# Riemannian stochastic quasi-Newton algorithm with variance reduction and its convergence analysis

arXiv: https://arxiv.org/abs/1703.04890 · HTML: https://mlsanity.com/paper/1703.04890/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Mar 2017.

## Built on by (most cited first, 2 of 2)

- [Riemannian Natural Gradient Methods](https://mlsanity.com/paper/2207.07287/) (2022)
- [Riemannian Stochastic Gradient Method for Nested Composition Optimization](https://mlsanity.com/paper/2207.09350/) (2022)

## Abstract (arXiv)

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions. The present paper proposes a Riemannian stochastic quasi-Newton algorithm with variance reduction (R-SQN-VR). The key challenges of averaging, adding, and subtracting multiple gradients are addressed with notions of retraction and vector transport. We present convergence analyses of R-SQN-VR on both non-convex and retraction-convex functions under retraction and vector transport operators. The proposed algorithm is evaluated on the Karcher mean computation on the symmetric positive-definite manifold and the low-rank matrix completion on the Grassmann manifold. In all cases, the proposed algorithm outperforms the state-of-the-art Riemannian batch and stochastic gradient algorithms.