On the Convergence of IRLS and Its Variants in Outlier-Robust Estimation

Outlier-robust estimation involves estimating some parameters (e.g., 3D rotations) from data samples in the presence of outliers, and is typically formulated as a non-convex and non-smooth problem. For this problem, the classical method called iteratively reweighted least-squares (IRLS) and its variants have shown impressive performance. This paper makes several contributions towards understanding why these algorithms work so well. First, we incorporate majorization and graduated non-convexity (GNC) into the IRLS framework and prove that the resulting IRLS variant is a convergent method for outlier-robust estimation. Moreover, in the robust regression context with a constant fraction of outliers, we prove this IRLS variant converges to the ground truth at a global linear and local quadratic rate for a random Gaussian feature matrix with high probability. Experiments corroborate our theory and show that the proposed IRLS variant converges within 5-10 iterations for typical problem instances of outlier-robust estimation, while state-of-the-art methods need at least 30 iterations. A basic implementation of our method is provided: https: //github.com/liangzu/IRLS-CVPR2023 ... attempts to analyze this difficulty [caused by infinite weights of IRLS for the ℓp-loss] have a long history of proofs and counterexamples to incorrect claims.