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

We advance both the theory and practice of robust \ell_p -quasinorm regression for p \in (0,1] by using novel variants of iteratively reweighted least-squares (IRLS) to solve the underlying non-smooth problem. 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 non-convex case, p\in(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\to 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.