تقرير
Optimization over bounded-rank matrices through a desingularization enables joint global and local guarantees
| العنوان: | Optimization over bounded-rank matrices through a desingularization enables joint global and local guarantees |
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| المؤلفون: | Rebjock, Quentin, Boumal, Nicolas |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Optimization and Control, Mathematics - Numerical Analysis |
| الوصف: | Convergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a non-smooth and non-convex algebraic variety. Existing techniques include 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 critical points) or fast local convergence (e.g., if the limit has non-maximal rank). We seek optimization algorithms that enjoy both. Khrulkov 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 secure global convergence to critical points with fast local rates, for a large range of algorithms. On matrix completion tasks, we find that this approach is comparable to others, while enjoying better general-purpose theoretical guarantees. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.14211 |
| رقم الأكسشن: | edsarx.2406.14211 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |