Supervised distance preserving projection using alternating direction method of multipliers

Sohana Jahan

doi:10.3934/jimo.2019029

Article Contents

2020, Volume 16, Issue 4: 1783-1799. Doi: 10.3934/jimo.2019029

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Supervised distance preserving projection using alternating direction method of multipliers

Sohana Jahan

School of Mathematics, University of Dhaka, Bangladesh, School of Mathematics, University of Southampton, UK

Received: February 2018

Revised: November 2018

Early access: May 2019

Published: May 2020

The research of the author was supported by the Commonwealth Scholarship Commission, UK BDCS-2012-44.

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Abstract

Supervised Distance Preserving Projection (SDPP) is a dimension reduction method in supervised setting proposed recently by Zhu et. al in [43]. The method learns a linear mapping from the input space to the reduced feature space. While the method showed very promising result in regression task, for classification problems the performance is not satisfactory. The preservation of distance relation with neighborhood points forces data to project very close to one another in the projected space irrespective of their classes which ends up with low classification rate. To avoid the crowdedness of SDPP approach we have proposed a modification of SDPP which deals both regression and classification problems and significantly improves the performance of SDPP. We have incorporated the total variance of the projected co-variates to the SDPP problem which is maximized to preserve the global structure. This approach not only facilitates efficient regression like SDPP but also successfully classifies data into different classes. We have formulated the proposed optimization problem as a Semidefinite Least Square (SLS) SDPP problem. A two block Alternating Direction Method of Multipliers have been developed to learn the transformation matrix solving the SLS-SDPP which can easily handle out of sample data.

Keywords:

Supervised distance preserving projection,
semidefinite programming,
Alternating Direction Method of Multipliers,
dimension reduction,
Euclidean distance.

Mathematics Subject Classification: Primary: 62H30, 68T10; Secondary: 90C25.

Citation:

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Figure 1. (a) SDPP: Solid lines indicate connection between neighbors. (b-c) Preservation scheme of the local geometry by SDPP

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Figure 2. Average Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) with error bars for prediction of test set of Parkinsons Telemonitoring Data Set obtained by SLS-SDPP, SDPP, PLS, SPCA and KDR. The bar diagram represents almost same performance for all the methods in terms of RMSE. In terms of MAE, SLS-SDPP outperforms all other methods

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Figure 3. Continuity measure with respect to different $ k $ and $ k_r $ for (a) Parkinsons Telemonitoring Data: Figure suggests to choose the neighborhood size $ k = 8 $ since highest continuity measure is obtained at $ k = 8 $ (b) Concrete Compressive Strength Data: Highest continuity measure is obtained at $ k = 10 $ therefore $ k = 10 $ is chosen as the neighborhood size

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Figure 4. Average RMSE and MAE for test data prediction of Concrete Compressive Strength Data set along different dimension obtained by SLS-SDPP, SDPP, PLS, SPCA and KDR. Best performance achieved by SLS-SDPP at D = 5. The small error bar implies the stability of SLS-SDPP method regardless of training data

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Figure 5. Classification error rates for different projection dimension computed by algorithm SLS-SDPP, SDPP, SPCA, KDR and FDA. (a)CTG data (b) Seismic bump data (c)Diabetic Retinopathy data (d) Mushroom data. Figures illustrate that best performance is achieved by SLS-SDPP. The error rate for this method remained consistently lower then other methods

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Table 1. List of datasets used in this article and their sources :

	Dataset	Dim	Class	no. of ins.	Source
Classification	Seismic bump	19	2	2584	UCI Repository
	Cardiotocography	21	3	2126	UCI Repository
	Diabetic Retinopathy	19	2	1115	UCI Repository
	Mushroom	22	2	8124	UCI Repository
Regression	Parkinson's Telemonitoring	16	-	5875	UCI Repository
Regression	Concrete Compressive Strength	8	-	1030	UCI Repository

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Table 2. Average RMSE and MAE for test set prediction of Parkinson Telemonitoring dataset

Method	RMSE (mean$ \pm $std)	MAE (mean$ \pm $std)
SLS-SDPP	10.6781$ \pm $1.1481	8.3503$ \pm $0.8525
SDPP	10.7934$ \pm $1.2371	8.7459$ \pm $0.8378
PLS	10.8133$ \pm $1.2806	8.7822$ \pm $0.8883
SPCA	10.8006$ \pm $ 1.2449	8.7714 $ \pm $0.8555
KDR	10.8478$ \pm $ 1.3032	8.8008$ \pm $0.9139

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Table 3. Average RMSE and MAE (mean$ \pm $std) for the test set prediction on Concrete Compressive Strength Data Set

Error	Dim	SLS-SDPP	SDPP	PLS	SPCA	KDR
RMSE	1	10.4649$ \pm $1.2072	10.5241$ \pm $1.4220	12.7666$ \pm $2.1539	12.8090$ \pm $2.0429	13.7423$ \pm $2.7046
	2	10.4540$ \pm $1.1356	10.4075$ \pm $1.5903	11.8629$ \pm $1.1837	12.8712$ \pm $1.9074	11.6399$ \pm $2.6641
	3	10.4629$ \pm $1.1648	10.4079$ \pm $1.5915	10.9379$ \pm $1.1096	12.8370$ \pm $1.9316	11.5163$ \pm $2.2178
	4	10.4450$ \pm $1.2848	10.5890$ \pm $1.3303	10.5108$ \pm $1.1943	12.8674$ \pm $1.8353	11.0320$ \pm $1.7599
	5	10.2932$ \pm $0.7866	10.7247$ \pm $1.0038	10.4432$ \pm $1.1883	12.9647$ \pm $1.5969	10.2933$ \pm $1.6597
	6	10.9910$ \pm $0.7200	10.4893$ \pm $0.7228	10.4359$ \pm $1.1480	10.4770$ \pm $1.1142	10.4473$ \pm $1.1485
	7	10.4495$ \pm $0.7052	10.5313$ \pm $0.7228	10.4480$ \pm $1.0746	10.4520$ \pm $1.0604	10.4514$ \pm $1.0952
	8	10.4349$ \pm $1.1134	10.4342$ \pm $1.0034	10.4342$ \pm $1.0034	10.4342$ \pm $1.0034	16.3019$ \pm $1.8078
MAE	1	8.3879$ \pm $1.0882	8.3687$ \pm $1.0994	10.3995$ \pm $1.7948	10.4451$ \pm $1.6868	10.8884$ \pm $2.3565
	2	8.2339$ \pm $1.2642	8.2338$ \pm $1.2918	9.3977$ \pm $0.8294	10.4484$ \pm $1.5463	9.2285$ \pm $2.3954
	3	8.2288$ \pm $1.3385	8.2342$ \pm $1.3380	8.3771$ \pm $0.6452	10.4125$ \pm $1.5656	8.9818$ \pm $1.6882
	4	8.5898$ \pm $1.1262	8.3935$ \pm $1.1461	8.2199$ \pm $0.7943	10.4553$ \pm $1.4677	8.6622$ \pm $1.3565
	5	8.0177$ \pm $0.7413	8.5133$ \pm $0.8507	8.1563$ \pm $0.8538	10.5221$ \pm $1.3204	8.0167$ \pm $1.2904
	6	8.3434$ \pm $0.7236	8.2713$ \pm $0.5728	8.1612$ \pm $0.8389	8.1860$ \pm $0.7925	8.1659$ \pm $0.8225
	7	8.2747$ \pm $0.5815	8.2820$ \pm $0.5787	8.1869$ \pm $0.7809	8.1872$ \pm $0.7735	8.1868$ \pm $0.7903
	8	8.1852$ \pm $0.7413	8.1842$ \pm $0.7477	8.1842$ \pm $0.7477	8.1842$ \pm $0.7477	13.1814$ \pm $1.5359

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Table 4. Average error rate of class prediction of test set for CTG data

Error	Dim	SLS-SDPP	SDPP	SPCA	KDR	FDA
Error Rate	2	0.2251	0.2865	0.2739	0.2593	0.208
	3	0.1940	0.2827	0.2661	0.2992	0.208
	4	0.1949	0.2943	0.3314	0.2427	0.208
	5	0.1969	0.2661	0.3372	0.2437	0.208
	6	0.1969	0.2749	0.2710	0.2115	0.208
	7	0.1988	0.2827	0.2768	0.2193	0.208
	8	0.1979	0.2768	0.2817	0.2300	0.208
	9	0.1988	0.2700	0.2612	0.2315	0.208
	10	0.1949	0.2690	0.2515	0.2412	0.208

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Table 5. Average classification error rate of test set for Seismic bump data

Error	Dim	SLS-SDPP	SDPP	SPCA	KDR	FDA
Error Rate	1	0.0328	0.0328	0.0328	0.0328	0.0328
	2	0.0701	0.0707	0.1004	0.0688	0.0328
	3	0.0701	0.0669	0.0694	0.0669	0.0328
	4	0.0701	0.0720	0.0676	0.0720	0.032
	5	0.0701	0.0720	0.0732	0.0726	0.0328
	6	0.0701	0.0713	0.0789	0.0728	0.0328
	7	0.0701	0.0713	0.0789	0.0727	0.0328
	8	0.0701	0.0713	0.0795	0.0729	0.0328
	9	0.0701	0.0713	0.0795	0.0730	0.0328
	10	0.0701	0.0713	0.0795	0.0730	0.0328

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Table 6. Average error rate of class prediction of test set for Diabetic Retinopathy data

Error	Dim	SLS-SDPP	SDPP	SPCA	KDR	FDA
Error Rate	1	0.6382	0.6382	0.6382	0.6382	0.6382
	2	0.2678	0.3390	0.2963	0.2934	0.6382
	3	0.2678	0.3105	0.3618	0.2963	0.6382
	4	0.2678	0.3191	0.2877	0.3048	0.6382
	5	0.2678	0.2934	0.2906	0.3134	0.6382
	6	0.2678	0.2849	0.2877	0.3048	0.6382
	7	0.2735	0.3191	0.2963	0.3048	0.6382
	8	0.2707	0.3191	0.2934	0.3134	0.6382
	9	0.2707	0.2906	0.2906	0.3134	0.6382
	10	0.2678	0.3020	0.3077	0.3048	0.6382

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Table 7. Average error rate of class prediction of test set for Mushroom data

Error	Dim	SLS-SDPP	SDPP	SPCA	KDR	FDA
Error Rate	1	0.2486	0.2486	0.2486	0.2486	0.2486
	2	0.3555	0.1318	0.2735	0.3037	0.2486
	3	0.2891	0.1266	0.1836	0.2741	0.2486
	4	0.1516	0.2076	0.2020	0.2018	0.2486
	5	0.1648	0.2524	0.1911	0.2311	0.2486
	6	0.1667	0.2524	0.3785	0.3815	0.2486
	7	0.1723	0.2693	0.3653	0.3544	0.2486
	8	0.1728	0.2255	0.3630	0.2587	0.2486
	9	0.1186	0.2655	0.1756	0.1615	0.2486
	10	0.1427	0.2665	0.3545	0.2812	0.2486

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References

[1]	E. Barshan, A. Ghodsi, Z. Azimifar and M. Z. Jahromi, Supervised principal component analysis: Visualization, classification and regression on subspaces and submanifolds, Pattern Recognit, 44 (2010), 1357-1371. doi: 10.1016/j.patcog.2010.12.015.
[2]	J. Borwein and A. S. Lewis, Convex Analysis and Non Linear Optimization: Theory and Examples, Springer, New York, 2006. doi: 10.1007/978-0-387-31256-9.
[3]	S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Machine Learning, 3 (2010), 1-122.
[4]	S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.
[5]	M. R. Brito, E. L. Chávez, A. J. Quiroz and J. E. Yukich, Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection, Stat. Probabil. Lett., 35 (1997), 33-42. doi: 10.1016/S0167-7152(96)00213-1.
[6]	I. Cheng Yeh, Modeling of strength of high performance concrete using artificial neural networks, Cement and Concrete Research, 28 (1998), 1797-1808. doi: 10.1016/S0008-8846(98)00165-3.
[7]	F. Coronaa, Z. Zhu, A. H. d. Souza Jr, M. Mulasd, E. Muruf, L. Sassuf, G. Barretob and R. Baratti, Supervised Distance Preserving Projections: Applications in the quantitative analysis of diesel fuels and light cycle oils from NIR spectra, Journal of Process Control, 30 (2015), 10-21. doi: 10.1016/j.jprocont.2014.11.005.
[8]	J. Eckstein and D. P. Bertsekas, On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204.
[9]	J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, Large Scale Optimization: State of the Art, (1993), 115–134.
[10]	J. Eckstein and W. Yao, Understanding the convergence of Alternating Direction Method of Multipliers, Theoritical and Computational Perspectives, RUTCOR Research Report, 2014.
[11]	R. A. Fisher, The use of multiple measurements in taxonomic problems, Annals of Eugenics, 7 (1936), 179-188. doi: 10.1111/j.1469-1809.1936.tb02137.x.
[12]	M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland Publishing Co., Amsterdam, 1983.
[13]	M. Fortin and R. Glowinski, On Decomposition-Coordination Methods Using an Augmented Lagrangian, Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, North-Holland: Amsterdam, 1983.
[14]	K. Fukumizu, F. R. Bach and M. Jordan, Kernel dimension reduction in regression, Annals of Statistics, 37 (2009), 1871-1905. doi: 10.1214/08-AOS637.
[15]	D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, studies in Mathematics, 15 (1983), 299-331.
[16]	D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinearvariational problems via Finite element approximation, Computers and Mathematics with Applications, 2 (1976), 17-40.
[17]	R. Glowinski, Lectures on Numerical Methods for Nonlinear Variational Problem, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay, Notes by M. G. Vijayasundaram and M. Adimurthi, 1980.
[18]	R. Glowinski and A. Marrocco, Sur l'approximation, par $\acute{e}l\acute{e}ments$ finis d'ordre un, et la $r\acute{e}$solution, par $p\acute{e}$nalisation-dualit$\acute{e}$, d'une classe de probl$\grave{e}$mes de dirichlet non lin$\acute{e}$ares, Revue Francaise d'Automatique, Informatique et Recherche Op$\acute{e}$rationelle, 9 (1975), 41–76.
[19]	R. Glowinski and P. L. Tallec, Augmented Lagrangian Methods for the Solution of Variational Problems, Studies in Applied and Numerical Mathematics, 1989. doi: 10.1137/1.9781611970838.ch3.
[20]	J. Han, M. Kamber and J. Pei, Data Mining: Concepts and Techniques, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-19721-5.
[21]	B. He, H. Yang and S. Wang, Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities, Journal of Optimization Theory and Applications, 106 (2000), 337–356. doi: 10.1023/A:1004603514434.
[22]	S. Jahan and H. D. Qi, Regularized Multidimensional Scaling with Radial Basis Functions, Journal of Industrial and Management Optimization, 12 (2016), 543-563. doi: 10.3934/jimo.2016.12.543.
[23]	K. Jiang, D. Sun and K.-C. Toh, Solving nuclear norm regularized and semidefinite matrix least square problems with linear equality constraints, Discrete Geometry and Optimization, 69 (2013), 133-162. doi: 10.1007/978-3-319-00200-2_9.
[24]	J. Lee and M. Verleysen, Nonlinear Dimensionality Reduction, Springer, New York, 2007. doi: 10.1007/978-0-387-39351-3.
[25]	K. Li, Sliced inverse regression for dimension reduction, J Am Stat Assoc, 86 (1991), 316-342. doi: 10.1080/01621459.1991.10475035.
[26]	X. Li, D. Sun and K.-C. Toh, A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions,, Math. Program., 155 (2016), Ser. A, 333–373. doi: 10.1007/s10107-014-0850-5.
[27]	L. J. P. Maaten, E. Postma and H. V. D. Herik, Dimensionality Reduction: A Comparative Review, Technical Report TiCC-TR 2009–005, Tilburg University Technical, Tilburg, 2009.
[28]	S. Mika, G. Ratsch, J. Weston, B. Scholkopf and K. Mullers, Fisher discriminant analysis with kernels, Neural Networks for Signal Processing IX, Proceedings of the IEEE Signal Processing Society Workshop, IEEE, Piscataway, (2002), 41–48. doi: 10.1109/NNSP.1999.788121.
[29]	H.-D. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem, IMA Journal of Numerical Analysis, 31 (2011), 491-511. doi: 10.1093/imanum/drp031.
[30]	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), 3815-3826. doi: 10.1109/TSP.2013.2264814.
[31]	R. T. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116. doi: 10.1287/moor.1.2.97.
[32]	S. Roweis and L. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science, 290 (2000), 2323-2326.
[33]	B. Schölkopf, A. Smola and K. Müller, Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation, 10 (1998), 1299-1319.
[34]	D. Sun, K.-C. Toh and L. Yang, A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-type of constraints, SIAM Journal on Optimization, 25 (2015), 882-915. doi: 10.1137/140964357.
[35]	J. Tenenbaum, V. Silva and J. Langford, A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), 2319-2323.
[36]	S. Theodoridis and K. Koutroumbas, An Introduction to Pattern Recognition, A MATLAB approach, Elsevier Inc., 2010.
[37]	S. Theodoridis and K. Koutroumbas, Pattern Recognition, Elsevier Inc., 2009.
[38]	A. Tsanas, M. A. Little, P. E. McSharry and L. O. Ramig, Accurate telemonitoring of Parkinson.s disease progression by non-invasive speech tests, IEEE Transactions on Biomedical Engineering, 57 (2010), 884-893. doi: 10.1109/TBME.2009.2036000.
[39]	J. Venna and S. Kaski, Comparison of visualization methods for an atlas of gene expression data sets, Inf Vis, 6 (2007), 139-154. doi: 10.1057/palgrave.ivs.9500153.
[40]	H. Wold, Soft modeling by latent variables: The nonlinear iterative partial least squares approach, Perspectives in Probability and Statistics, Papers in Honour of MS Bartlett, 1975,117–142. doi: 10.1017/s0021900200047604.
[41]	H. Wold, Partial Least Squares, Encyclopedia of Statistical Sciences, 2009.
[42]	Y. Yeh Y, S. Huang and Y. Lee, Nonlinear dimension reduction with kernel sliced inverse regression, IEEE Trans Knowl Data Eng, 21 (2009), 1590-1603.
[43]	Z. Zhu, T. Simil$\ddot{a}$ and F. Corona, Supervised distance preserving projection, Neural Processing Letters, 38 (2013), 445-463. doi: 10.1007/s11063-013-9285-x.