Homogeneity and sparsity pursuit using robust adaptive fused lasso

Fused lasso regression is a popular method for identifying homogeneous groups and sparsity patterns in regression coefficients based on either the presumed order or a more general graph structure of the covariates. However, the traditional fused lasso may yield misleading outcomes in the presence of outliers. In this paper, we propose an extension of the fused lasso, namely the robust adaptive fused lasso (RAFL), which pursues homogeneity and sparsity patterns in regression coefficients while accounting for potential outliers within the data. By using Huber's loss or Tukey's biweight loss, RAFL can resist outliers in the responses or in both the responses and the covariates. We also demonstrate that when the adaptive weights are properly chosen, the proposed RAFL achieves consistency in variable selection, consistency in grouping and asymptotic normality. Furthermore, a novel optimization algorithm, which employs the alternating direction method of multipliers, embedded with an accelerated proximal gradient algorithm, is developed to solve RAFL efficiently. Our simulation study shows that RAFL offers substantial improvements in terms of both grouping accuracy and prediction accuracy compared with the fused lasso, particularly when dealing with contaminated data. Additionally, a real analysis of cookie data demonstrates the effectiveness of RAFL.