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Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels
1. | Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, China |
2. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001 |
References:
[1] |
J. P. Aubin and A. Cellina, Differential Inclusions,, Springer-Verlag, (1984).
doi: 10.1007/978-3-642-69512-4. |
[2] |
J. P. Aubin and H. Frankowska, Set-Valued Analysis,, MA: Birkauser, (1990).
|
[3] |
E. K. P. Chong, S. Hui and S. H. Zak, An analysis of a class of neural networks for solving linear programming problems,, Automatic Control, 44 (1999), 1995.
doi: 10.1109/9.802909. |
[4] |
J. Chen and J. Ye, Training SVM with indefinite kernels,, In Proceedings of the 25th International Conference on Machine Learning, (2008), 136.
doi: 10.1145/1390156.1390174. |
[5] |
F. H. Clarke, Optimization and Non-smooth Analysis,, Wiley, (1969). Google Scholar |
[6] |
L. Fengqiu and X. Xiaoping, Design of natural classification kernels using prior knowledge,, Fuzzy Systems, 20 (2012), 135. Google Scholar |
[7] |
A. F. Filippove, Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications (Soviet Series),, Boston, (1988). Google Scholar |
[8] |
M. Forti, P. Nistri and M. Quincampoix, Convergence of neural networks for programming problems via a nonsmooth Łojasiewicz inequality,, Neural Networks, 17 (2006), 1471. Google Scholar |
[9] |
M. Forti and A. Tesi, Absolute stability of analytic neural networks: An approach based on finite trajectory length,, Circuits System I, 51 (2004), 2460.
doi: 10.1109/TCSI.2004.838143. |
[10] |
B. Haasdonk, Feature space interpretation of svms with indefinite kernels,, Pattern Analysis and Machine Intelligence, 27 (2005), 482.
doi: 10.1109/TPAMI.2005.78. |
[11] |
M. Hein and O. Bousquet, Hilbertian metrics and positive definite kernels on probability measures,, In Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics, (2005), 136. Google Scholar |
[12] |
J. J. Hopfield and D. W. Tank, "Neural" computation of decisions in optimization problems,, Biological cybernetics, 52 (1985), 141.
|
[13] |
R. Luss and A. d'Aspremon, Support vector machine classification with indefinite kernels,, Mathematical Programming Computation, 1 (2009), 97.
doi: 10.1007/s12532-009-0005-5. |
[14] |
I. Mierswa, Making Indefinite Kernel Learning Practical,, Technical report, (2006). Google Scholar |
[15] |
J. C. Platt, Fast training of support vector machines using sequential minimal optimization,, In Advances in Kernel Methods, (1999), 185. Google Scholar |
[16] |
T. Rockafellar and R. Wets, Variational Analysis,, Springer-Verlag, (1998).
doi: 10.1007/978-3-642-02431-3. |
[17] |
S. Shalev-Shwartz, Y. Singer, N. Srebro and A. Cotter, Pegasos: Primal estimated sub-gradient solver for SVM,, Math. Program., 127 (2011), 3.
doi: 10.1007/s10107-010-0420-4. |
[18] |
H. Saigo, J. P. Vert, N. Ueda and T. Akutsu, Protein homology detection using string alignment kernels,, Bioinformatics, 20 (2004), 1682.
doi: 10.1093/bioinformatics/bth141. |
[19] |
B. Schölkopf and A. J. Smola, Learning with Kernels,, MIT, (2002). Google Scholar |
[20] |
B. Schölkopf, P. Simard, A. J. Smola and V. N. Vapnik, Prior Knowledge in Support Vector Kernels,, MIT, (1998). Google Scholar |
[21] |
V. N. Vapnik, Statistical Learning Theory,, Wiley, (1998).
|
[22] |
B. Wei and X. Xiaoping, Subgradient-based neural networks for nonsmooth nonconvex optimization problems,, Circuits and Systems I: Regular Papers, 55 (2008), 2378.
doi: 10.1109/TCSI.2008.920131. |
[23] |
Y. Xia and J. Wang, A one-layer recurrent neural network for support vector machine learning,, Systems, 34 (2004), 1621.
doi: 10.1109/TSMCB.2003.822955. |
[24] |
S. Ziye and J. Qingwei, Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems,, Journal of Industrial and Management Optimization, 10 (2014), 871.
doi: 10.3934/jimo.2014.10.871. |
show all references
References:
[1] |
J. P. Aubin and A. Cellina, Differential Inclusions,, Springer-Verlag, (1984).
doi: 10.1007/978-3-642-69512-4. |
[2] |
J. P. Aubin and H. Frankowska, Set-Valued Analysis,, MA: Birkauser, (1990).
|
[3] |
E. K. P. Chong, S. Hui and S. H. Zak, An analysis of a class of neural networks for solving linear programming problems,, Automatic Control, 44 (1999), 1995.
doi: 10.1109/9.802909. |
[4] |
J. Chen and J. Ye, Training SVM with indefinite kernels,, In Proceedings of the 25th International Conference on Machine Learning, (2008), 136.
doi: 10.1145/1390156.1390174. |
[5] |
F. H. Clarke, Optimization and Non-smooth Analysis,, Wiley, (1969). Google Scholar |
[6] |
L. Fengqiu and X. Xiaoping, Design of natural classification kernels using prior knowledge,, Fuzzy Systems, 20 (2012), 135. Google Scholar |
[7] |
A. F. Filippove, Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications (Soviet Series),, Boston, (1988). Google Scholar |
[8] |
M. Forti, P. Nistri and M. Quincampoix, Convergence of neural networks for programming problems via a nonsmooth Łojasiewicz inequality,, Neural Networks, 17 (2006), 1471. Google Scholar |
[9] |
M. Forti and A. Tesi, Absolute stability of analytic neural networks: An approach based on finite trajectory length,, Circuits System I, 51 (2004), 2460.
doi: 10.1109/TCSI.2004.838143. |
[10] |
B. Haasdonk, Feature space interpretation of svms with indefinite kernels,, Pattern Analysis and Machine Intelligence, 27 (2005), 482.
doi: 10.1109/TPAMI.2005.78. |
[11] |
M. Hein and O. Bousquet, Hilbertian metrics and positive definite kernels on probability measures,, In Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics, (2005), 136. Google Scholar |
[12] |
J. J. Hopfield and D. W. Tank, "Neural" computation of decisions in optimization problems,, Biological cybernetics, 52 (1985), 141.
|
[13] |
R. Luss and A. d'Aspremon, Support vector machine classification with indefinite kernels,, Mathematical Programming Computation, 1 (2009), 97.
doi: 10.1007/s12532-009-0005-5. |
[14] |
I. Mierswa, Making Indefinite Kernel Learning Practical,, Technical report, (2006). Google Scholar |
[15] |
J. C. Platt, Fast training of support vector machines using sequential minimal optimization,, In Advances in Kernel Methods, (1999), 185. Google Scholar |
[16] |
T. Rockafellar and R. Wets, Variational Analysis,, Springer-Verlag, (1998).
doi: 10.1007/978-3-642-02431-3. |
[17] |
S. Shalev-Shwartz, Y. Singer, N. Srebro and A. Cotter, Pegasos: Primal estimated sub-gradient solver for SVM,, Math. Program., 127 (2011), 3.
doi: 10.1007/s10107-010-0420-4. |
[18] |
H. Saigo, J. P. Vert, N. Ueda and T. Akutsu, Protein homology detection using string alignment kernels,, Bioinformatics, 20 (2004), 1682.
doi: 10.1093/bioinformatics/bth141. |
[19] |
B. Schölkopf and A. J. Smola, Learning with Kernels,, MIT, (2002). Google Scholar |
[20] |
B. Schölkopf, P. Simard, A. J. Smola and V. N. Vapnik, Prior Knowledge in Support Vector Kernels,, MIT, (1998). Google Scholar |
[21] |
V. N. Vapnik, Statistical Learning Theory,, Wiley, (1998).
|
[22] |
B. Wei and X. Xiaoping, Subgradient-based neural networks for nonsmooth nonconvex optimization problems,, Circuits and Systems I: Regular Papers, 55 (2008), 2378.
doi: 10.1109/TCSI.2008.920131. |
[23] |
Y. Xia and J. Wang, A one-layer recurrent neural network for support vector machine learning,, Systems, 34 (2004), 1621.
doi: 10.1109/TSMCB.2003.822955. |
[24] |
S. Ziye and J. Qingwei, Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems,, Journal of Industrial and Management Optimization, 10 (2014), 871.
doi: 10.3934/jimo.2014.10.871. |
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