Outlier Detection Using Nonconvex Penalized Regression

TL;DR

This paper introduces a fast, thresholding-based iterative method called -IPOD for outlier detection in regression, which outperforms existing methods in speed and accuracy, especially in high-dimensional settings.

Contribution

The paper proposes -IPOD, a novel outlier detection algorithm using nonconvex penalized regression with hard thresholding, improving speed and robustness over traditional methods.

Findings

-IPOD accurately detects outliers in various datasets.

It is significantly faster than iteratively reweighted least squares.

The method performs well in high-dimensional sparse settings.

Abstract

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.