Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms

[2404.12828] Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Optimization and Control arXiv:2404.12828 (math) [Submitted on 19 Apr 2024 ( v1 ), last revised 22 Apr 2025 (this version, v2)] Title: Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms Authors: Andrew D. McRae View a PDF of the paper titled Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms, by Andrew D. McRae View PDF HTML (experimental) Abstract: We show that solutions to the popular convex matrix LASSO problem (nuclear-norm--penalized linear least-squares) have low rank under similar assumptions as required by classical low-rank matrix sensing error bounds. Although the purpose of the nuclear norm penalty is to promote low solution rank, a proof has not yet (to our knowledge) been provided outside very specific circumstances. Furthermore, we show that this result has significant theoretical consequences for nonconvex rank-constrained optimization approaches. Specifically, we show that if (a) the ground truth matrix has low rank, (b) the (linear) measurement operator has the matrix restricted isometry property (RIP), and (c) the measurement error is small enough relative to the nuclear norm penalty, then the (unique) LASSO solution has rank (approximately) bounded by that of the ground truth. From this, we show (a) that a low-rank--projected proximal gradient descent algorithm will converge linearly to the LASSO solution from any initialization, and (b) that the nonconvex landscape of the low-rank Burer-Monteiro--factored problem formulation is benign in the sense that all second-order critical points are globally optimal and yield the LASSO solution. Subjects: Optimization and Control (math.OC) ; Statistics Theory (math.ST) MSC classes: 62J07, 90C25, 90C26, 90C31, 90C46 Cite as: arXiv:2404.12828 [math.OC] (or arXiv:2404.12828v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2404.12828 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s10107-025-02236-x Focus to learn more DOI(s) linking to related resources Submission history From: Andrew McRae [ view email ] [v1] Fri, 19 Apr 2024 12:05:50 UTC (22 KB) [v2] Tue, 22 Apr 2025 13:57:52 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms, by Andrew D. McRae View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2024-04 Change to browse by: math math.ST stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark 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? ) 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? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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