翻訳待ち：Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS
AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。ソース概要：arXiv:2608.23765v1 Announce Type: new Abstract: Iteratively reweighted least squares (IRLS) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low-rank recovery. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic-mean weight operator defines a valid global quadratic majorizer. Furthermore, we show that this weight operator is optimal within the family of power-mean weights, clarifying why it improves over classical one-sided reweighting schemes that use only row- or column-space information. Under a Schatten-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic-mean weights. For IRLS with harmonic-mean weights, we prove a dimension-independent, locally linear convergence rate. We provide a counterexample showing that this dimension-independent local rate cannot in general be obtained for IRLS algorithms using one-sided weight operators, which predominate in the literature. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic-mean reweighting across square, rectangular, and adversarially initialized recovery problems.
AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。
--> [Submitted on 24 Aug 2026] Title:Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS View a PDF of the paper titled Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS, by Christian K\"ummerle and 2 other authors View PDF HTML (experimental) Abstract:Iteratively reweighted least squares (IRLS) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low-rank recovery. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic-mean weight operator defines a valid global quadratic majorizer. Furthermore, we show that this weight operator is optimal within the family of power-mean weights, clarifying why it improves over classical one-sided reweighting schemes that use only row- or column-space information. Under a Schatten-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic-mean weights. For IRLS with harmonic-mean weights, we prove a dimension-independent, locally linear convergence rate. We provide a counterexample showing that this dimension-independent local rate cannot in general be obtained for IRLS algorithms using one-sided weight operators, which predominate in the literature. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic-mean reweighting across square, rectangular, and adversarially initialized recovery problems. Comments: 97 pages, 9 figures, 2 tables Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA); Optimization and Control (math.OC) MSC classes: 65F55, 65K05, 68T09, 90C25, 15A03 Cite as: arXiv:2608.23765 [cs.LG] (or arXiv:2608.23765v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2608.23765 arXiv-issued DOI via DataCite (pending registration) Submission history From: Christian Kümmerle [view email] [v1] Mon, 24 Aug 2026 18:56:13 UTC (546 KB) Full-text links: Access Paper: View a PDF of the paper titled Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS, by Christian K\"ummerle and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-08 Change to browse by: cs cs.NA math math.NA math.OC References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)