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<p>Partially linear regression (PLR) models provide a powerful framework for interpreting key variables while adjusting for complex confounders in modern data analysis. However, substantial challenges arise for traditional PLR methods in high-dimensional settings, particularly when both the linear and nonlinear components involve many predictors. Existing approaches typically tackle high dimensionality in only one component-either linear or nonlinear-often facing the curse of dimensionality or a loss of interpretability when both components are high-dimensional. To overcome these limitations, we propose a Double High-dimensional Partially Linear Regression (DH-PLR) framework that integrates the sparsity-inducing SCAD penalty for variable selection in the linear component with deep neural networks for flexible nonlinear modeling. Our unified loss-based formulation accommodates a broad class of loss functions, including the least-squares, quantile, and Huber loss functions. From a theoretical perspective, we establish consistency in coefficient estimation and variable selection for the linear component, as well as consistency in estimation for the nonlinear component. Furthermore, we develop an efficient algorithm and demonstrate the superior performance of DH-PLR through simulation studies and an application to breast cancer data.</p>

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Penalized partially linear regression with deep neural networks for high-dimensional learning

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Gefei Li,
  • Riquan Zhang,
  • Qiao Zheng,
  • Yan Zhong

摘要

Partially linear regression (PLR) models provide a powerful framework for interpreting key variables while adjusting for complex confounders in modern data analysis. However, substantial challenges arise for traditional PLR methods in high-dimensional settings, particularly when both the linear and nonlinear components involve many predictors. Existing approaches typically tackle high dimensionality in only one component-either linear or nonlinear-often facing the curse of dimensionality or a loss of interpretability when both components are high-dimensional. To overcome these limitations, we propose a Double High-dimensional Partially Linear Regression (DH-PLR) framework that integrates the sparsity-inducing SCAD penalty for variable selection in the linear component with deep neural networks for flexible nonlinear modeling. Our unified loss-based formulation accommodates a broad class of loss functions, including the least-squares, quantile, and Huber loss functions. From a theoretical perspective, we establish consistency in coefficient estimation and variable selection for the linear component, as well as consistency in estimation for the nonlinear component. Furthermore, we develop an efficient algorithm and demonstrate the superior performance of DH-PLR through simulation studies and an application to breast cancer data.