SIAM Journal on Optimization· 2026Q1
Optimization over Bounded-Rank Matrices through a Desingularization Enables Joint Global and Local Guarantees
- 1citations
- Q1SCImago
- 2026year
Short summary
A novel Riemannian geometry on a desingularized manifold allows optimization algorithms to achieve both global convergence to stationary points and fast local convergence for bounded-rank matrices.
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Key points
- Introduces a Riemannian geometry on a desingularized manifold for bounded-rank matrix optimization.
- Enables algorithms to achieve both global convergence to stationary points and fast local convergence.
- Addresses limitations of existing methods that compromise either global or local convergence.
- Demonstrates comparable performance to existing approaches on matrix completion tasks.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. Convergence 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 nonmaximal rank). We study a lifted geometry that allows algorithms to enjoy both. Khrulkov and Oseledets [ SIAM J. Matrix Anal. Appl., 39 (2018), pp. 451–471] 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.
The authors' abstract, as published at the source. SIAM Journal on Optimization, 2026 · DOI ↗
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Field: Computational Mechanics
Computational MechanicsEngineering