  For some special data in reality, such as the genetic data, adjacent genes
may have the similar function. Thus ensuring the smoothness between adjacent
genes is highly necessary. But, in this case, the standard lasso penalty just
doesn't seem appropriate anymore. On the other hand, in high-dimensional
statistics, some datasets are easily contaminated by outliers or contain
variables with heavy-tailed distributions, which makes many conventional
methods inadequate. To address both issues, in this paper, we propose an
adaptive Huber regression for robust estimation and inference, in which, the
fused lasso penalty is used to encourage the sparsity of the coefficients as
well as the sparsity of their differences, i.e., local constancy of the
coefficient profile. Theoretically, we establish its nonasymptotic estimation
error bounds under $\ell_2$-norm in high-dimensional setting. The proposed
estimation method is formulated as a convex, nonsmooth and separable
optimization problem, hence, the alternating direction method of multipliers
can be employed. In the end, we perform on simulation studies and real cancer
data studies, which illustrate that the proposed estimation method is more
robust and predictive.
