  In many statistical modeling problems, such as classification and regression,
it is common to encounter sparse and blocky coefficients. Sparse fused Lasso is
specifically designed to recover these sparse and blocky structured features,
especially in cases where the design matrix has ultrahigh dimensions, meaning
that the number of features significantly surpasses the number of samples.
Quantile loss is a well-known robust loss function that is widely used in
statistical modeling. In this paper, we propose a new sparse fused lasso
classification model, and develop a unified multi-block linearized alternating
direction method of multipliers algorithm that effectively selects sparse and
blocky features for regression and classification. Our algorithm has been
proven to converge with a derived linear convergence rate. Additionally, our
algorithm has a significant advantage over existing methods for solving
ultrahigh dimensional sparse fused Lasso regression and classification models
due to its lower time complexity. Note that the algorithm can be easily
extended to solve various existing fused Lasso models. Finally, we present
numerical results for several synthetic and real-world examples, which
demonstrate the robustness, scalability, and accuracy of the proposed
classification model and algorithm
