A class of optimization problems motivated by rank estimators in robust regression

TL;DR

This paper investigates optimization methods for rank estimators in robust regression, introducing efficient algorithms for two classes of objective functions, including a polynomial-time algorithm for the general case with few regressors.

Contribution

It presents novel polynomial-time algorithms for optimizing rank estimators, especially for the general class with a small number of regressors, improving over existing heuristic methods.

Findings

The enumerative algorithm is polynomial-time for O(1) regressors.

The algorithms outperform existing methods in efficiency and accuracy.

Exact minimizers are found for the continuous and convex class.

Abstract

A rank estimator in robust regression is a minimizer of a function which depends (in addition to other factors) on the ordering of residuals but not on their values. Here we focus on the optimization aspects of rank estimators. We distinguish two classes of functions: the class with a continuous and convex objective function (CCC), which covers the class of rank estimators known from statistics, and also another class (GEN), which is far more general. We propose efficient algorithms for both classes. For GEN we propose an enumerative algorithm that works in polynomial time as long as the number of regressors is O(1). The proposed algorithm utilizes the special structure of arrangements of hyperplanes that occur in our problem and is superior to other known algorithms in this area. For the continuous and convex case, we propose an unconditionally polynomial algorithm finding the exact minimizer, unlike the heuristic or approximate methods implemented in statistical packages.