Comparisons of different methods for balanced data classification under the discrete non-local total variational framework

Shixiu Zheng; Zhilei Xu; Huan Yang; Jintao Song; Zhenkuan Pan

doi:10.3934/mfc.2019002

Article Contents

2019, Volume 2, Issue 1: 11-28. Doi: 10.3934/mfc.2019002

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Comparisons of different methods for balanced data classification under the discrete non-local total variational framework

1.
College of Computer Science and Technology, Qingdao University, Qingdao 266071, China
2.
College of Business, Qingdao University, Qingdao 266071, China

^* Corresponding author: Zhenkuan Pan
^* Corresponding author: Zhenkuan Pan

Published: March 2019

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Abstract

Because balanced constraints can overcome the problems of trivial solutions of data classification via minimum cut method, many techniques with different balanced strategies have been proposed to improve data classification accuracy. However, their performances have not been compared comprehensively so far. In this paper, we investigate seven balanced classification methods under the discrete non-local total variational framework and compare their accuracy performances on graph. The two-class classification problem with equality constraints, inequality constraints and Ratio Cut, Normalized Cut, Cheeger Cut models are investigated. For cases of equality constraint, we firstly compare the Penalty Function Method (PFM) and the Augmented Lagrangian Method (ALM), which can transform the constrained problems into unconstrained ones, to show the advantages of ALM. The other cases are all solved using the ALM also. In order to make the comparison fairly, we solve all models using ALM method and using the same proportion of fidelity points and the same neighborhood size on graph. Experimental results demonstrate ALM with the equality balanced constraint has the best classification accuracy compared with other six constraints. 200 words.

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Mathematics Subject Classification: Primary: 65K10; Secondary: 65F22, 65N20.

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Figure 1. Reference basis of three data sets

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Figure 2. Reference basis of three data sets

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Figure 3. Different results for two-moon dataset classification

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Figure 4. Different results for handwritten digits classification of 3 & 8 dataset

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Figure 5. Different results for handwritten digit classification of 4 & 9 dataset

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Table 0. Different algorithm's parameters

	EC-PFM	EC-ALM	SDIC	DDIC	RC	CC	NC
$ \mu_0 $	0.01	0.005	$ \frac{3k}{n} $	$ \frac{3k}{n}\cdot n $		0.01
$ \mu_1 $	15	25	20	15	15	12	15

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Table 1. Accuracy comparisons ofdifferent algorithms for two-moon dataset classification

Method	$ V_1 $.class	$ V_2 $.class	Error	Ranking
Solution	1000	1000
EC-PFM	1014	986	1.7$ \% $	5
EC-ALM	1005	995	1.25$ \% $	1
SDIC	1016	984	1.50$ \% $	2
DDIC	1015	985	1.55$ \% $	3
RC	1030	970	1.70$ \% $	5
CC	1021	979	1.65$ \% $	4
NC	986	1014	1.90$ \% $	7

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Table 2. Accuracy comparisons of different algorithms for handwritten digit classification of 3 & 8 dataset

Method	$ V_1 $.class	$ V_2 $.class	Error	Ranking
Solution	7141	6825
EC-PFM	7165	6801	1.2745$ \% $	6
EC-ALM	7139	6827	1.1385$ \% $	1
SDIC	7128	6838	1.1671$ \% $	3
DDIC	7161	6805	1.2244$ \% $	4
RC	7173	6793	1.2387$ \% $	5
CC	7134	6832	1.1600$ \% $	2
NC	7167	6799	1.5538$ \% $	7

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Table 3. Accuracy comparisons of different algorithms for handwritten digit classification of 4 & 9 dataset

Method	$ V_1 $.class	$ V_2 $.class	Error	Ranking
Solution	6824	6958
EC-PFM	6814	6968	1.2988$ \% $	4
EC-ALM	6818	6964	1.2335$ \% $	1
SDIC	6799	6983	1.3133$ \% $	5
DDIC	6811	6971	1.3206$ \% $	6
RC	6836	6946	1.2698$ \% $	3
CC	6787	6995	1.2407$ \% $	2
NC	6807	6975	1.4729$ \% $	7

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Table 4. The dependences of error rate on fidelity set size

Two-moons		3&8
Fidelity set size(per class)	Error rate	Fidelity set size(per class)	Error rate
100	1.20$ \% $	350	1.1287$ \% $
50	1.25$ \% $	300	1.1385 $ \% $
40	1.65$ \% $	250	1.2888$ \% $
30	1.80$ \% $	200	1.3604$ \% $
26	1.85$ \% $	150	1.6397$ \% $
20	2.45$ \% $	100	3.8665$ \% $
16	2.55$ \% $	90	4.1386$ \% $
10	3.25$ \% $	70	9.8597$ \% $
6	4.20$ \% $	50	11.2201$ \% $

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Table 5. Dependence of error rate on DDIC range parameter

$ \varepsilon $value	$ V_1 $.class	$ V_2 $.class	Error rate
5	7153	6813	1.2316$ \% $
10	7160	6806	1.2387$ \% $
15	7121	6845	1.2459$ \% $
20	7159	6807	1.2602$ \% $
30	7160	6806	1.2576$ \% $
50	7161	6805	1.2724$ \% $

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