In many applications, sparse and blocky coefficients often occur in regression and classification problems. The fused Lasso was designed to recover these sparse structured features especially when the design matrix encounters the situation of ultrahigh dimension. Quantile loss is well known as a robust loss function in regression and classification. In this paper, we combine quantile loss and fused Lasso penalty together to produce quantile fused Lasso which can achieve sparse and blocky feature selection in both regression and classification. Interestingly, our proposed model has the unified optimization formula for regression and classification. For ultrahigh dimensional collected data, we derive multi-block linearized alternating direction method of multipliers (LADMM) to deal with it. Moreover, we prove convergence and derive convergence rates of the proposed LADMM algorithm through an elegant method. Note that the algorithm can be easily extended to solve many existing fused Lasso models. Finally, we present some numerical results for several synthetic and real world examples, which illustrate the robustness, scalability, and accuracy of the proposed method.

翻译：在许多应用中，回归和分类问题中常出现稀疏且块状的系数。融合Lasso旨在恢复这些稀疏结构化特征，尤其是在设计矩阵面临超高维情形时。分位数损失作为回归和分类中的稳健损失函数广为人知。本文通过结合分位数损失与融合Lasso惩罚项，提出分位数融合Lasso方法，可在回归与分类中同时实现稀疏与块状特征选择。有趣的是，我们提出的模型具有回归与分类的统一优化公式。针对超高维采集数据，我们推导了多块线性化交替方向乘子法（LADMM）进行处理。此外，通过一种精巧的方法，我们证明了所提LADMM算法的收敛性并推导了其收敛速率。该算法可便捷地扩展至求解现有多种融合Lasso模型。最后，我们通过合成数据与真实世界实例的数值结果，展示了所提方法的稳健性、可扩展性与准确性。