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Smooth augmented Lagrangian method for twin bounded support vector machine

  • * Corresponding author: Saeed Ketabchi

    * Corresponding author: Saeed Ketabchi 
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  • In this paper, we propose a method for solving the twin bounded support vector machine (TBSVM) for the binary classification. To do so, we use the augmented Lagrangian (AL) optimization method and smoothing technique, to obtain new unconstrained smooth minimization problems for TBSVM classifiers. At first, the augmented Lagrangian method is recruited to convert TBSVM into unconstrained minimization programming problems called as AL-TBSVM. We attempt to solve the primal programming problems of AL-TBSVM by converting them into smooth unconstrained minimization problems. Then, the smooth reformulations of AL-TBSVM, which we called AL-STBSVM, are solved by the well-known Newton's algorithm. Finally, experimental results on artificial and several University of California Irvine (UCI) benchmark data sets are provided along with the statistical analysis to show the superior performance of our method in terms of classification accuracy and learning speed.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  Illustration of TWSVM

    Figure 2.  Results of linear TWSVM, TBSVM, I$ \nu $-TBSVM and AL-STBSVM on generated data set

    Table 1.  Descriptions of the data sets from the UCI repository

    Data set $ \# $ Cases $ \# $ Features $ \# $ Classes Source
    Sonar 208 60 2 UCI
    Cancer 699 9 2 UCI
    Diabet 768 8 2 UCI
    Wdbc 569 30 2 UCI
    Ionosphere 351 34 2 UCI
    Australian 690 14 2 UCI
    Heart 270 14 2 UCI
    Haberman 306 3 2 UCI
    German 1000 24 2 UCI
    House Votes 435 16 2 UCI
    Spect 237 22 2 UCI
    Splice 1000 60 2 UCI
    Lung-cancer 32 56 2 UCI
    F-diagnosis 100 9 2 UCI
    Breast-cancer 116 9 2 UCI
    Bupa 345 6 2 UCI
    Pima 768 9 2 UCI
    Housing 506 14 2 UCI
     | Show Table
    DownLoad: CSV

    Table 2.  Comparison of linear TWSVM, TBSVM, I$ \nu $-TBSVM and proposed model (AL-STBSVM) on UCI benchmark data sets

    Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
    size Acc($ \% $), Time(s) $ c_{1}=c_{2} $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $ Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $
    Sonar 76.88, 0.73 77.45, 1.53 70.29, 1.53 79.14, 0.52
    208$ \times $ 60 $ 2^{6} $ $ 2^{6} $, $ 2^{-3} $ $ 2^{6} $, 0.2 1, $ 2^{-8} $
    Cancer 96.13, 1.63 96.28, 2.63 90.13, 2.28 96.15, 1.99
    699$ \times $9 $ 2^{-2} $ $ 2^{-5},2^{-3} $ 1, 0.1 $ 2^{-5},2^{-8} $
    Diabet 69.13, 1.90 73.58, 2.68 62.24, 3.39 71.74, 27.35
    768$ \times $8 $ 2^{3} $ $ 2^{5},2^{-4} $ $ 2^{4} $, 0.5 $ 2^{3},2^{-7} $
    Wdbc 93.66, 1.6 94.74, 2.37 90.87, 2 95.62, 6.16
    569$ \times $30 $ 2^{6} $ $ 2^{3},2^{-3} $ $ 2^{5} $, 0.8 $ 2^{-5},2^{-4} $
    Ionosphere 84.89, 0.86 86.30, 1.71 84.03, 1.67 85.21, 0.62
    351$ \times $34 $ 2^{-3} $ $ 2^{-2},2^{-2} $ $ 2^{-2} $, 0.7 $ 2^{3},2^{-7} $
    Australian 83.05, 1.62 84.05, 2.68 67.66, 3.30 83.90, 2.93
    690 $ \times $ 14 $ 2^{5} $ $ 2^{5} $, 1 $ 2^{3} $, 0.6 $ 2,2^{4} $
    Heart 84.44, 0.98 84.44, 1.57 83.70, 1.64 85.19, 0.63
    270$ \times $14 $ 2^{2} $ $ 2,2^{3} $ $ 2^{2} $, 0.9 $ 2^{5},2^{-7} $
    Haberman 74.51, 0.86 75.91, 1.64 72.91, 1.66 77.05, 0.55
    306$ \times $3 $ 2^{-2} $ $ 2^{3},2^{2} $ 1, 0.6 $ 2^{4},2^{4} $
    German 73.9, 1.78 75.4, 3.94 61.8, 6.59 75.1, 11.42
    1000$ \times $24 $ 2^{-2} $ $ 2^{-5},2^{-3} $ $ 2^{-3} $, 0.4 $ 2^{10},2^{2} $
    House Votes 95.62, 0.85 95.85, 1.90 93.11, 1.67 96.08, 0.65
    435$ \times $16 $ 2^{8} $ $ 2,2^{-3} $ 1, 0.1 $ 2^{2},2^{2} $
    Spect 68.60, 0.75 70.44, 1.59 73.57, 1.6 71.24, 0.55
    237$ \times $22 $ 2^{-5} $ $ 2^{-5},2^{-6} $ $ 2^{9} $, 0.9 $ 2^{3},2^{4} $
    Splice 75.40, 2.44 79.70, 3.77 78.20, 5.63 79.18, 10.19
    1000$ \times $60 $ 2^{-5} $ $ 2^{-5},2^{-6} $ $ 2^{-5} $, 0.7 $ 2^{5},2^{4} $
    Lung-cancer 82.5, 0.62 82.5, 1.40 85, 1.42 85.83, 0.42
    32$ \times $56 $ 2^{-1} $ $ 2^{-2},2^{-3} $ $ 2^{-3} $, 0.5 $ 2^{6},2^{-8} $
    F-diagnosis 76.49, 0.62 82.34, 1.40 68.35, 1.48 75.01, 0.49
    100$ \times $9 $ 2^{5} $ $ 2^{8},2^{5} $ $ 2^{10} $, 0.2 $ 2^{4},2^{-7} $
    Breast-cancer 72.52, 0.61 71.84, 1.38 57.04, 1.52 73.81, 0.45
    116$ \times $9 $ 2^{6} $ $ 2^{3},2^{-2} $ $ 2^{6} $, 0.9 $ 2^{5},2^{-7} $
    Bupa 64.35, 0.73 65.23, 1.69 69.29, 1.72 66.68, 0.58
    345$ \times $6 $ 2^{-4} $ $ 2^{-3} $, 2 $ 2^{-4} $, 0.2 $ 2^{2},2^{4} $
    Pima 71.36, 0.86 73.30, 1.64 64.72, 2.65 73.62, 2.04
    768$ \times $9 $ 2^{-7} $ $ 2^{-5},2^{-3} $ $ 2^{-7} $, 0.2 $ 2^{2},2^{4} $
    Housing 80.08, 0.96 80.24, 2.89 61.11, 2.01 93.09, 1.5
    506$ \times $14 $ 2^{-8} $ $ 2^{-7},2^{-6} $ $ 2^{-7} $, 0.2 $ 2^{6},2^{-8} $
    Avg.acc 79.08 80.53 74.11 83.31
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison of nonlinear TWSVM, TBSVM, I$ \nu $-TBSVM and proposed model (AL-STBSVM) on UCI benchmark data sets

    Dataset TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
    size Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \gamma $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $ Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $, $ \gamma $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $
    Sonar 84.53, 0.80 86.14, 1.62 83.27, 1.67 87.54, 0.84
    208$ \times $ 60 $ 2^{5},2^{-6} $ $ 2^{6},2^{-3},2^{-3} $ $ 2^{5} $, 0.2, $ 2^{-3} $ $ 2^{2},2^{-8},2^{-6} $
    Cancer 96.42, 4.71 96.42, 6.59 96.02, 2.90 96.85, 3.85
    699$ \times $9 $ 2^{-2},2^{-6} $ $ 2^{-3},2^{-3},2^{-6} $ $ 2 $, 0.1, $ 2^{-4} $ $ 2^{-5},2^{-8},2^{-3} $
    Diabet 65.49, 10.18 66.28, 8.21 65.11, 5.33 69.14, 21.28
    768$ \times $8 $ 2^{5},2^{-6} $ $ 2^{5},2^{-4},2^{-6} $ $ 2^{4} $, 0.4, $ 2^{-3} $ $ 2^{5},2^{-6},2^{-3} $
    Wdbc 62.74, 2.64 62.74, 3.53 91.76, 2.17 87.18, 10.69
    569$ \times $30 $ 2^{5},2^{-6} $ $ 2^{4},2^{-3},2^{-6} $ $ 2^{-9} $, 0.3, $ 2^{-8} $ $ 2^{-5},2^{-6},2^{-1} $
    Ionosphere 94.89, 1.42 94.54, 2.15 87.17, 1.88 94.89, 2.96
    351$ \times $34 $ 2^{3},2^{-6} $ $ 2^{-4},2^{-2},2^{-3} $ $ 2^{-2} $, 0.2, $ 2^{-2} $ $ 2^{4},2^{-6},2^{-6} $
    Australian 55.65, 4.89 55.22, 5.47 55.51, 6.94 65.21, 17.95
    690$ \times $14 $ 2^{3},2^{-3} $ $ 2^{5},2,2^{-6} $ $ 2^{3} $, 0.5, $ 2^{-3} $ $ 2^{2},2^{4},2^{-5} $
    Heart 82.59, 0.92 83.33, 1.86 81.48, 1.76 83.33, 1.58
    270$ \times $14 $ 2^{2},2^{-3} $ $ 2,2,2^{-6} $ $ 2^{2} $, 0.9, $ 2^{9} $ $ 2^{5},2^{-7},2^{-6} $
    Haberman 73.85, 1.28 73.52, 2.01 73.19, 1.88 73.52, 2.67
    306$ \times $3 $ 2^{-1},2^{-6} $ $ 2,2,2^{-6} $ $ 1 $, 0.6, $ 2^{9} $ $ 2^{3},2^{4},2^{-6} $
    German 70.1, 14.69 70.2, 14.34 70, 7.33 71.5, 35.33
    1000$ \times $24 $ 2^{-2},2^{-6} $ $ 2^{-4},2^{-4},2^{-6} $ $ 2^{-3} $, 0.4, $ 2^{7} $ $ 2^{9},2^{4},2^{-6} $
    House Votes 92.64, 1.22 93.55, 2.20 91.70, 1.78 94.71, 4.12
    435$ \times $16 $ 2^{8},2^{-6} $ $ 2,2^{-2},2^{-6} $ $ 1 $, 0.1, $ 2^{-5} $ $ 2^{3},2^{4},2^{-3} $
    Spect 71.89, 0.86 73.76, 1.66 68.90, 1.66 74.17, 1.45
    237$ \times $22 $ 2^{-5},2^{-6} $ $ 2^{-3},2^{-6},2^{-4} $ $ 2^{-7} $, 0.6, $ 2^{-5} $ $ 2^{3},2^{4},2^{-1} $
    Splice 76.71, 15.98 75.20, 16.58 74.41, 14.66 87.20, 42.10
    1000$ \times $60 $ 2^{-6},2^{-6} $ $ 2^{-5},2^{-5},2^{-6} $ $ 2^{-4} $, 0.2, $ 2^{-3} $ $ 2^{5},2^{5},2^{-6} $
    Lung-caner 80.83, 0.67 85.83, 1.41 84.16, 1.51 85, 0.35
    32$ \times $56 $ 2,2^{-6} $ $ 2^{-2},2^{-3},2^{-6} $ $ 2^{-5} $, 0.1, $ 2^{-3} $ $ 2^{5},2^{-8},2^{-6} $
    F-diagnosis 88.14, 0.64 88.14, 1.42 87.16, 1.59 89.25, 0.44
    100$ \times $9 $ 2^{5},2^{-3} $ $ 2^{6},2^{5},2^{-3} $ $ 2^{10} $, 0.2, $ 2^{7} $ $ 2^{4},2^{-7},2^{-2} $
    Breast-cancer 55.16, 0.65 55.16, 1.48 55.15, 1.56 60.31, 0.48
    116$ \times $9 $ 2^{5},2^{-3} $ $ 2^{3},2^{-5},2^{-3} $ $ 2^{7} $, 0.9, $ 2^{-6} $ $ 2^{4},2^{-7},2^{-6} $
    Bupa 64.37, 1.19 64.35, 2.02 65.80, 1.94 66.36, 2.62
    345$ \times $6 $ 2^{-4},2^{-6} $ $ 2^{3},2,2^{-3} $ $ 2^{-5} $, 0.2, $ 2^{-3} $ $ 2^{3},2^{4},2^{-6} $
    Pima 65.63, 9.66 66.15, 8.08 65.12, 4.74 69, 20.07
    768$ \times $9 $ 2^{-3},2^{-6} $ $ 2^{-6},2^{-6},2^{-6} $ $ 2^{-3} $, 0.6, $ 2^{-6} $ $ 2^{2},2^{5},2^{-3} $
    Housing 93.09, 2.12 93.09, 3.65 93.09, 4.30 93.09, 7.01
    506$ \times $14 $ 2^{-6},2^{6} $ $ 2^{-5},2^{-3},2^{7} $ $ 2^{10} $, 0.7, $ 2^{7} $ $ 2^{7},2^{-8},2^{-6} $
    Avg.acc 76.37 76.86 77.16 80.46
     | Show Table
    DownLoad: CSV

    Table 4.  Rank of accuracy linear classifiers on UCI benchmark data sets

    Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
    Sonar 3 2 4 1
    Cancer 3 1 4 2
    Diabet 3 1 4 2
    Wdbc 3 2 4 1
    Ionosphere 3 1 4 2
    Australian 3 1 4 2
    Heart 2.5 2.5 4 1
    Haberman 3 2 4 1
    German 3 1 4 2
    House Votes 3 2 4 1
    Spect 4 3 1 2
    Splice 4 1 3 2
    Lung-cancer 3.5 3.5 2 1
    F-diagnosis 2 1 4 3
    Breast-cancer 2 3 4 1
    Bupa 4 3 1 2
    Pima 3 2 4 1
    Housing 3 2 4 1
    Average rank 3.06 1.89 3.5 1.56
     | Show Table
    DownLoad: CSV

    Table 5.  Rank of accuracy nonlinear classifiers on UCI benchmark data sets

    Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
    Sonar 3 2 4 1
    Cancer 2.5 2.5 4 1
    Diabet 3 2 4 1
    Wdbc 3.5 3.5 1 2
    Ionosphere 1.5 3 4 1.5
    Australian 2 4 3 1
    Heart 3 1.5 4 1.5
    Haberman 1 2.5 4 2.5
    German 3 2 4 1
    House Votes 3 2 4 1
    Spect 3 2 4 1
    Splice 2 3 4 1
    Lung-cancer 4 1 3 2
    F-diagnosis 2.5 2.5 4 1
    Breast-cancer 2.5 2.5 4 1
    Bupa 3 4 2 1
    Pima 2 3 4 1
    Housing 2.5 2.5 2.5 2.5
    Average rank 2.61 2.53 3.53 1.33
     | Show Table
    DownLoad: CSV
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