# Manifold Free Riemannian Optimization

arXiv: https://arxiv.org/abs/2209.03269 · HTML: https://mlsanity.com/paper/2209.03269/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 45 references that are in the map.
- First posted on arXiv 7 Sep 2022.

## Builds on (most cited first, 9 of 9)

- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [Manifold Approximation by Moving Least-Squares Projection (MMLS)](https://mlsanity.com/paper/1606.07104/) (2016)
- [Inferring manifolds using Gaussian processes](https://mlsanity.com/paper/2110.07478/) (2021)
- [Approximation of Functions over Manifolds: A Moving Least-Squares Approach](https://mlsanity.com/paper/1711.00765/) (2017)
- [Approximating the Riemannian Metric from Point Clouds via Manifold Moving Least Squares](https://mlsanity.com/paper/2007.09885/) (2020)
- [Stochastic Zeroth-order Riemannian Derivative Estimation and Optimization](https://mlsanity.com/paper/2003.11238/) (2020)

## Built on by (most cited first, 4 of 4)

- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [Changes from Classical Statistics to Modern Statistics and Data Science](https://mlsanity.com/paper/2211.03756/) (2022)

## Abstract (arXiv)

Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold M. Algorithms designed in this framework usually require some geometrical description of the manifold, which typically includes tangent spaces, retractions, and gradients of the cost function. However, in many cases, only a subset (or none at all) of these elements can be accessed due to lack of information or intractability. In this paper, we propose a novel approach that can perform approximate Riemannian optimization in such cases, where the constraining manifold is a submanifold of RD. At the bare minimum, our method requires only a noiseless sample set of the cost function (xi, yi)∈ M × ℝ and the intrinsic dimension of the manifold M. Using the samples, and utilizing the Manifold-MLS framework (Sober and Levin 2020), we construct approximations of the missing components entertaining provable guarantees and analyze their computational costs. In case some of the components are given analytically (e.g., if the cost function and its gradient are given explicitly, or if the tangent spaces can be computed), the algorithm can be easily adapted to use the accurate expressions instead of the approximations. We analyze the global convergence of Riemannian gradient-based methods using our approach, and we demonstrate empirically the strength of this method, together with a conjugate-gradients type method based upon similar principles.