Global Convergence of Iteratively Reweighted Least Squares for Robust Subspace Recovery

TL;DR

This paper proves that a variant of IRLS algorithm converges linearly to the true subspace in robust subspace recovery, providing the first global convergence guarantees for IRLS in this context and in nonconvex Riemannian settings.

Contribution

It establishes the first global convergence guarantees for IRLS in robust subspace recovery and extends these results to affine subspace estimation.

Findings

IRLS converges linearly under deterministic conditions

Guarantees extend to affine subspace estimation

Demonstrated practical benefits in neural network training

Abstract

Robust subspace estimation is fundamental to many machine learning and data analysis tasks. Iteratively Reweighted Least Squares (IRLS) is an elegant and empirically effective approach to this problem, yet its theoretical properties remain poorly understood. This paper establishes that, under deterministic conditions, a variant of IRLS with dynamic smoothing regularization converges linearly to the underlying subspace from any initialization. We extend these guarantees to affine subspace estimation, a setting that lacks prior recovery theory. Additionally, we illustrate the practical benefits of IRLS through an application to low-dimensional neural network training. Our results provide the first global convergence guarantees for IRLS in robust subspace recovery and, more broadly, for nonconvex IRLS on a Riemannian manifold.