Regularized Least Absolute Deviations Regression and an Efficient Algorithm for Parameter Tuning

Linear regression is one of the most important and widely used techniques for data analysis. However, sometimes people are not satisfied with it because of the following two limitations: 1) its results are sensitive to outliers, so when the error terms are not normally distributed, especially when they have heavy-tailed distributions, linear regression often works badly; 2) its estimated coefficients tend to have high variance, although their bias is low. To reduce the influence of outliers, robust regression models were developed. Least absolute deviation (LAD) regression is one of them. LAD minimizes the mean absolute errors, instead of mean squared errors, so its results are more robust. To address the second limitation, shrinkage methods were proposed, which add a penalty on the size of the coefficients. The LASSO is one of these methods and it uses the L1-norm penalty, which not only reduces the prediction error and the variance of estimated coefficients, but also provides an automatic feature selection function. In this paper, we propose the regularized least absolute deviation (RLAD) regression model, which combines the nice features of the LAD and the LASSO together. The RLAD is a regularization method, whose objective function has the form of "loss + penalty." The "loss" is the sum of the absolute deviations and the "penalty" is the L1-norm of the coefficient vector. Furthermore, to facilitate parameter tuning, we develop an efficient algorithm which can solve the entire regularization path in one pass. Simulations with various settings are performed to demonstrate its performance. Finally, we apply the algorithm to solve the image reconstruction problem and find interesting results.