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Retraction-Based Gradient Projection Algorithms on Manifolds

arXiv 2609.30885 ↗First posted 25 Sep 2026math.OC cs.LGcited by 0

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Abstract (arXiv)

We introduce a framework for retraction-based convex optimization on Riemannian manifolds, which includes a notion of retraction-specific convex sets and retraction-based gradient projection algorithms. The standard theory of gradient projection algorithms generalizes easily to this framework. Within this framework, we establish convergence results for retraction-based gradient projection algorithms with various stepsize rules. As an application, we use our framework to study the weighted low-rank approximation. We also provide numerical validation of our convergence results on the image completion task.

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References · 20

  1. 2018Riemannian Adaptive Optimization Methodscited by 194
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  3. 2013Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimizationcited by 143
  4. 2014Matrix Completion and Low-Rank SVD via Fast Alternating Least Squarescited by 104
  5. 2020Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transformcited by 80
  6. 2017Riemannian Optimization via Frank-Wolfe Methodscited by 21
  7. 2013A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifoldcited by 13
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  9. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  10. 2013R3MC: A Riemannian three-factor algorithm for low-rank matrix completioncited by 11
  11. 2019Regularized Weighted Low Rank Approximationcited by 10
  12. 2020Global Riemannian Acceleration in Hyperbolic and Spherical Spacescited by 10
8 more
  1. 2015Matrix Completion Under Monotonic Single Index Modelscited by 8
  2. 2022Convergence of First-Order Methods for Constrained Nonconvex Optimization with Dependent Datacited by 7
  3. 2021Matrix Completion with Model-free Weightingcited by 6
  4. 2025Implicit Riemannian Optimism with Applications to Min-Max Problemscited by 4
  5. 2020Median Matrix Completion: from Embarrassment to Optimalitycited by 1
  6. 2024Computing Augustin Information via Hybrid Geodesically Convex Optimizationcited by 1
  7. 2024Structured Regularization for Constrained Optimization on the SPD Manifoldcited by 1
  8. 2025Weighted Low-rank Approximation via Stochastic Gradient Descent on Manifoldscited by 1

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