The Statistical Analysis And Optimization Algorithm For Fused-LASSO Penalized Least Absolute Deviation Regression

The least absolute shrinkage and selection operator (LASSO) has been playing an important role in variable selection and dimensionality re-duction for high-dimensional linear regression under the Gaussian assumption. However, the Gaussian assumption may not hold in practice. In this case, the least absolute deviation is a popular and useful method. In this article we focus on the least absolute deviation via fused-LASSO, called robust fused-LASSO, un-der the assumption that the unknown vector is sparsity for both the coefficients and its successive differences. Robust fused-LASSO estimator does not need any knowledge of standard deviation of the noises or any moment assumptions of the noises. We show that the robust fused-LASSO estimator possesses near oracle performance, i.e. with large probability, the L2-norm of the estimation error is of order O(√k(log p)/n). The result is true for a wide range of noise distributions, even for the Cauchy distribution. In addition, we apply the linearized alternating direction method of multipliers to find the Robust fused-LASSO estimator, which possesses the global convergence. Numerical results are reported to demonstrate the efficiency of our proposed method.