Fast Approximate L_infty Minimization: Speeding Up Robust Regression

TL;DR

This paper introduces a fast, scalable method for $L_ Infty$ norm minimization that significantly accelerates robust regression, enabling application to large, high-dimensional datasets with many outliers.

Contribution

The paper presents a novel, efficient algorithm for $L_ Infty$ minimization that drastically reduces computation time and extends robust regression to larger, more complex problems.

Findings

Achieves multiple orders of magnitude speedup in high-dimensional data.

Demonstrates robustness against large outlier contamination.

Enables application of robust regression to previously inaccessible problem sizes.

Abstract

Minimization of the $L_\infty$ norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving the problem at the heart of $L_\infty$ norm minimization are slow, and therefore cannot scale to large problems. A new method for the minimization of the $L_\infty$ norm is presented here, which provides a speedup of multiple orders of magnitude for data with high dimension. This method, termed Fast $L_\infty$ Minimization, allows robust regression to be applied to a class of problems which were previously inaccessible. It is shown how the $L_\infty$ norm minimization problem can be broken up into smaller sub-problems, which can then be solved extremely efficiently. Experimental results demonstrate the radical reduction in computation time, along with robustness against large numbers of outliers in a few model-fitting problems.