  It is of importance to develop statistical techniques to analyze
high-dimensional data in the presence of both complex dependence and possible
outliers in real-world applications such as imaging data analyses. We propose a
new robust high-dimensional regression with coefficient thresholding, in which
an efficient nonconvex estimation procedure is proposed through a thresholding
function and the robust Huber loss. The proposed regularization method accounts
for complex dependence structures in predictors and is robust against outliers
in outcomes. Theoretically, we analyze rigorously the landscape of the
population and empirical risk functions for the proposed method. The fine
landscape enables us to establish both {statistical consistency and
computational convergence} under the high-dimensional setting. The
finite-sample properties of the proposed method are examined by extensive
simulation studies. An illustration of real-world application concerns a
scalar-on-image regression analysis for an association of psychiatric disorder
measured by the general factor of psychopathology with features extracted from
the task functional magnetic resonance imaging data in the Adolescent Brain
Cognitive Development study.
