This paper studies the outlier detection problem from the point of view of
penalized regressions. Our regression model adds one mean shift parameter for
each of the $n$ data points. We then apply a regularization favoring a sparse
vector of mean shift parameters. The usual $L_1$ penalty yields a convex
criterion, but we find that it fails to deliver a robust estimator. The $L_1$
penalty corresponds to soft thresholding. We introduce a thresholding (denoted
by $\Theta$) based iterative procedure for outlier detection ($\Theta$-IPOD). A
version based on hard thresholding correctly identifies outliers on some hard
test problems. We find that $\Theta$-IPOD is much faster than iteratively
reweighted least squares for large data because each iteration costs at most
$O(np)$ (and sometimes much less) avoiding an $O(np^2)$ least squares estimate.
We describe the connection between $\Theta$-IPOD and $M$-estimators. Our
proposed method has one tuning parameter with which to both identify outliers
and estimate regression coefficients. A data-dependent choice can be made based
on BIC. The tuned $\Theta$-IPOD shows outstanding performance in identifying
outliers in various situations in comparison to other existing approaches. This
methodology extends to high-dimensional modeling with $p\gg n$, if both the
coefficient vector and the outlier pattern are sparse.

本文使用惩罚回归的方法对异常值检测问题进行研究，提出了一种基于阈值迭代的异常值检测方法，并通过大量实验表明，在各种情况下，该方法可以非常有效且准确地识别出异常值。