# Accelerated Stochastic Quasi-Newton Optimization on Riemann Manifolds

## TL;DR

This paper introduces an accelerated stochastic quasi-Newton method on Riemannian manifolds that achieves fast convergence without line searches, supported by new theoretical proofs and empirical validation on matrix and eigenvalue problems.

## Contribution

It presents a novel L-BFGS algorithm on Riemannian manifolds with variance reduction, providing convergence proofs without curvature assumptions and demonstrating effectiveness in practical experiments.

## Key findings

- Fast convergence on synthetic experiments
- Effective computation of Karcher means and eigenvalues
- Outperforms VR-PCA and Riemannian SVRG in tests

## Abstract

We propose an L-BFGS optimization algorithm on Riemannian manifolds using minibatched stochastic variance reduction techniques for fast convergence with constant step sizes, without resorting to linesearch methods designed to satisfy Wolfe conditions. We provide a new convergence proof for strongly convex functions without using curvature conditions on the manifold, as well as a convergence discussion for nonconvex functions. We discuss a couple of ways to obtain the correction pairs used to calculate the product of the gradient with the inverse Hessian, and empirically demonstrate their use in synthetic experiments on computation of Karcher means for symmetric positive definite matrices and leading eigenvalues of large scale data matrices. We compare our method to VR-PCA for the latter experiment, along with Riemannian SVRG for both cases, and show strong convergence results for a range of datasets.