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For some special data in reality, such as the genetic data, adjacent genesmay have the similar function. Thus ensuring the smoothness between adjacentgenes is highly necessary. But, in this case, the standard lasso penalty justdoesn't seem appropriate anymore. On the other hand, in high-dimensionalstatistics, some datasets are easily contaminated by outliers or containvariables with heavy-tailed distributions, which makes many conventionalmethods inadequate. To address both issues, in this paper, we propose anadaptive Huber regression for robust estimation and inference, in which, thefused lasso penalty is used to encourage the sparsity of the coefficients aswell as the sparsity of their differences, i.e., local constancy of thecoefficient profile. Theoretically, we establish its nonasymptotic estimationerror bounds under $\ell_2$-norm in high-dimensional setting. The proposedestimation method is formulated as a convex, nonsmooth and separableoptimization problem, hence, the alternating direction method of multiplierscan be employed. In the end, we perform on simulation studies and real cancerdata studies, which illustrate that the proposed estimation method is morerobust and predictive.

Robust Fused Lasso Penalized Huber Regression with Nonasymptotic Property and Implementation Studies