# Projection-free nonconvex stochastic optimization on Riemannian manifolds

arXiv: https://arxiv.org/abs/1910.04194 · HTML: https://mlsanity.com/paper/1910.04194/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

## Built on by (most cited first, 3 of 3)

- [No-go Theorem for Acceleration in the Hyperbolic Plane](https://mlsanity.com/paper/2101.05657/) (2021)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penalties](https://mlsanity.com/paper/2305.16186/) (2023)

## Abstract (arXiv)

We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank-Wolfe methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: The computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.