
-
Previous Article
Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations
- JIMO Home
- This Issue
-
Next Article
The skewness for uncertain random variable and application to portfolio selection problem
Quadratic surface support vector machine with L1 norm regularization
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 |
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.
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. |
[3] |
D. P. Bertsekas,
Nonlinear programming, Journal of the Operational Research Society, 48 (1997), 334-334.
|
[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. |
[5] |
C. Cortes and V. Vapnik,
Support-vector networks, Machine Learning, 20 (1995), 273-297.
doi: 10.1007/BF00994018. |
[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.![]() |
[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. |
[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. |
[9] |
N. Deng, Y. Tian and C. Zhang, Support Vector Machines: Optimization Based Theory, Algorithms, and Extensions, Chapman and Hall/CRC, 2012. |
[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. |
[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. |
[13] |
Z. Gao and G. Petrova, Rescaled pure greedy algorithm for convex optimization, Calcolo, 56 (2019), 15.
doi: 10.1007/s10092-019-0311-x. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[28] |
F. Pedregosa,
Scikit-learn: Machine learning in Python, Journal of Machine Learning Research, 12 (2011), 2825-2830.
|
[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. |
[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. |
[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.![]() |
[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. |
[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. |
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. |
[3] |
D. P. Bertsekas,
Nonlinear programming, Journal of the Operational Research Society, 48 (1997), 334-334.
|
[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. |
[5] |
C. Cortes and V. Vapnik,
Support-vector networks, Machine Learning, 20 (1995), 273-297.
doi: 10.1007/BF00994018. |
[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.![]() |
[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. |
[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. |
[9] |
N. Deng, Y. Tian and C. Zhang, Support Vector Machines: Optimization Based Theory, Algorithms, and Extensions, Chapman and Hall/CRC, 2012. |
[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. |
[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. |
[13] |
Z. Gao and G. Petrova, Rescaled pure greedy algorithm for convex optimization, Calcolo, 56 (2019), 15.
doi: 10.1007/s10092-019-0311-x. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[28] |
F. Pedregosa,
Scikit-learn: Machine learning in Python, Journal of Machine Learning Research, 12 (2011), 2825-2830.
|
[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. |
[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. |
[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.![]() |
[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. |
[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. |





Data set | Artificial I | Artificial II | Artificial III | Artificial IV | Artificial 3-D |
3 | 3 | 5 | 10 | 3 | |
Sample size ( |
67/58 | 79/71 | 106/81 | 204/171 | 99/101 |
Data set | Artificial I | Artificial II | Artificial III | Artificial IV | Artificial 3-D |
3 | 3 | 5 | 10 | 3 | |
Sample size ( |
67/58 | 79/71 | 106/81 | 204/171 | 99/101 |
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 |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
Training Rate |
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 |
![]() |
L1-QSSVM | L1-SQSSVM |
Linearly Separable |
● Solution existence ● z* is almost always unique ● Equivalence with SVM for large enough |
● Solution existence ● z* is almost always unique ● Equivalence with SSVM for large enough ● Solution is almost always unique with |
Quadratically Separable |
● Solution existence ● z* is almost always unique ● Capturing possible sparsity of |
● Solution existence ● z* is almost always unique ● Solution is almost always unique with ● Capturing possible sparsity of |
![]() |
L1-QSSVM | L1-SQSSVM |
Linearly Separable |
● Solution existence ● z* is almost always unique ● Equivalence with SVM for large enough |
● Solution existence ● z* is almost always unique ● Equivalence with SSVM for large enough ● Solution is almost always unique with |
Quadratically Separable |
● Solution existence ● z* is almost always unique ● Capturing possible sparsity of |
● Solution existence ● z* is almost always unique ● Solution is almost always unique with ● Capturing possible sparsity of |
[1] |
Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 |
[2] |
Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 |
[3] |
Jiangang Qi, Bing Xie. Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243 |
[4] |
Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021063 |
[5] |
Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011 |
[6] |
A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 |
[7] |
Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047 |
[8] |
Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220 |
[9] |
Qing Liu, Bingo Wing-Kuen Ling, Qingyun Dai, Qing Miao, Caixia Liu. Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1993-2011. doi: 10.3934/jimo.2020055 |
[10] |
Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374 |
[11] |
Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389 |
[12] |
Andreas Neubauer. On Tikhonov-type regularization with approximated penalty terms. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021027 |
[13] |
F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605 |
[14] |
Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019 |
[15] |
Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021008 |
[16] |
Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, 2021, 15 (3) : 415-443. doi: 10.3934/ipi.2020074 |
[17] |
Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027 |
[18] |
Linyao Ge, Baoxiang Huang, Weibo Wei, Zhenkuan Pan. Semi-Supervised classification of hyperspectral images using discrete nonlocal variation Potts Model. Mathematical Foundations of Computing, 2021 doi: 10.3934/mfc.2021003 |
[19] |
Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 |
[20] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]