Regularized multidimensional scaling with radial basis functions

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April 2016, 12(2): 543-563. doi: 10.3934/jimo.2016.12.543

Regularized multidimensional scaling with radial basis functions

1.	School of Mathematics, University of Southampton, United Kingdom, United Kingdom

Received August 2014 Revised January 2015 Published June 2015

Abstract
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The classical Multi-Dimensional Scaling (MDS) is an important method for data dimension reduction. Nonlinear variants have been developed to improve its performance. One of them is the MDS with Radial Basis Functions (RBF). A key issue that has not been well addressed in MDS-RBF is the effective selection of its centers. This paper treats this selection problem as a multi-task learning problem, which leads us to employ the $(2,1)$-norm to regularize the original MDS-RBF objective function. We then study its two reformulations: Diagonal and spectral reformulations. Both can be effectively solved through an iterative block-majorization method. Numerical experiments show that the regularized models can improve the original model significantly.

Keywords: support vector machine., regularization, radial basis function, Multi-dimensional scaling, iterative majorization.

Mathematics Subject Classification: Primary: 62H30, 68T10; Secondary: 90C2.

Citation: Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543

References:

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[3]	J. Bénasséni, Partial additive constant,, J. Statist. Comput. Simul., 49 (1994), 179. Google Scholar
[4]	I. Borg and P. J. F. Groenen, Modern Multidimensional Scaling. Theory and Applications,, $2^{nd}$ edition, (2005). Google Scholar
[5]	F. Cailliez, The analytical solution of the additive constant problem,, Psychometrika, 48 (1983), 305. doi: 10.1007/BF02294026. Google Scholar
[6]	H. G. Chew and C. C. Lim, On regularisation parameter transformation of support vector machines,, Journal of Industrial and Management Optimization, 5 (2009), 403. doi: 10.3934/jimo.2009.5.403. Google Scholar
[7]	L. G. Cooper, A new solution to the additive constant problem in metric and multidimensional scaling,, Psychometrika, 37 (1972), 311. Google Scholar
[8]	T. F. Cox and M. A. Cox, Multidimensional Scaling,, $2^{nd}$ edition, (2002). doi: 10.1007/978-3-540-33037-0_14. Google Scholar
[9]	J. de Leeuw, Applications of convex analysis to multidimensional scaling,, in Recent Developments in Statistics* (eds. J. Barra*, (): 133. Google Scholar
[10]	J. de Leeuw, Block relaxation algorithms in statistics,, in Information Systems and Data Analysis* (eds. Bock*, (1994), 308. doi: 10.1007/978-3-642-46808-7_28. Google Scholar
[11]	W. Glunt, T. L. Hayden, S. Hong and J. Wells, An alternating projection algorithm for computing the nearest Euclidean distance matrix,, SIAM J. Matrix Anal. Appl., 11 (1990), 589. doi: 10.1137/0611042. Google Scholar
[12]	W. Glunt, T. L. Hayden and R. Raydan, Molecular conformations from distance matrices,, J. Computational Chemistry, 14 (1993), 114. doi: 10.1002/jcc.540140115. Google Scholar
[13]	J. C. Gower, Some distance properties of latent rootand vector methods in multivariate analysis,, Biometrika, 53 (1966), 315. doi: 10.1093/biomet/53.3-4.325. Google Scholar
[14]	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
[15]	J. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis,, Psychometrika, 29 (1964), 1. doi: 10.1007/BF02289565. Google Scholar
[16]	K. V. Mardia, J. T. Kent and J. M. Bibby, Multivariate Analysis,, $10^{th}$ printing, (1995). Google Scholar
[17]	S. J. Messick and R. P Abelson, The additive constant problem in multidimensional scaling,, Psychometrika, 21 (1956), 1. Google Scholar
[18]	E. Pękalaska and R. P. W. Duin, The Dissimilarity Representation for Pattern Recognition: Foundations and Application,, Series in Machine Perception Artificial Intelligence 64, (2005). Google Scholar
[19]	H.-D. Qi, A semismooth Newton method for the nearest Euclidean distance matrix problem,, SIAM Journal Matrix Analysis and Applications, 34 (2013), 67. doi: 10.1137/110849523. Google Scholar
[20]	H.-D. Qi and N. Xiu, A convex quadratic semidefinite programming approach to the partial additive constant problem in multidimensional scaling,, Journal of Statistical Computation and Simulation, 82 (2012), 1317. doi: 10.1080/00949655.2011.579970. Google Scholar
[21]	H.-D. Qi, N. H. Xiu and X. M. Yuan, A Lagrangian dual approach to the single source localization problem,, IEEE Transactions on Signal Processing, 61 (2013), 3815. doi: 10.1109/TSP.2013.2264814. Google Scholar
[22]	H.-D. Qi and X. M. Yuan, Computing the nearest Euclidean distance matrix with low embedding dimensions,, Mathematical Programming, (): 10107. doi: 10.1007/s10107-013-0726-0. Google Scholar
[23]	K. Schittkowski, Optimal parameter selection in support vector machines,, Journal of Industrial and Management Optimization, 1 (2005), 465. doi: 10.3934/jimo.2005.1.465. Google Scholar
[24]	I. J. Schoenberg, Remarks to Maurice Fréchet's article "Sur la définition axiomatque d'une classe d'espaces vectoriels distanciés applicbles vectoriellement sur l'espace de Hilbet'',, Ann. Math., 36 (1935), 724. doi: 10.2307/1968654. Google Scholar
[25]	S. Theodoridis and K. Koutroumbas, Pattern Recognition,, Elsevier Inc., (2009). doi: 10.1016/B0-12-227240-4/00132-5. Google Scholar
[26]	S. Theodoridis and K. Koutroumbas, An Introduction to Pattern Recognition, A MATLAB approach,, Elsevier Inc., (2010). Google Scholar
[27]	W. S. Torgerson, Theory and Methods for Scaling,, Wiley, (1958). Google Scholar
[28]	A. R. Webb, Multidimensional Scaling by iterative majorization using radial basis functions,, Pattern Recognition, 28 (1995), 753. doi: 10.1016/0031-3203(94)00135-9. Google Scholar
[29]	A. R. Webb, Nonlinear feature extraction with radial basis functions using a weighted multidimensional scaling stress measure,, Pattern Recognition, 4 (1996), 635. doi: 10.1109/ICPR.1996.547642. Google Scholar
[30]	A. R. Webb, An approach to nonlinear principal component analysis using radially-symmetric kernel functions,, Statistics and Computing, 6 (1996), 159. Google Scholar
[31]	G. Young and A. S. Householder, Discussion of a set of points in terms of their mutual distances,, Psychometrika, 3 (1938), 19. doi: 10.1007/BF02287916. Google Scholar
[32]	Y. Yuan, W. Fan and D. Pu, Spline function smooth support vector machine for classification,, Journal of Industrial and Management Optimization, 3 (2007), 529. doi: 10.3934/jimo.2007.3.529. Google Scholar

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References:

[1]	A. Argyriou, T. Evgeniou and M. Pontil, Multi-task Feature Learning,, in Advances in Neural Information Processing Systems* (eds. B. Schoelkopf*, (2007). Google Scholar
[2]	A. Argyriou, T. Evgeniou and M. Pontil, Convex Multi-task Feature Learning,, Machine Learning, 73 (2008), 243. doi: 10.2139/ssrn.1031158. Google Scholar
[3]	J. Bénasséni, Partial additive constant,, J. Statist. Comput. Simul., 49 (1994), 179. Google Scholar
[4]	I. Borg and P. J. F. Groenen, Modern Multidimensional Scaling. Theory and Applications,, $2^{nd}$ edition, (2005). Google Scholar
[5]	F. Cailliez, The analytical solution of the additive constant problem,, Psychometrika, 48 (1983), 305. doi: 10.1007/BF02294026. Google Scholar
[6]	H. G. Chew and C. C. Lim, On regularisation parameter transformation of support vector machines,, Journal of Industrial and Management Optimization, 5 (2009), 403. doi: 10.3934/jimo.2009.5.403. Google Scholar
[7]	L. G. Cooper, A new solution to the additive constant problem in metric and multidimensional scaling,, Psychometrika, 37 (1972), 311. Google Scholar
[8]	T. F. Cox and M. A. Cox, Multidimensional Scaling,, $2^{nd}$ edition, (2002). doi: 10.1007/978-3-540-33037-0_14. Google Scholar
[9]	J. de Leeuw, Applications of convex analysis to multidimensional scaling,, in Recent Developments in Statistics* (eds. J. Barra*, (): 133. Google Scholar
[10]	J. de Leeuw, Block relaxation algorithms in statistics,, in Information Systems and Data Analysis* (eds. Bock*, (1994), 308. doi: 10.1007/978-3-642-46808-7_28. Google Scholar
[11]	W. Glunt, T. L. Hayden, S. Hong and J. Wells, An alternating projection algorithm for computing the nearest Euclidean distance matrix,, SIAM J. Matrix Anal. Appl., 11 (1990), 589. doi: 10.1137/0611042. Google Scholar
[12]	W. Glunt, T. L. Hayden and R. Raydan, Molecular conformations from distance matrices,, J. Computational Chemistry, 14 (1993), 114. doi: 10.1002/jcc.540140115. Google Scholar
[13]	J. C. Gower, Some distance properties of latent rootand vector methods in multivariate analysis,, Biometrika, 53 (1966), 315. doi: 10.1093/biomet/53.3-4.325. Google Scholar
[14]	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
[15]	J. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis,, Psychometrika, 29 (1964), 1. doi: 10.1007/BF02289565. Google Scholar
[16]	K. V. Mardia, J. T. Kent and J. M. Bibby, Multivariate Analysis,, $10^{th}$ printing, (1995). Google Scholar
[17]	S. J. Messick and R. P Abelson, The additive constant problem in multidimensional scaling,, Psychometrika, 21 (1956), 1. Google Scholar
[18]	E. Pękalaska and R. P. W. Duin, The Dissimilarity Representation for Pattern Recognition: Foundations and Application,, Series in Machine Perception Artificial Intelligence 64, (2005). Google Scholar
[19]	H.-D. Qi, A semismooth Newton method for the nearest Euclidean distance matrix problem,, SIAM Journal Matrix Analysis and Applications, 34 (2013), 67. doi: 10.1137/110849523. Google Scholar
[20]	H.-D. Qi and N. Xiu, A convex quadratic semidefinite programming approach to the partial additive constant problem in multidimensional scaling,, Journal of Statistical Computation and Simulation, 82 (2012), 1317. doi: 10.1080/00949655.2011.579970. Google Scholar
[21]	H.-D. Qi, N. H. Xiu and X. M. Yuan, A Lagrangian dual approach to the single source localization problem,, IEEE Transactions on Signal Processing, 61 (2013), 3815. doi: 10.1109/TSP.2013.2264814. Google Scholar
[22]	H.-D. Qi and X. M. Yuan, Computing the nearest Euclidean distance matrix with low embedding dimensions,, Mathematical Programming, (): 10107. doi: 10.1007/s10107-013-0726-0. Google Scholar
[23]	K. Schittkowski, Optimal parameter selection in support vector machines,, Journal of Industrial and Management Optimization, 1 (2005), 465. doi: 10.3934/jimo.2005.1.465. Google Scholar
[24]	I. J. Schoenberg, Remarks to Maurice Fréchet's article "Sur la définition axiomatque d'une classe d'espaces vectoriels distanciés applicbles vectoriellement sur l'espace de Hilbet'',, Ann. Math., 36 (1935), 724. doi: 10.2307/1968654. Google Scholar
[25]	S. Theodoridis and K. Koutroumbas, Pattern Recognition,, Elsevier Inc., (2009). doi: 10.1016/B0-12-227240-4/00132-5. Google Scholar
[26]	S. Theodoridis and K. Koutroumbas, An Introduction to Pattern Recognition, A MATLAB approach,, Elsevier Inc., (2010). Google Scholar
[27]	W. S. Torgerson, Theory and Methods for Scaling,, Wiley, (1958). Google Scholar
[28]	A. R. Webb, Multidimensional Scaling by iterative majorization using radial basis functions,, Pattern Recognition, 28 (1995), 753. doi: 10.1016/0031-3203(94)00135-9. Google Scholar
[29]	A. R. Webb, Nonlinear feature extraction with radial basis functions using a weighted multidimensional scaling stress measure,, Pattern Recognition, 4 (1996), 635. doi: 10.1109/ICPR.1996.547642. Google Scholar
[30]	A. R. Webb, An approach to nonlinear principal component analysis using radially-symmetric kernel functions,, Statistics and Computing, 6 (1996), 159. Google Scholar
[31]	G. Young and A. S. Householder, Discussion of a set of points in terms of their mutual distances,, Psychometrika, 3 (1938), 19. doi: 10.1007/BF02287916. Google Scholar
[32]	Y. Yuan, W. Fan and D. Pu, Spline function smooth support vector machine for classification,, Journal of Industrial and Management Optimization, 3 (2007), 529. doi: 10.3934/jimo.2007.3.529. Google Scholar

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