

[14]

B. Ghaddar and J. Naoum-Sawaya, High dimensional data classification and feature selection using support vector machines, European Journal of Operational Research, 265 (2018), 993-1004.  doi: 10.1016/j.ejor.2017.08.040.  Google Scholar

[15]

Y. Hao and F. Meng, A new method on gene selection for tissue classification, Journal of Industrial and Management Optimization, 3 (2007), 739. doi: 10.3934/jimo. 2007.3.739.  Google Scholar

[16]

T. K. Ho and M. Basu, Complexity measures of supervised classification problems, IEEE Transactions on Pattern Analysis & Machine Intelligence, (2002), 289-300. Google Scholar

[17]

D. S. Kim, N. N. Tam and and N. D. Yen, Solution existence and stability of quadratically constrained convex quadratic programs, Optimization Letters, 6 (2012), 363-373.  doi: 10.1007/s11590-011-0300-8.  Google Scholar

[18]

P. Langley and H. A. Simon, Applications of machine learning and rule induction, Communications of the ACM, 38 (1995), 54-64.  doi: 10.21236/ADA292607.  Google Scholar

[19]

K. Lounici, M. Pontil, A. B. Tsybakov and S. Van De Geer, Taking Advantage of Sparsity in Multi-Task Learning, arXiv preprint, arXiv: 0903.1468, 2009. Google Scholar

[20]

J. Luo, S.-C. Fang, Y. Bai and Z. Deng, Fuzzy quadratic surface support vector machine based on Fisher discriminant analysis, Journal of Industrial and Management Optimization, 12 (2016), 357-373.  doi: 10.3934/jimo.2016.12.357.  Google Scholar

[21]

J. Luo, S. -C. Fang, Z. Deng and X. Guo, Soft quadratic surface support vector machine for binary classification, Asia-Pacific Journal of Operational Research, 33 (2016), 1650046. doi: 10.1142/S0217595916500469.  Google Scholar

[22]

J. Luo, T. Hong and S.-C. Fang, Benchmarking robustness of load forecasting models under data integrity attacks, International Journal of Forecasting, 34 (2018), 89-104.  doi: 10.1016/j.ijforecast.2017.08.004.  Google Scholar

[23]

J. R. Magnus and H. Neudecker, The elimination matrix: Some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1 (1980), 422-449.  doi: 10.1137/0601049.  Google Scholar

[24]

O. L. Mangasarian, Uniqueness of solution in linear programming, Linear Algebra and its Applications, 25 (1979), 151-162.  doi: 10.1016/0024-3795(79)90014-4.  Google Scholar

[25]

L. Monostori, A. Márkus, H. Van Brussel and E. Westkämpfer, Machine learning approaches to manufacturing, CIRP Annals, 45 (1996), 675-712.  doi: 10.1016/S0007-8506(L1-QSSVM")30216-6.  Google Scholar

[26]

A. Mousavi, M. Rezaee and R. Ayanzadeh, A survey on compressive sensing: classical results and recent advancements, Journal of Mathematical Modeling, 8 (2020), 309-344.   Google Scholar

[27]

S. Mousavi and J. Shen, Solution uniqueness of convex piecewise affine functions based optimization with applications to constrained $\ell_1$ minimization, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), 56. doi: 10.1051/cocv/2018061.  Google Scholar

[28]

F. Pedregosa, Scikit-learn: Machine learning in Python, Journal of Machine Learning Research, 12 (2011), 2825-2830.   Google Scholar

[29]

H. Qiu, X. Chen, W. Liu, G. Zhou, Y. Wang and J. Lai, A fast $\ell_1$-solver and its applications to robust face recognition, Journal of Industrial and Management Optimization, 8 (2012), 163-178.  doi: 10.3934/jimo.2012.8.163.  Google Scholar

[30]

R. Saab, R. Chartrand and O. Yilmaz, Stable sparse approximations via nonconvex optimization, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 3885-3888. doi: 10.1109/ICASSP. 2008.4518502.  Google Scholar

[31] B. Scholkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT press, 2001.  doi: 10.7551/mitpress/4175.001.0001.  Google Scholar
[32]

J. Shen and S. Mousavi, Least sparsity of $p$-norm based optimization problems with $p>1$, SIAM Journal on Optimization, 28 (2018), 2721-2751.  doi: 10.1137/17M1140066.  Google Scholar

[33]

J. Shen and S. Mousavi, Exact Support and Vector Recovery of Constrained Sparse VVectors via Constrained Matching Pursuit, arXiv preprint, arXiv: 1903.07236, 2019. Google Scholar

[34]

C. Zhang, J. Wang and N. Xiu, Robust and sparse portfolio model for index tracking, Journal of Industrial and Management Optimization, 15 (2019), 1001-1015.  doi: 10.3934/jimo.  Google Scholar

show all references

References:
[1]

Arthur Asuncion and David Newman, UCI Machine Learning Repository, 2007., Google Scholar

[2]

Y. Bai, X. Han, T. Chen and H. Yu, Quadratic kernel-free least squares support vector machine for target diseases classification, Journal of Combinatorial Optimization, 30 (2015), 850-870.  doi: 10.1007/s10878-015-9848-z.  Google Scholar

[3]

D. P. Bertsekas, Nonlinear programming, Journal of the Operational Research Society, 48 (1997), 334-334.   Google Scholar

[4]

J. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer Science & Business Media, 2010. doi: 10.1007/978-0-387-31256-9.  Google Scholar

[5]

C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, 20 (1995), 273-297.  doi: 10.1007/BF00994018.  Google Scholar

[6] N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods, Cambridge University Press, 2000.  doi: 10.1017/CBO9780511801389.  Google Scholar
[7]

I. Dagher, Quadratic kernel-free non-linear support vector machine, Journal of Global Optimization, 41 (2008), 15-30.  doi: 10.1007/s10898-007-9162-0.  Google Scholar

[8]

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, Journal of Industrial and Management Optimization, 14 (2018), 1651-1666.  doi: 10.3934/jimo.2018025.  Google Scholar

[9]

N. Deng, Y. Tian and C. Zhang, Support Vector Machines: Optimization Based Theory, Algorithms, and Extensions, Chapman and Hall/CRC, 2012.  Google Scholar

[10]

M. Di and E. M. Joo, A survey of machine learning in wireless sensor networks from networking and application perspectives, 2007 6th International Conference on Information, Communications & Signal Processing, IEEE, (2007), 1-5. Google Scholar

[11]

J. Gallier, Schur complements and applications, Geometric Methods and Applications, Springer, (2011), 431-437. doi: 10.1007/978-1-4419-9961-0.  Google Scholar

[12]

Z. Gao, S.-C. Fang, J. Luo and and N. Medhin, A Kernel-Free Double Well Potential Support Vector Machine with Applications, European Journal of Operational Research, 290 (2021), 248-262.  doi: 10.1016/j.ejor.2020.10.040.  Google Scholar

[13]

Z. Gao and G. Petrova, Rescaled pure greedy algorithm for convex optimization, Calcolo, 56 (2019), 15. doi: 10.1007/s10092-019-0311-x.  Google Scholar

[14]

B. Ghaddar and J. Naoum-Sawaya, High dimensional data classification and feature selection using support vector machines, European Journal of Operational Research, 265 (2018), 993-1004.  doi: 10.1016/j.ejor.2017.08.040.  Google Scholar

[15]

Y. Hao and F. Meng, A new method on gene selection for tissue classification, Journal of Industrial and Management Optimization, 3 (2007), 739. doi: 10.3934/jimo. 2007.3.739.  Google Scholar

[16]

T. K. Ho and M. Basu, Complexity measures of supervised classification problems, IEEE Transactions on Pattern Analysis & Machine Intelligence, (2002), 289-300. Google Scholar

[17]

D. S. Kim, N. N. Tam and and N. D. Yen, Solution existence and stability of quadratically constrained convex quadratic programs, Optimization Letters, 6 (2012), 363-373.  doi: 10.1007/s11590-011-0300-8.  Google Scholar

[18]

P. Langley and H. A. Simon, Applications of machine learning and rule induction, Communications of the ACM, 38 (1995), 54-64.  doi: 10.21236/ADA292607.  Google Scholar

[19]

K. Lounici, M. Pontil, A. B. Tsybakov and S. Van De Geer, Taking Advantage of Sparsity in Multi-Task Learning, arXiv preprint, arXiv: 0903.1468, 2009. Google Scholar

[20]

J. Luo, S.-C. Fang, Y. Bai and Z. Deng, Fuzzy quadratic surface support vector machine based on Fisher discriminant analysis, Journal of Industrial and Management Optimization, 12 (2016), 357-373.  doi: 10.3934/jimo.2016.12.357.  Google Scholar

[21]

J. Luo, S. -C. Fang, Z. Deng and X. Guo, Soft quadratic surface support vector machine for binary classification, Asia-Pacific Journal of Operational Research, 33 (2016), 1650046. doi: 10.1142/S0217595916500469.  Google Scholar

[22]

J. Luo, T. Hong and S.-C. Fang, Benchmarking robustness of load forecasting models under data integrity attacks, International Journal of Forecasting, 34 (2018), 89-104.  doi: 10.1016/j.ijforecast.2017.08.004.  Google Scholar

[23]

J. R. Magnus and H. Neudecker, The elimination matrix: Some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1 (1980), 422-449.  doi: 10.1137/0601049.  Google Scholar

[24]

O. L. Mangasarian, Uniqueness of solution in linear programming, Linear Algebra and its Applications, 25 (1979), 151-162.  doi: 10.1016/0024-3795(79)90014-4.  Google Scholar

[25]

L. Monostori, A. Márkus, H. Van Brussel and E. Westkämpfer, Machine learning approaches to manufacturing, CIRP Annals, 45 (1996), 675-712.  doi: 10.1016/S0007-8506(L1-QSSVM")30216-6.  Google Scholar

[26]

A. Mousavi, M. Rezaee and R. Ayanzadeh, A survey on compressive sensing: classical results and recent advancements, Journal of Mathematical Modeling, 8 (2020), 309-344.   Google Scholar

[27]

S. Mousavi and J. Shen, Solution uniqueness of convex piecewise affine functions based optimization with applications to constrained $\ell_1$ minimization, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), 56. doi: 10.1051/cocv/2018061.  Google Scholar

[28]

F. Pedregosa, Scikit-learn: Machine learning in Python, Journal of Machine Learning Research, 12 (2011), 2825-2830.   Google Scholar

[29]

H. Qiu, X. Chen, W. Liu, G. Zhou, Y. Wang and J. Lai, A fast $\ell_1$-solver and its applications to robust face recognition, Journal of Industrial and Management Optimization, 8 (2012), 163-178.  doi: 10.3934/jimo.2012.8.163.  Google Scholar

[30]

R. Saab, R. Chartrand and O. Yilmaz, Stable sparse approximations via nonconvex optimization, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 3885-3888. doi: 10.1109/ICASSP. 2008.4518502.  Google Scholar

[31] B. Scholkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT press, 2001.  doi: 10.7551/mitpress/4175.001.0001.  Google Scholar
[32]

J. Shen and S. Mousavi, Least sparsity of $p$-norm based optimization problems with $p>1$, SIAM Journal on Optimization, 28 (2018), 2721-2751.  doi: 10.1137/17M1140066.  Google Scholar

[33]

J. Shen and S. Mousavi, Exact Support and Vector Recovery of Constrained Sparse VVectors via Constrained Matching Pursuit, arXiv preprint, arXiv: 1903.07236, 2019. Google Scholar

[34]

C. Zhang, J. Wang and N. Xiu, Robust and sparse portfolio model for index tracking, Journal of Industrial and Management Optimization, 15 (2019), 1001-1015.  doi: 10.3934/jimo.  Google Scholar

Figure 1.  L1-QSSVM performance on linearly and quadratically separable data sets
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Figure 2.  Influence of the parameter $ \lambda $ on the curvature of the optimal solution of L1-SQSSVM
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Figure 3.  Influence of the parameter $ \mu $ on the behavior of the optimal solution of L1-SQSSVM
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Figure 4.  Accuracy score against the parameter $ \lambda $
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Figure 5.  Sparsity pattern detection using L1-SQSSVM as parameter $ \lambda $ varies
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Table 1.  Basic information of artificial data sets
Data set Artificial I Artificial II Artificial III Artificial IV Artificial 3-D
$ n $ 3 3 5 10 3
Sample size ($ N_1 $/$ N_2 $) 67/58 79/71 106/81 204/171 99/101
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Data set Artificial I Artificial II Artificial III Artificial IV Artificial 3-D
$ n $ 3 3 5 10 3
Sample size ($ N_1 $/$ N_2 $) 67/58 79/71 106/81 204/171 99/101
Table 2.  Description of 2-class data sets used
Data set # of features name of class sample size
Iris 4 versicolour 50
virginica 50
Car Evaluation 6 unacc 1210
acc 384
Diabetes 8 yes 268
no 500
German Credit Data 20 creditworthy 700
non-creditworthy 300
Ionosphere 34 good 225
bad 126
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Data set # of features name of class sample size
Iris 4 versicolour 50
virginica 50
Car Evaluation 6 unacc 1210
acc 384
Diabetes 8 yes 268
no 500
German Credit Data 20 creditworthy 700
non-creditworthy 300
Ionosphere 34 good 225
bad 126
Table 3.  Iris results
Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 91.93 5.49 63.33 98.89 0.058
SQSSVM 89.33 4.07 81.11 96.67 0.050
SVM-Quad 89.49 4.91 80.00 97.78 0.003
SVM 89.62 4.10 78.89 97.78 0.001
20 L1-SQSSVM 94.33 2.20 90.00 98.75 0.063
SQSSVM 92.60 2.57 82.50 96.25 0.055
SVM-Quad 93.03 2.72 86.25 98.75 0.002
SVM 93.00 3.01 82.50 97.50 0.002
40 L1-SQSSVM 95.40 2.76 86.67 100.00 0.075
SQSSVM 93.97 3.73 78.33 100.00 0.062
SVM-Quad 94.30 3.38 81.67 98.33 0.002
SVM 94.50 3.29 85.00 100.00 0.002
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Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 91.93 5.49 63.33 98.89 0.058
SQSSVM 89.33 4.07 81.11 96.67 0.050
SVM-Quad 89.49 4.91 80.00 97.78 0.003
SVM 89.62 4.10 78.89 97.78 0.001
20 L1-SQSSVM 94.33 2.20 90.00 98.75 0.063
SQSSVM 92.60 2.57 82.50 96.25 0.055
SVM-Quad 93.03 2.72 86.25 98.75 0.002
SVM 93.00 3.01 82.50 97.50 0.002
40 L1-SQSSVM 95.40 2.76 86.67 100.00 0.075
SQSSVM 93.97 3.73 78.33 100.00 0.062
SVM-Quad 94.30 3.38 81.67 98.33 0.002
SVM 94.50 3.29 85.00 100.00 0.002
Table 4.  Car evaluation results
Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 90.48 2.13 83.48 95.05 0.961
SQSSVM 90.48 2.35 80.98 94.49 0.937
SVM-Quad 88.32 2.70 80.98 93.45 0.023
SVM 84.40 1.09 81.88 86.90 0.001
20 L1-SQSSVM 92.81 1.17 89.50 95.30 1.109
SQSSVM 92.77 1.21 89.58 95.30 1.117
SVM-Quad 92.30 1.14 88.56 94.83 0.001
SVM 85.08 0.91 83.23 86.91 0.008
40 L1-SQSSVM 95.80 0.73 93.83 97.07 1.501
SQSSVM 95.76 0.77 93.83 97.28 1.521
SVM-Quad 93.69 0.83 91.43 95.72 0.087
SVM 85.26 1.09 81.71 87.36 0.003
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Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 90.48 2.13 83.48 95.05 0.961
SQSSVM 90.48 2.35 80.98 94.49 0.937
SVM-Quad 88.32 2.70 80.98 93.45 0.023
SVM 84.40 1.09 81.88 86.90 0.001
20 L1-SQSSVM 92.81 1.17 89.50 95.30 1.109
SQSSVM 92.77 1.21 89.58 95.30 1.117
SVM-Quad 92.30 1.14 88.56 94.83 0.001
SVM 85.08 0.91 83.23 86.91 0.008
40 L1-SQSSVM 95.80 0.73 93.83 97.07 1.501
SQSSVM 95.76 0.77 93.83 97.28 1.521
SVM-Quad 93.69 0.83 91.43 95.72 0.087
SVM 85.26 1.09 81.71 87.36 0.003
Table 5.  Diabetes results
Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 74.21 1.53 71.24 76.01 0.692
SQSSVM 64.38 3.65 57.80 71.68 0.679
SVM-Quad 66.07 4.53 57.66 71.53 0.102
SVM 72.95 3.49 65.61 76.16 0.003
20 L1-SQSSVM 76.28 0.63 75.12 77.07 0.924
SQSSVM 69.40 2.49 65.85 72.52 0.950
SVM-Quad 70.28 2.30 65.85 73.82 9.080
SVM 74.86 1.68 71.54 77.07 0.009
40 L1-SQSSVM 76.62 1.83 73.97 79.61 1.459
SQSSVM 74.34 1.99 71.15 77.01 1.490
SVM-Quad 75.21 1.23 73.54 77.22 86.561
SVM 76.29 2.15 73.10 80.26 0.006
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Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 74.21 1.53 71.24 76.01 0.692
SQSSVM 64.38 3.65 57.80 71.68 0.679
SVM-Quad 66.07 4.53 57.66 71.53 0.102
SVM 72.95 3.49 65.61 76.16 0.003
20 L1-SQSSVM 76.28 0.63 75.12 77.07 0.924
SQSSVM 69.40 2.49 65.85 72.52 0.950
SVM-Quad 70.28 2.30 65.85 73.82 9.080
SVM 74.86 1.68 71.54 77.07 0.009
40 L1-SQSSVM 76.62 1.83 73.97 79.61 1.459
SQSSVM 74.34 1.99 71.15 77.01 1.490
SVM-Quad 75.21 1.23 73.54 77.22 86.561
SVM 76.29 2.15 73.10 80.26 0.006
Table 6.  German Credit Data results
Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 71.86 1.85 68.44 75.00 1.596
SQSSVM 67.00 3.02 63.67 71.67 1.598
SVM-Quad 68.29 2.61 64.00 72.44 0.006
SVM 69.49 3.58 61.89 74.33 0.002
20 L1-SQSSVM 73.88 1.29 71.38 75.88 2.572
SQSSVM 67.55 2.78 62.88 72.88 2.541
SVM-Quad 67.78 2.75 64.13 72.13 0.005
SVM 73.86 1.22 71.25 75.88 0.005
40 L1-SQSSVM 74.86 1.25 72.00 77.00 4.622
SQSSVM 65.99 2.66 61.17 69.83 4.456
SVM-Quad 65.13 1.19 63.50 67.00 0.262
SVM 74.73 1.07 73.50 77.00 0.005
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Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 71.86 1.85 68.44 75.00 1.596
SQSSVM 67.00 3.02 63.67 71.67 1.598
SVM-Quad 68.29 2.61 64.00 72.44 0.006
SVM 69.49 3.58 61.89 74.33 0.002
20 L1-SQSSVM 73.88 1.29 71.38 75.88 2.572
SQSSVM 67.55 2.78 62.88 72.88 2.541
SVM-Quad 67.78 2.75 64.13 72.13 0.005
SVM 73.86 1.22 71.25 75.88 0.005
40 L1-SQSSVM 74.86 1.25 72.00 77.00 4.622
SQSSVM 65.99 2.66 61.17 69.83 4.456
SVM-Quad 65.13 1.19 63.50 67.00 0.262
SVM 74.73 1.07 73.50 77.00 0.005
Table 7.  Ionosphere results
Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 82.75 3.69 76.27 88.29 4.141
SQSSVM 79.24 3.15 74.37 83.86 3.945
SVM-Quad 83.48 2.39 78.48 78.48 0.003
SVM 80.09 2.24 75.95 82.28 0.006
20 L1-SQSSVM 87.90 3.72 80.07 92.53 5.096
SQSSVM 87.19 4.32 77.94 91.81 4.854
SVM-Quad 86.16 1.24 84.34 84.34 0.005
SVM 82.03 5.40 67.97 86.83 0.002
40 L1-SQSSVM 90.28 3.33 83.41 94.31 7.063
SQSSVM 89.53 4.23 81.99 94.31 6.781
SVM-Quad 86.40 3.03 81.04 91.00 0.007
SVM 83.60 3.46 76.78 88.63 0.006
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Training Rate $k$% Model Accuracy score (%) CPU time (s)
mean std min max
10 L1-SQSSVM 82.75 3.69 76.27 88.29 4.141
SQSSVM 79.24 3.15 74.37 83.86 3.945
SVM-Quad 83.48 2.39 78.48 78.48 0.003
SVM 80.09 2.24 75.95 82.28 0.006
20 L1-SQSSVM 87.90 3.72 80.07 92.53 5.096
SQSSVM 87.19 4.32 77.94 91.81 4.854
SVM-Quad 86.16 1.24 84.34 84.34 0.005
SVM 82.03 5.40 67.97 86.83 0.002
40 L1-SQSSVM 90.28 3.33 83.41 94.31 7.063
SQSSVM 89.53 4.23 81.99 94.31 6.781
SVM-Quad 86.40 3.03 81.04 91.00 0.007
SVM 83.60 3.46 76.78 88.63 0.006
Table 8.  Summary of obtained theoretical results in this paper
L1-QSSVM L1-SQSSVM
Linearly
Separable		 ● Solution existence
● z* is almost always unique
● Equivalence with SVM for large enough $\lambda$		 ● Solution existence
● z* is almost always unique
● Equivalence with SSVM for large enough $\lambda$
● Solution is almost always unique with $\xi^*=\boldsymbol 0$ for large enough $\mu$
Quadratically
Separable		 ● Solution existence
● z* is almost always unique
● Capturing possible sparsity of $\textbf W^*$ for large enough $\lambda$		 ● Solution existence
● z* is almost always unique
● Solution is almost always unique with $\boldsymbol \xi^*=\boldsymbol 0$ for large enough $\mu$
● Capturing possible sparsity of $\textbf W^*$ for large enough $\lambda$
$ \begin{array}{c} \rm{Neither} \end{array} $ $ \begin{array}{l} \bullet \rm{ Solution existence}\\ \bullet \boldsymbol z^* \rm{ is almost always unique} \end{array} $
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L1-QSSVM L1-SQSSVM
Linearly
Separable		 ● Solution existence
● z* is almost always unique
● Equivalence with SVM for large enough $\lambda$		 ● Solution existence
● z* is almost always unique
● Equivalence with SSVM for large enough $\lambda$
● Solution is almost always unique with $\xi^*=\boldsymbol 0$ for large enough $\mu$
Quadratically
Separable		 ● Solution existence
● z* is almost always unique
● Capturing possible sparsity of $\textbf W^*$ for large enough $\lambda$		 ● Solution existence
● z* is almost always unique
● Solution is almost always unique with $\boldsymbol \xi^*=\boldsymbol 0$ for large enough $\mu$
● Capturing possible sparsity of $\textbf W^*$ for large enough $\lambda$
$ \begin{array}{c} \rm{Neither} \end{array} $ $ \begin{array}{l} \bullet \rm{ Solution existence}\\ \bullet \boldsymbol z^* \rm{ is almost always unique} \end{array} $
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