# Retraction-Based Gradient Projection Algorithms on Manifolds

arXiv: https://arxiv.org/abs/2609.30885 · HTML: https://mlsanity.com/paper/2609.30885/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 61 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares](https://mlsanity.com/paper/1410.2596/) (2014)
- [Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform](https://mlsanity.com/paper/2002.01113/) (2020)
- [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017)
- [A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold](https://mlsanity.com/paper/1301.0172/) (2013)
- [Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity](https://mlsanity.com/paper/1806.06373/) (2018)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [R3MC: A Riemannian three-factor algorithm for low-rank matrix completion](https://mlsanity.com/paper/1306.2672/) (2013)
- [Regularized Weighted Low Rank Approximation](https://mlsanity.com/paper/1911.06958/) (2019)
- [Global Riemannian Acceleration in Hyperbolic and Spherical Spaces](https://mlsanity.com/paper/2012.03618/) (2020)
- [Matrix Completion Under Monotonic Single Index Models](https://mlsanity.com/paper/1512.08787/) (2015)
- [Convergence of First-Order Methods for Constrained Nonconvex Optimization with Dependent Data](https://mlsanity.com/paper/2203.15797/) (2022)
- [Matrix Completion with Model-free Weighting](https://mlsanity.com/paper/2106.05850/) (2021)
- [Implicit Riemannian Optimism with Applications to Min-Max Problems](https://mlsanity.com/paper/2501.18381/) (2025)
- [Median Matrix Completion: from Embarrassment to Optimality](https://mlsanity.com/paper/2006.10400/) (2020)
- [Computing Augustin Information via Hybrid Geodesically Convex Optimization](https://mlsanity.com/paper/2402.02731/) (2024)
- [Structured Regularization for Constrained Optimization on the SPD Manifold](https://mlsanity.com/paper/2410.09660/) (2024)
- [Weighted Low-rank Approximation via Stochastic Gradient Descent on Manifolds](https://mlsanity.com/paper/2502.14174/) (2025)

## 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.