mlsanityData to 7 Oct 2026

First-order Methods for Geodesically Convex Optimization

arXiv 1602.06053 ↗First posted 19 Feb 2016math.OC cs.LG stat.MLcited by 92

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Abstract (arXiv)

Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for several first-order algorithms on Hadamard manifolds. Specifically, we prove upper bounds for the global complexity of deterministic and stochastic (sub)gradient methods for optimizing smooth and nonsmooth g-convex functions, both with and without strong g-convexity. Our analysis also reveals how the manifold geometry, especially sectional curvature, impacts convergence rates. To the best of our knowledge, our work is the first to provide global complexity analysis for first-order algorithms for general g-convex optimization.

Key facts. Cited by 92 papers in the map (data through 7 Oct 2026). First posted on arXiv 19 Feb 2016.

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Citing papers per year: 2021: 18; 2022: 17; 2023: 12; 2024: 20; 2025: 12; 2026: 13.

92 papers.

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  1. 2021Hyperbolic Deep Neural Networks: A Surveycited by 138
  2. 2021Fast and accurate optimization on the orthogonal manifold without retractioncited by 35
  3. 2021Decentralized Riemannian Gradient Descent on the Stiefel Manifoldcited by 33
  4. 2021On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometrycited by 32
  5. 2021Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descentcited by 32
  6. 2021Universal Approximation Under Constraints is Possible with Transformerscited by 32
  7. 2024NRDF: Neural Riemannian Distance Fields for Learning Articulated Pose Priorscited by 27
  8. 2025Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometriescited by 15
  9. 2024PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programmingcited by 14
  10. 2024Guiding Neural Collapse: Optimising Towards the Nearest Simplex Equiangular Tight Framecited by 14
  11. 2022Federated Learning on Riemannian Manifoldscited by 13
  12. 2023Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifoldcited by 13
38 more
  1. 2021On Riemannian Stochastic Approximation Schemes with Fixed Step-Sizecited by 12
  2. 2022Differentially private Riemannian optimizationcited by 12
  3. 2023Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Spacecited by 11
  4. 2021Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometrycited by 10
  5. 2022Momentum Stiefel Optimizer, with Applications to Suitably-Orthogonal Attention, and Optimal Transportcited by 10
  6. 2022First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spacescited by 10
  7. 2022Lifting Weak Supervision To Structured Predictioncited by 10
  8. 2025Riemannian Optimization for LoRA on the Stiefel Manifoldcited by 10
  9. 2022Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithmcited by 9
  10. 2024Geodesic Optimization for Predictive Shift Adaptation on EEG datacited by 9
  11. 2021On the Local Linear Rate of Consensus on the Stiefel Manifoldcited by 8
  12. 2023Particle-based Online Bayesian Samplingcited by 7
  13. 2024Optimization without Retraction on the Random Generalized Stiefel Manifoldcited by 7
  14. 2021From the Greene--Wu Convolution to Gradient Estimation over Riemannian Manifoldscited by 6
  15. 2022Online Learning to Transport via the Minimal Selection Principlecited by 6
  16. 2022Shape And Structure Preserving Differential Privacycited by 6
  17. 2023Minimizing Dynamic Regret on Geodesic Metric Spacescited by 6
  18. 2024Efficient Sampling on Riemannian Manifolds via Langevin MCMCcited by 6
  19. 2021Optimization on manifolds: A symplectic approachcited by 5
  20. 2022Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agentscited by 5
  21. 2022Riemannian accelerated gradient methods via extrapolationcited by 5
  22. 2023Efficient displacement convex optimization with particle gradient descentcited by 5
  23. 2024Federated Learning on Riemannian Manifolds with Differential Privacycited by 5
  24. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  25. 2021Accelerating Training of Batch Normalization: A Manifold Perspectivecited by 4
  26. 2021No-go Theorem for Acceleration in the Hyperbolic Planecited by 4
  27. 2021Universal Regular Conditional Distributionscited by 4
  28. 2022Stochastic and Private Nonconvex Outlier-Robust PCAcited by 4
  29. 2022Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penaltiescited by 4
  30. 2023Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penaltiescited by 4
  31. 2023Riemannian Projection-free Online Learningcited by 4
  32. 2024Riemannian Federated Learning via Averaging Gradient Streamscited by 4
  33. 2025Decentralized Online Riemannian Optimization Beyond Hadamard Manifoldscited by 4
  34. 2021Understanding Riemannian Acceleration via a Proximal Extragradient Frameworkcited by 3
  35. 2024A Framework for Bilevel Optimization on Riemannian Manifoldscited by 3
  36. 2025Multiple Wasserstein Gradient Descent Algorithm for Multi-Objective Distributional Optimizationcited by 3
  37. 2025Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectivescited by 3
  38. 2021Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problemscited by 2

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