# Riemannian Optimization via Frank-Wolfe Methods

arXiv: https://arxiv.org/abs/1710.10770 · HTML: https://mlsanity.com/paper/1710.10770/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2017.

## Built on by (most cited first, 21 of 21)

- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds](https://mlsanity.com/paper/2605.09238/) (2026)
- [Riemannian Optimization on Relaxed Indicator Matrix Manifold](https://mlsanity.com/paper/2503.20505/) (2025)
- [Riemannian Projection-free Online Learning](https://mlsanity.com/paper/2305.19349/) (2023)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Low-complexity subspace-descent over symmetric positive definite manifold](https://mlsanity.com/paper/2305.02041/) (2023)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
- [Manifold learning and optimization using tangent space proxies](https://mlsanity.com/paper/2501.12678/) (2025)
- [Faster Convergence of Riemannian Stochastic Gradient Descent with Increasing Batch Size](https://mlsanity.com/paper/2501.18164/) (2025)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [On a class of geodesically convex optimization problems solved via Euclidean MM methods](https://mlsanity.com/paper/2206.11426/) (2022)
- [Invex Programs: First Order Algorithms and Their Convergence](https://mlsanity.com/paper/2307.04456/) (2023)
- [Disciplined Geodesically Convex Programming](https://mlsanity.com/paper/2407.05261/) (2024)
- [Structured Regularization for Constrained Optimization on the SPD Manifold](https://mlsanity.com/paper/2410.09660/) (2024)
- [Adaptive Lipschitz-Free Conditional Gradient Methods for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/2603.06369/) (2026)
- [Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds](https://mlsanity.com/paper/2601.08039/) (2026)
- [Riemannian Dueling Optimization](https://mlsanity.com/paper/2603.00023/) (2026)
- [Information Theoretic Bayesian Optimization over the Probability Simplex](https://mlsanity.com/paper/2603.09793/) (2026)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [Retraction-Based Gradient Projection Algorithms on Manifolds](https://mlsanity.com/paper/2609.30885/) (2026)

## Abstract (arXiv)

We study projection-free methods for constrained Riemannian optimization. In particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize RFW to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian "linear" oracle required by RFW admits a closed-form solution; this result may be of independent interest. We further specialize RFW to the special orthogonal group and show that here too, the Riemannian "linear" oracle can be solved in closed form. Here, we describe an application to the synchronization of data matrices (Procrustes problem). We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods and observe that RFW performs competitively on the task of computing Riemannian centroids.