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This paper advances the theoretical foundations of machine learning by formalizing gradient projection algorithms on Riemannian manifolds, offering rigorous convergence proofs for complex optimization landscapes. The work provides a generalized framework applicable to tasks like weighted low-rank approximation and image completion, bridging advanced mathematical control theory with practical ML applications.
Retraction-Based Gradient Projection Algorithms on Manifolds
arXiv:2609.30885v1 Announce Type: cross Abstract: 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.
Abstract: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.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2609.30885 [math.OC]
(or arXiv:2609.30885v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2609.30885 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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