Convergence of Iteratively Re-weighted Least Squares to Robust M-Estimators

This paper presents a way of using the Iteratively Reweighted Least Squares (IRLS) method to minimize several robust cost functions such as the Huber function, the Cauchy function and others. It is known that IRLS (otherwise known as Weiszfeld) techniques are generally more robust to outliers than the corresponding least squares methods, but the full range of robust M-estimators that are amenable to IRLS has not been investigated. In this paper we address this question and show that IRLS methods can be used to minimize most common robust M-estimators. An exact condition is given and proved for decrease of the cost, from which convergence follows. In addition to the advantage of increased robustness, the proposed algorithm is far simpler than the standard L1 Weiszfeld algorithm. We show the applicability of the proposed algorithm to the rotation averaging, triangulation and point cloud alignment problems.