[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84415-en":3,"doc-seo-84415-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84415,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Optimization over Bounded-Rank Matrices Through a Desingularization Enables Joint Global and Local Guarantees","Optimization over bounded-rank matrices is challenging because the feasible set forms a nonsmooth, nonconvex algebraic variety. Existing approaches—direct optimization on the rank-bounded set, fixed-rank optimization on the maximal-rank stratum, and LR parameterization—cannot simultaneously provide global convergence to stationary points and fast local rates. The work studies a lifted geometry based on desingularization to build a Riemannian structure, enabling standard algorithms to achieve both global convergence and rapid local convergence. Numerical considerations and comparisons on matrix completion confirm practical competitiveness.","arXiv :2406 . 14211v2 [math .OC] 12 Jul 2026  \nOptimization over bounded-rank matrices through adesingularization enables joint global and local guarantees  \nQuentin Rebjock and Nicolas Boumal ∗  \nJuly 14, 2026  \nAbstract  \nConvergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a nonsmooth and nonconvex algebraic variety. Existing techniques include direct optimization over bounded-rank matrices (e.g., projected gradient descent), fixed-rank optimization (over the maximal-rank stratum), and the LR parameterization. They all lack either global guarantees (the ability to accumulate only at stationary points) or fast local convergence (e.g., if the limit has non-maximal rank) . We study a lifted geometry that allows algorithms to enjoy both.  \nKhrulkov and Oseledets [2018] parameterize the bounded-rank variety via a desingularization to recast the optimization problem onto a smooth manifold. Building on their ideas, we develop a Riemannian geometry for this desingularization, also with care for numerical considerations. We use it to ensure conditions that, for many standard algorithms, yield global convergence to stationary points with fast local rates. On matrix completion tasks, we find that this approach is comparable to others.  \n1 Introduction  \nWe aim to minimize a continuously differentiable function f : Rm×n → R over the set of bounded-rank matrices:  \nmin f (X) subject to rank X ≤ r, (P)  \nX∈Rm× n  \nfor some r \u003C min(m, n) . This is a classical problem: see Section 1.2 for a literature review. The feasible set is a nonconvex, nonsmooth algebraic variety:  \nRr×n = {X ∈ Rm×n : rank(X) ≤ r} .  \nSuch problems are computationally hard in general [Gillis and Glineur, 2011], but we can still aim for local solutions. There exist at least three paradigms to do so.  \nThe first consists of methods that optimize directly over Rr×n. This notably includes the projected gradient descent (PGD) algorithm [Goldfarb and Ma, 2011 , Jain et al. , 2010 , 2014] and variants [Schneider and Uschmajew, 2015] . These are first-order methods with limited local convergence rates (at most linear) .  \n∗ Ecole Polytechnique Fédérale de Lausanne (EPFL), Institute of Mathematics. This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22 .00027.  \nAnother approach is to optimize only over the maximal rank stratum  \nRmr×n = {X ∈ Rm×n : rank(X) = r},  \nthereby ignoring matrices of rank strictly less than r. That stratum accounts for almost all of Rr×n and it is a smooth manifold. Thus, it is possible to endow it with a Riemannian structure and roll out off-the-shelf second-order optimization algorithms [Absil et al. , 2008 , Shalit et al. , 2012 , Vandereycken, 2013] . Those algorithms generate sequences of matrices of rank r. These may converge to matrices of rank strictly less than r, in which case they“fall off” the manifold: theory for optimization on manifolds breaks down in that event, which makes it difficult to provide a priori guarantees. This is likely to happen when the parameter r overestimates the actual rank of the solution. Additionally, the Hessian can grow unbounded near lesser rank matrices [Vandereycken, 2013, §2.3] .  \nThe third, and perhaps most popular, class of techniques is to smoothly parameterize the feasible set Rr×n. Specifically, let M be a smooth manifold and let φ : M → Rm×n be a smooth map such that φ (M) = Rr×n. Then minimizing f over Rr×n is equivalent to minimizing the lifted function g = f ◦ φ over M, for which we can apply standard (Riemannian) optimization algorithms: this last paradigm is what we explore here.  \nUnfortunately, even when f has favorable properties, the lifted cost function g = f ◦ φ may not. As an example, consider the prominent parameterization φ (L, R) = LR⊤ with domain M = Rm×r ×Rn×r . We refer to it as the LR parameterization (see also Appendix A) . It reduces the feasi","cbCaipZ9IFdJiWEP","https://ap.wps.com/l/cbCaipZ9IFdJiWEP","pdf",778268,1,38,"English","en",105,"# Introduction\n## Bounded-rank optimization problem\n## Three solution paradigms\n## Direct optimization and PGD\n## Fixed-rank manifold optimization\n## Smooth parameterization and LR limitations\n## Desingularization via kernel subspaces","[{\"question\":\"Why is optimization over bounded-rank matrices difficult?\",\"answer\":\"The feasible set of rank-constrained matrices is a nonsmooth, nonconvex algebraic variety, which makes global and local convergence guarantees delicate to obtain.\"},{\"question\":\"What are the main limitations of existing methods like PGD, fixed-rank optimization, and the LR parameterization?\",\"answer\":\"Direct methods such as PGD have limited local convergence rates, fixed-rank manifold methods can “fall off” the manifold and lose theory when solutions have lower rank, and LR parameterization leads to issues such as unbounded sublevel sets and the lack of local PŁ conditions.\"},{\"question\":\"How does the proposed desingularization-based lifted geometry help achieve both global and local guarantees?\",\"answer\":\"By developing a Riemannian geometry for a desingularization lifting, the approach provides conditions under which many standard algorithms achieve global convergence to stationary points with fast local rates, supported by comparisons on matrix completion 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is optimization over bounded-rank matrices difficult?","Question",{"text":75,"@type":76},"The feasible set of rank-constrained matrices is a nonsmooth, nonconvex algebraic variety, which makes global and local convergence guarantees delicate to obtain.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are the main limitations of existing methods like PGD, fixed-rank optimization, and the LR parameterization?",{"text":80,"@type":76},"Direct methods such as PGD have limited local convergence rates, fixed-rank manifold methods can “fall off” the manifold and lose theory when solutions have lower rank, and LR parameterization leads to issues such as unbounded sublevel sets and the lack of local PŁ conditions.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed desingularization-based lifted geometry help achieve both global and local guarantees?",{"text":84,"@type":76},"By developing a Riemannian geometry for a desingularization lifting, the approach provides 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