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The paper gives theoretical convergence guarantees for a robust subspace estimation method, extending them to affine subspaces and to nonconvex optimization on Riemannian manifolds. Its neural network experiment is a small-scale illustration, so the practical impact depends on whether the guarantees hold under less idealized, noisy conditions. For ML readers, the main value is a clearer theoretical footing for a method that has mostly been used empirically.
Global Convergence of Iteratively Reweighted Least Squares for Robust Subspace Recovery
arXiv:2506.20533v5 Announce Type: replace-cross 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.
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.
Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2506.20533 [stat.ML]
(or arXiv:2506.20533v5 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2506.20533 Focus to learn more arXiv-issued DOI via DataCite
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