A General Family of Trimmed Estimators for Robust High-dimensional Data Analysis

TL;DR

This paper introduces a broad class of trimmed estimators for robust high-dimensional data analysis, providing theoretical guarantees and practical algorithms, with applications demonstrated on genomics data.

Contribution

It develops a unified framework for analyzing and optimizing trimmed M-estimators in high-dimensional settings, including extensions of Lasso and Graphical Lasso.

Findings

Trimmed estimators achieve robustness against outliers.

Theoretical convergence rates and consistency are established.

Numerical experiments show competitive performance on real data.

Abstract

We consider the problem of robustifying high-dimensional structured estimation. Robust techniques are key in real-world applications which often involve outliers and data corruption. We focus on trimmed versions of structurally regularized M-estimators in the high-dimensional setting, including the popular Least Trimmed Squares estimator, as well as analogous estimators for generalized linear models and graphical models, using possibly non-convex loss functions. We present a general analysis of their statistical convergence rates and consistency, and then take a closer look at the trimmed versions of the Lasso and Graphical Lasso estimators as special cases. On the optimization side, we show how to extend algorithms for M-estimators to fit trimmed variants and provide guarantees on their numerical convergence. The generality and competitive performance of high-dimensional trimmed estimators are illustrated numerically on both simulated and real-world genomics data.