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Volume 30, Issue 1
Truncated $L_1$ Regularized Linear Regression: Theory and Algorithm

Mingwei Dai, Shuyang Dai, Junjun Huang, Lican Kang & Xiliang Lu

DOI: 10.4208/cicp.OA-2020-0250

Commun. Comput. Phys., 30 (2021), pp. 190-209.

Published online: 2021-04

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  • Abstract

Truncated $L_1$ regularization proposed by Fan in [5], is an approximation to the $L_0$ regularization in high-dimensional sparse models. In this work, we prove the non-asymptotic error bound for the global optimal solution to the truncated $L_1$ regularized linear regression problem and study the support recovery property. Moreover, a primal dual active set algorithm (PDAS) for variable estimation and selection is proposed. Coupled with continuation by a warm-start strategy leads to a primal dual active set with continuation algorithm (PDASC). Data-driven parameter selection rules such as cross validation, BIC or voting method can be applied to select a proper regularization parameter. The application of the proposed method is demonstrated by applying it to simulation data and a breast cancer gene expression data set (bcTCGA).

  • Keywords

High-dimensional linear regression, sparsity, truncated $L_1$ regularization, primal dual active set algorithm.

  • AMS Subject Headings

62J99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-30-190, author = {Dai , MingweiDai , ShuyangHuang , JunjunKang , Lican and Lu , Xiliang}, title = {Truncated $L_1$ Regularized Linear Regression: Theory and Algorithm}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {1}, pages = {190--209}, abstract = {

Truncated $L_1$ regularization proposed by Fan in [5], is an approximation to the $L_0$ regularization in high-dimensional sparse models. In this work, we prove the non-asymptotic error bound for the global optimal solution to the truncated $L_1$ regularized linear regression problem and study the support recovery property. Moreover, a primal dual active set algorithm (PDAS) for variable estimation and selection is proposed. Coupled with continuation by a warm-start strategy leads to a primal dual active set with continuation algorithm (PDASC). Data-driven parameter selection rules such as cross validation, BIC or voting method can be applied to select a proper regularization parameter. The application of the proposed method is demonstrated by applying it to simulation data and a breast cancer gene expression data set (bcTCGA).

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0250}, url = {http://global-sci.org/intro/article_detail/cicp/18878.html} }
TY - JOUR T1 - Truncated $L_1$ Regularized Linear Regression: Theory and Algorithm AU - Dai , Mingwei AU - Dai , Shuyang AU - Huang , Junjun AU - Kang , Lican AU - Lu , Xiliang JO - Communications in Computational Physics VL - 1 SP - 190 EP - 209 PY - 2021 DA - 2021/04 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0250 UR - https://global-sci.org/intro/article_detail/cicp/18878.html KW - High-dimensional linear regression, sparsity, truncated $L_1$ regularization, primal dual active set algorithm. AB -

Truncated $L_1$ regularization proposed by Fan in [5], is an approximation to the $L_0$ regularization in high-dimensional sparse models. In this work, we prove the non-asymptotic error bound for the global optimal solution to the truncated $L_1$ regularized linear regression problem and study the support recovery property. Moreover, a primal dual active set algorithm (PDAS) for variable estimation and selection is proposed. Coupled with continuation by a warm-start strategy leads to a primal dual active set with continuation algorithm (PDASC). Data-driven parameter selection rules such as cross validation, BIC or voting method can be applied to select a proper regularization parameter. The application of the proposed method is demonstrated by applying it to simulation data and a breast cancer gene expression data set (bcTCGA).

Dai , MingweiDai , ShuyangHuang , JunjunKang , Lican and Lu , Xiliang. (2021). Truncated $L_1$ Regularized Linear Regression: Theory and Algorithm. Communications in Computational Physics. 30 (1). 190-209. doi:10.4208/cicp.OA-2020-0250
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