Robust Mean Estimation in High Dimensions via $\ell_0$ Minimization

TL;DR

This paper introduces a new approach for robust mean estimation in high-dimensional data with a significant fraction of outliers, using $ ext{l}_0$ minimization inspired by compressive sensing, and demonstrates its effectiveness through experiments.

Contribution

It formulates the robust mean estimation as an $ ext{l}_0$ minimization problem and provides a computational framework leveraging $ ext{l}_1$ and $ ext{l}_p$ techniques for practical solutions.

Findings

The proposed methods outperform existing robust mean estimation techniques.

The $ ext{l}_0$ minimization approach is shown to be order optimal.

Experimental results on synthetic and real data validate the effectiveness.

Abstract

We study the robust mean estimation problem in high dimensions, where $\alpha <0.5$ fraction of the data points can be arbitrarily corrupted. Motivated by compressive sensing, we formulate the robust mean estimation problem as the minimization of the $\ell_0$-`norm' of the outlier indicator vector, under second moment constraints on the inlier data points. We prove that the global minimum of this objective is order optimal for the robust mean estimation problem, and we propose a general framework for minimizing the objective. We further leverage the $\ell_1$ and $\ell_p$ $(0<p<1)$, minimization techniques in compressive sensing to provide computationally tractable solutions to the $\ell_0$ minimization problem. Both synthetic and real data experiments demonstrate that the proposed algorithms significantly outperform state-of-the-art robust mean estimation methods.