Global Linear and Local Superlinear Convergence of IRLS for Non-Smooth Robust Regression

In the convex case, p = 1, we prove that this IRLS variant converges globally at a linear rate under a mild, deterministic condition on the feature matrix called the stable range space property. In the nonconvex case, p (0, 1), we prove that under a similar condition, IRLS converges locally to the global minimizer at a superlinear rate of order 2 p; the rate becomes quadratic as p 0. We showcase the proposed methods in three applications: real phase retrieval, regression without correspondences, and robust face restoration. The results show that (1) IRLS can handle a larger number of outliers than other methods, (2) it is faster than competing methods at the same level of accuracy, (3) it restores a sparsely corrupted face image with satisfactory visual quality.