A Bregman ADMM for Robust Fused Lasso Estimation with Doubly Nonconvex Regularizers

The fused lasso method has emerged as crucial for variable selection in high-dimensional linear regression. It can effectively deal with the case where adjacent variables exhibit strong correlation and gain sparse solutions under the Gaussian noise. However, it exhibits poor robustness in scenarios involving non-Gaussian noise, especially in heavy-tail distributions. Moreover, comparing to use a convex relaxation with the ℓ1-norm, the near unbiasedness of sparse solutions can be enhanced by employing appropriate nonconvex regularization. In this paper, we preserve the structural features of the fused lasso by proposing a robust fused lasso model with ℓ1-norm loss function and doubly nonconvex regularizers. Furthermore, we develop a customized three-block Bregman alternating direction method of multipliers (ADMM) to effectively solve the proposed model, and provide the convergence analysis for the developed algorithm under some mild conditions. Theoretically, we present a smoothing technique for nonconvex regularizers to expand the choice space. This approach aims to ensure the convergence guarantees of the three-block Bregman ADMM. Extensive experiments demonstrate both the robustness of the proposed model and the effectiveness of the developed algorithm.