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High-dimensional linear regression with hard thresholding regularization: Theory and algorithm

  • *Corresponding author: Jing Zhang

    *Corresponding author: Jing Zhang
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  • Variable selection and parameter estimation are fundamental and important problems in high dimensional data analysis. In this paper, we employ the hard thresholding regularization method [1] to handle these issues under the framework of high-dimensional and sparse linear regression model. Theoretically, we establish a sharp non-asymptotic estimation error for the global solution and further show that the support of the global solution coincides with the target support with high probability. Motivated by the KKT condition, we propose a primal dual active set algorithm (PDAS) to solve the minimization problem, and show that the proposed PDAS algorithm is essentially a generalized Newton method, which guarantees that the proposed PDAS algorithm will converge fast if a good initial value is provided. Furthermore, we propose a sequential version of the PDAS algorithm (SPDAS) with a warm-start strategy to choose the initial value adaptively. The most significant advantage of the proposed procedure is its fast calculation speed. Extensive numerical studies demonstrate that the proposed method performs well on variable selection and estimation accuracy. It has favorable exhibition over the existing methods in terms of computational speed. As an illustration, we apply the proposed method to a breast cancer gene expression data set.

    Mathematics Subject Classification: Primary: 62J05, 62H12; Secondary: 68Q25.

    Citation:

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  • Figure 1.  Plot of $ P_{3}(\cdot) $

    Figure 2.  Time versus $ n, p, \rho $ and $ \sigma $

    Figure 3.  RP versus $ n, p, \rho $ and $ \sigma $

    Table 1.  Numerical results with $ n = 300 $, $ p = 5000 $, $ T = 15 $, $ R = 10 $, $ \sigma = 0.2\; \mbox{and}\; 1 $, $ \rho = 0.2:0.2:0.8 $ and $ {\bf{X}} $ follows (I)

    $ \rho $ $ \sigma $ Method AE RE ($ 10^{-2} $) Time(s) RP MSES
    0.2 0.2 Lasso 0.635 10.71 3.36 0.88 15.13
    MCP 0.125 0.94 3.49 1 15
    SCAD 0.315 2.56 3.56 1 15
    GraHTP 0.025 0.28 0.10 1 15
    SDAR 0.024 0.27 0.61 1 15
    SPDAS 0.024 0.27 0.53 1 15
    1 Lasso 0.662 10.80 4.12 0.52 16.27
    MCP 0.184 1.79 3.93 0.99 15.01
    SCAD 0.362 3.13 4.42 1 15
    GraHTP 0.124 1.37 0.15 1 15
    SDAR 0.122 1.36 0.56 1 15
    SPDAS 0.122 1.36 0.45 1 15
    0.4 0.2 Lasso 0.648 10.88 3.39 0.9 15.1
    MCP 0.115 0.87 3.60 1 15
    SCAD 0.305 2.44 3.47 1 15
    GraHTP 0.023 0.27 0.11 1 15
    SDAR 0.024 0.27 0.61 1 15
    SPDAS 0.023 0.27 0.81 1 15
    1 Lasso 0.683 11.04 3.61 0.32 16.27
    MCP 0.187 1.78 4.04 1 15
    SCAD 0.348 3.04 4.44 0.99 14.99
    GraHTP 0.118 1.35 0.17 1 15
    SDAR 0.121 1.36 0.57 1 15
    SPDAS 0.118 1.35 0.73 1 15
    0.6 0.2 Lasso 0.665 10.88 3.51 0.5 15.6
    MCP 0.130 1.01 4.34 0.99 14.98
    SCAD 0.314 2.60 3.92 0.99 14.99
    GraHTP 0.088 0.80 0.15 0.96 15
    SDAR 0.061 0.57 0.62 0.98 15
    SPDAS 0.024 0.27 0.85 1 15
    1 Lasso 0.695 11.06 3.55 0.13 17.18
    MCP 0.198 1.91 4.09 0.99 14.98
    SCAD 0.362 3.25 5.17 0.99 14.99
    GraHTP 0.166 1.75 0.20 0.97 15
    SDAR 0.136 1.48 0.60 0.99 15
    SPDAS 0.121 1.36 0.76 1 15
    0.8 0.2 Lasso 0.738 11.75 3.50 0.1 17.95
    MCP 0.133 1.04 4.13 0.99 14.98
    SCAD 0.343 2.86 3.58 0.99 14.98
    GraHTP 0.648 5.48 0.22 0.75 15
    SDAR 0.265 2.27 0.67 0.89 15
    SPDAS 0.037 0.38 0.85 0.99 14.98
    1 Lasso 0.784 12.17 3.87 0.02 20.29
    MCP 0.192 1.92 3.97 0.99 14.98
    SCAD 0.383 3.41 4.62 0.98 14.99
    GraHTP 0.781 6.57 0.25 0.67 15
    SDAR 0.368 3.41 0.62 0.86 15
    SPDAS 0.138 1.49 0.72 0.99 14.98
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical results with $ n = 300 $, $ p = 5000 $, $ T = 15 $, $ R = 10 $, $ \sigma = 0.2~ \mbox{and} ~ 1 $, $ C = 2:2:8 $ and $ {\bf{X}} $ follows (II)

    C $ \sigma $ Method AE RE ($ 10^{-2} $) Time RP MSES
    2 0.2 Lasso 0.655 10.87 4.31 0.84 15.24
    MCP 0.119 0.94 4.02 1 15
    SCAD 0.310 2.57 5.45 1 15
    GraHTP 0.025 0.28 0.12 1 15
    SDAR 0.025 0.27 0.64 1 15
    SPDAS 0.024 0.27 0.67 1 15
    1 Lasso 0.681 11.02 4.87 0.4 16.65
    MCP 0.191 1.88 5.34 1 15
    SCAD 0.359 3.21 3.05 1 15
    GraHTP 0.123 1.38 0.14 1 15
    SDAR 0.121 1.36 0.65 1 15
    SPDAS 0.122 1.38 0.50 1 15
    4 0.2 Lasso 0.648 10.86 4.89 0.88 15.12
    MCP 0.115 0.90 5.19 1 15
    SCAD 0.307 2.54 5.43 1 15
    GraHTP 0.024 0.27 0.13 1 15
    SDAR 0.024 0.27 0.60 1 15
    SPDAS 0.024 0.26 0.70 1 15
    1 Lasso 0.674 10.97 4.68 0.41 16.48
    MCP 0.174 1.72 5.39 1 15
    SCAD 0.333 3.03 3.17 1 15
    GraHTP 0.122 1.35 0.13 1 15
    SDAR 0.120 1.34 0.58 1 15
    SPDAS 0.121 1.34 0.50 1 15
    6 0.2 Lasso 0.646 10.73 5.30 0.89 15.11
    MCP 0.121 0.95 4.96 1 15
    SCAD 0.308 2.58 5.74 1 15
    GraHTP 0.024 0.27 0.14 1 15
    SDAR 0.024 0.27 0.61 1 15
    SPDAS 0.024 0.27 0.74 1 15
    1 Lasso 0.668 10.85 4.60 0.38 16.35
    MCP 0.183 1.80 4.77 1 15
    SCAD 0.349 3.15 3.44 1 15
    GraHTP 0.120 1.34 0.15 1 15
    SDAR 0.120 1.35 0.55 1 15
    SPDAS 0.119 1.33 0.55 1 15
    8 0.2 Lasso 0.645 10.76 4.63 0.95 15.05
    MCP 0.124 0.96 3.47 1 15
    SCAD 0.308 2.57 4.92 1 15
    GraHTP 0.025 0.27 0.14 1 15
    SDAR 0.024 0.27 0.61 1 15
    SPDAS 0.024 0.27 0.89 1 15
    1 Lasso 0.663 10.82 4.51 0.48 16.23
    MCP 0.186 1.81 4.83 0.99 15.01
    SCAD 0.343 3.09 3.13 0.99 15.01
    GraHTP 0.124 1.35 0.13 1 15
    SDAR 0.121 1.33 0.62 1 15
    SPDAS 0.124 1.35 0.59 1 15
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical results with $ n = 600 $, $ p = 10000 $, $ T = 30 $, $ R = 10 $, $ \sigma = 0.2~ \mbox{and}~ 1 $, $ \rho = 0.2:0.2:0.8 $ and $ {\bf{X}} $ follows (III)

    $ \rho $ $ \sigma $ Method AE RE ($ 10^{-2} $) Time RP MSES
    0.2 0.2 Lasso 0.763 11.49 13.04 0.81 30.21
    MCP 0.222 1.39 12.89 1 30
    SCAD 0.485 3.62 13.11 1 30
    GraHTP 0.018 0.17 0.46 1 30
    SDAR 0.019 0.17 3.37 1 30
    SPDAS 0.018 0.17 4.84 1 30
    1 Lasso 0.769 11.51 12.25 0.64 30.45
    MCP 0.246 1.73 18.84 1 30
    SCAD 0.506 3.77 12.37 1 30
    GraHTP 0.092 0.87 0.42 1 30
    SDAR 0.094 0.87 3.51 1 30
    SPDAS 0.092 0.87 3.69 1 30
    0.4 0.2 Lasso 0.794 11.61 14.01 0.42 31.13
    MCP 0.241 1.49 13.49 0.99 30.01
    SCAD 0.508 3.76 13.96 0.99 30.01
    GraHTP 0.057 0.38 0.55 0.98 30
    SDAR 0.033 0.24 3.58 0.99 30
    SPDAS 0.017 0.15 4.96 1 30
    1 Lasso 0.802 11.64 12.23 0.24 31.44
    MCP 0.262 1.78 18.58 0.99 30
    SCAD 0.525 3.89 12.18 0.99 30.01
    GraHTP 0.135 1.05 0.52 0.97 30
    SDAR 0.098 0.85 3.54 0.99 30
    SPDAS 0.084 0.78 3.60 1 30
    0.6 0.2 Lasso 0.817 11.75 13.31 0.19 32.03
    MCP 0.461 2.93 18.34 0.92 30.05
    SCAD 0.612 4.45 13.06 0.94 30.06
    GraHTP 0.304 1.63 0.62 0.84 30
    SDAR 1.364 8.87 4.06 0.06 30
    SPDAS 0.015 0.14 4.96 1 30
    1 Lasso 0.825 11.79 12.08 0.11 32.51
    MCP 0.475 3.15 15.58 0.91 30.05
    SCAD 0.626 4.55 12.33 0.93 30.05
    GraHTP 0.282 1.73 0.58 0.88 30
    SDAR 1.379 9.08 3.99 0.05 30
    SPDAS 0.089 0.76 3.26 0.99 30
    0.8 0.2 Lasso 0.814 11.76 13.15 0.14 32.13
    MCP 0.266 1.60 18.20 0.98 29.99
    SCAD 0.535 3.92 12.63 0.97 30.01
    GraHTP 0.178 0.95 0.56 0.89 30
    SDAR 2.496 20.08 4.18 0 30
    SPDAS 0.105 0.85 4.30 0.98 30.47
    1 Lasso 0.822 11.80 12.37 0.12 32.55
    MCP 0.285 1.81 18.87 0.96 29.99
    SCAD 0.545 4.00 12.69 0.95 30.01
    GraHTP 0.257 1.51 0.55 0.89 30
    SDAR 2.501 19.98 4.03 0 30
    SPDAS 0.179 1.32 3.27 0.97 31.36
     | Show Table
    DownLoad: CSV

    Table 4.  The estimation of bcTCGA

    Gene name number Lasso MCP SCAD SPDAS
    ABHD13 82 - -0.022 - -
    C17orf53 1743 0.082 - 0.091 -
    CCDC56 2739 0.056 - 0.039 -
    CDC25C 2964 0.028 - 0.027 -
    CDC6 2987 0.011 - 0.005 0.069
    CEACAM6 3076 - - - 0.023
    CENPK 3105 0.018 - 0.011 -
    CRBN 3676 - -0.057 - -
    DTL 4543 0.091 0.355 0.089 -
    FABP1 5081 - - - -0.126
    FAM77C 5261 - - - 0.017
    FGFRL1 5481 - -0.022 - -
    HBG1 6616 0.069
    HIST2H2BE 6811 - -0.012 - -
    KHDRBS1 7709 - 0.112 - -
    KIAA0101 7719 0.007 - - -
    KLHL13 8002 - -0.013 - -
    LSM12 8782 0.006 - - -
    MFGE8 9230 -0.005 - - -
    MIA 9359 -0.006
    NBR2 9941 0.273 0.504 0.235 0.555
    NPY1R 10311 0.008
    PSME3 12146 0.085 - 0.074 -
    RDM1 12615 0.058
    SETMAR 13518 - -0.063 - -
    SLC25A22 13833 - 0.017 - -
    SLC6A4 14021 - - - 0.013
    SPAG5 14296 0.024 0.048 0.013 0.180
    SPRY2 14397 -0.012 - -0.005 -
    TIMELESS 15122 0.033 - 0.036 -
    TMPRSS4 15432 - - - 0.031
    TOP2A 15535 0.035 - 0.035 0.128
    TUBA1B 15882 0.021 - - -
    TYR 15953 - - - 0.132
    UHRF1 16087 0.003 - - -
    VPS25 16315 0.106 0.307 0.108 -
     | Show Table
    DownLoad: CSV
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