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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