Semi-proximal ADMM for fused Lasso penalized least absolute deviation in partially linear model

The partially linear model is a type of semiparametric regression model characterized by unknown linear regression coefficients, an unknown nonparametric function, and random errors. This model is particularly useful when the relationship between the dependent variable and some predictors is not strictly linear, allowing for more flexibility in capturing complex relationships in the data. In this paper, we propose a novel variable selection model using fused lasso penalized least absolute deviation in partially linear model (PLM-RFlasso), while simultaneously estimating both parametric and nonparametric components. The proposed model exhibits superior robustness and applicability compared to the least squares model. It is capable of handling high-dimensional data with correlated adjacent variables and with outliers or containing variables with heavy tailed distributions. Then we design a semi-proximal alternating direction method of multipliers (sPADMM) to solve the dual problem of PLM-RFlasso, and give its convergence guarantees and statistical property analysis. Finally, numerical experiments on simulated and real datasets illustrate the robustness and the effectiveness of the proposed model and algorithm.