  Challenges with data in the big-data era include (i) the dimension $p$ is
often larger than the sample size $n$ (ii) outliers or contaminated points are
frequently hidden and more difficult to detect. Challenge (i) renders most
conventional methods inapplicable. Thus, it attracts tremendous attention from
statistics, computer science, and bio-medical communities. Numerous penalized
regression methods have been introduced as modern methods for analyzing
high-dimensional data. Disproportionate attention has been paid to the
challenge (ii) though. Penalized regression methods can do their job very well
and are expected to handle the challenge (ii) simultaneously. Most of them,
however, can break down by a single outlier (or single adversary contaminated
point) as revealed in this article.
  The latter systematically examines leading penalized regression methods in
the literature in terms of their robustness, provides quantitative assessment,
and reveals that most of them can break down by a single outlier. Consequently,
a novel robust penalized regression method based on the least sum of squares of
depth trimmed residuals is proposed and studied carefully. Experiments with
simulated and real data reveal that the newly proposed method can outperform
some leading competitors in estimation and prediction accuracy in the cases
considered.
