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Regularized multidimensional scaling with radial basis functions
1. | School of Mathematics, University of Southampton, United Kingdom, United Kingdom |
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 |
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A. Argyriou, T. Evgeniou and M. Pontil, Convex Multi-task Feature Learning,, Machine Learning, 73 (2008), 243.
doi: 10.2139/ssrn.1031158. |
[3] |
J. Bénasséni, Partial additive constant,, J. Statist. Comput. Simul., 49 (1994), 179. Google Scholar |
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I. Borg and P. J. F. Groenen, Modern Multidimensional Scaling. Theory and Applications,, $2^{nd}$ edition, (2005).
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[5] |
F. Cailliez, The analytical solution of the additive constant problem,, Psychometrika, 48 (1983), 305.
doi: 10.1007/BF02294026. |
[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. |
[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. |
[9] |
J. de Leeuw, Applications of convex analysis to multidimensional scaling,, in Recent Developments in Statistics (eds. J. Barra, (): 133.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
J. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis,, Psychometrika, 29 (1964), 1.
doi: 10.1007/BF02289565. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
S. Theodoridis and K. Koutroumbas, Pattern Recognition,, Elsevier Inc., (2009).
doi: 10.1016/B0-12-227240-4/00132-5. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[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).
|
[5] |
F. Cailliez, The analytical solution of the additive constant problem,, Psychometrika, 48 (1983), 305.
doi: 10.1007/BF02294026. |
[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. |
[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. |
[9] |
J. de Leeuw, Applications of convex analysis to multidimensional scaling,, in Recent Developments in Statistics (eds. J. Barra, (): 133.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
J. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis,, Psychometrika, 29 (1964), 1.
doi: 10.1007/BF02289565. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
S. Theodoridis and K. Koutroumbas, Pattern Recognition,, Elsevier Inc., (2009).
doi: 10.1016/B0-12-227240-4/00132-5. |
[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. |
[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. |
[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. |
[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. |
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