Global Convergence of Iteratively Reweighted Least Squares for Robust Subspace Recovery
This work addresses a fundamental problem in machine learning and data analysis by providing the first theoretical guarantees for IRLS in robust subspace recovery, which is incremental as it builds on existing empirical methods but lacks prior theory.
The paper tackles the problem of robust subspace estimation by establishing global convergence guarantees for a variant of Iteratively Reweighted Least Squares (IRLS) with dynamic smoothing regularization, showing linear convergence to the underlying subspace from any initialization under deterministic conditions and extending these results to affine subspace estimation.
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.