Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels

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January 2016, 12(1): 285-301. doi: 10.3934/jimo.2016.12.285

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

Received June 2012 Revised January 2015 Published April 2015

Abstract
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Support vector machines (SVMs) with positive semidefinite kernels yield convex quadratic programming problems. SVMs with indefinite kernels yield nonconvex quadratic programming problems. Most optimization methods for SVMs rely on the convexity of objective functions and are not efficient for solving such nonconvex problems. In this paper, we propose a subgradient-based neural network (SGNN) for the problems cast by SVMs with indefinite kernels. It is shown that the state of the proposed neural network has finite length, and as a consequence it converges toward a singleton. The coincidence between the solution and the slow solution of SGNN is also proved starting from the initial value of SGNN. Moreover, we employ the Łojasiewicz inequality to exploit the convergence rate of trajectory of SGNN. The obtained results show that each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points through a quantitative evaluation of the Łojasiewicz exponent at the equilibrium points. This method is easy to implement without adding any new parameters. Three benchmark data sets from the University of California, Irvine machine learning repository are used in the numerical tests. Experimental results show the efficiency of the proposed neural network.

Keywords: support vector machine., indefinite kernels, neural networks, Sub-gradient, Łojasiewicz inequality.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Fengqiu Liu, Xiaoping Xue. Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels. Journal of Industrial & Management Optimization, 2016, 12 (1) : 285-301. doi: 10.3934/jimo.2016.12.285

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).
[6]	L. Fengqiu and X. Xiaoping, Design of natural classification kernels using prior knowledge,, Fuzzy Systems, 20 (2012), 135.
[7]	A. F. Filippove, Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications (Soviet Series),, Boston, (1988).
[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.
[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.
[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).
[15]	J. C. Platt, Fast training of support vector machines using sequential minimal optimization,, In Advances in Kernel Methods, (1999), 185.
[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).
[20]	B. Schölkopf, P. Simard, A. J. Smola and V. N. Vapnik, Prior Knowledge in Support Vector Kernels,, MIT, (1998).
[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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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).
[6]	L. Fengqiu and X. Xiaoping, Design of natural classification kernels using prior knowledge,, Fuzzy Systems, 20 (2012), 135.
[7]	A. F. Filippove, Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications (Soviet Series),, Boston, (1988).
[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.
[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.
[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).
[15]	J. C. Platt, Fast training of support vector machines using sequential minimal optimization,, In Advances in Kernel Methods, (1999), 185.
[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).
[20]	B. Schölkopf, P. Simard, A. J. Smola and V. N. Vapnik, Prior Knowledge in Support Vector Kernels,, MIT, (1998).
[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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